18. Modeling Mortgage Prepayments and Valuing Mortgages

18. Modeling Mortgage Prepayments and Valuing Mortgages

Prof: So we're talking
now about mortgages and how to value them,
and if you remember now a mortgage–
so the first mortgages, by the way,
that we know of, come from Babylonian times.
It's not like some American
invented the mortgage or something.
This was 3,500-3,800 years old
and we have on these cuneiform tablets these mortgages.
And so the idea of a mortgage
is you make a promise, you back your promise with
collateral, so if you don't keep the
promise they can take your house,
and there's some way of getting out of the promise because
everybody knows the collateral, you might want to leave the
home, and then you have to have some way of dissolving the
promise because the promise involves many payments over

So it's making a promise,
backing it with collateral, and finding a way to dissolve
the promise at prearranged terms in case you want to end it by
prepaying. And that prepaying is called
the refinancing option. And because there's a
refinancing option it makes the mortgage a much more complicated
thing, and a much more interesting
thing, and something that, for example,
a hedge fund could imagine that it could make money trading.
So I just want to give you a
slight indication of how that could happen.
So as we said if you have a
typical mortgage, say the mortgage rate is 8
percent– maybe this is a different
answer than I did– so here we have an 8 percent
mortgage with a 6 percent interest rate to begin with.
Now, if it's an 8 percent
mortgage the guy's going to have to pay much more than 8 percent
a year because a mortgage, remember, there are level

We're talking about fixed rate
mortgages. You pay the same amount every
single year for 30 years, now you're really paying
monthly and I've ignored the monthly business because it's
just too many months and there are 360 of them.
So I'm thinking of it as an
annual payment. You have to pay,
of course, more than 8 dollars a year because if the mortgage
rate were 8 percent and you had a balloon payment on the end,
you'd pay 8,8, 8,108. That's the way they used to
work, but they were changed.

So you could imagine the old
fashioned mortgage would pay 8,8, 8,8, 8,108;
if you didn't pay your 8 somewhere along the line they'd
confiscate your whole house and then take what was owed out of
it and you could get out of it by paying 100.
The new mortgages instead of
paying 8 every year for 30 years you pay 8.88 every year for 30
years because if you discount payments of 8.8 for 30 years at
8 percent you get 100. So the present value is 100 at
the agreed upon discounting rate or mortgage rate 8 percent.
And so you see how important
this discount rate is.

And the remaining balance,
however, goes down because every time you're paying you're
paying more than the 8 percent interest.
You're paying in the first year
8.8 instead of 8 and so that gap of .88 is used to reduce the
balance from 100 to 99.117. And so you see the balance is
going down over time and making the lender safer and safer
because the same house is backing it.
So it's called an amortizing

Now, why is it difficult to
value? Because you have the option,
any time you want, and there's a good reason for
that option, any time you want you have the
option of getting out of the mortgage and just saying,
"Okay, I've paid 3 payments of 8.88,
I don't want to do it anymore. I want to pay off 97.13 and
then let's call it quits." And they say,
"Okay," and there's nothing they can do
about it. Now, when are you going to
exercise that option? You're going to exercise that
option either because you have to move,
that's the intention of it, or you'll exercise it when it's
most advantageous to you. Now, why could it become
advantageous to exercise it? Well, you don't really want to
exercise the option and this is the way most people think of it
backwards. They think, "Oh,
the interest rates are going down.
That means I'll get a new
mortgage with a lower interest rate."
They're hoping for exactly the
wrong thing.

If the interest rates go up
what they've got is a much better mortgage because they're
continuing to buy at the same 8 percent interest and maybe
interest rates in the economy have become 12 percent and
they're actually making money. So people who borrow in times
of high inflation do better. When there are times of
deflation the borrowers get crushed.
Irving Fisher said one of the
main reasons for the Depression being so bad is all the
entrepreneurial people in the country,
as usual, were borrowing, and then there was a deflation
and so they were getting crushed.
And the very people who drive
the economy were being hurt the most.
And so that feedback,
he said, was responsible for part of the severity of the

So you see interest rates can
go up or down and what happens? When they go up,
if they go up high enough to 19 percent you think,
"My, gosh, I've made a fortune holding
this mortgage. I'm still borrowing at 8
percent and I can invest my money at 19 percent."
So you've made a fortune and
the poor lender's gotten crushed.
On the other hand if the
interest rates go way down here, so the present value of what
you owe if you kept paying it becomes huge,
you don't have to face that big loss because you just prepay at
whatever the remaining balance is there and then you've
protected your downside. So by paying attention and
deciding when the optimal time to prepay is,
you can save yourself a lot of money and thereby cost the bank
a lot of money. So when exactly should you
prepay? When should you exercise your
options? Well, in this example if you
never exercised it you'd be handing the bank,
effectively, 120 dollars even though they
lent you 20 [correction: lent you 100].
So the bank would have made a
20 percent profit on you.

But if you exercise your option
optimally you're going to make not 100–
the bank is not going to get 100 dollars out of you,
they're going to even get less than 100 dollars.
They're going to get 98 dollars
out of you. So when exactly should you be
exercising your option? Well, we went over this last
time. I'll do it once again.
So remember,
the payment you owed was 8.88,8.88, blah,
blah, blah, 8.88.

The remaining balance started,
of course, at 100 and then it went down to 99.11 and then it
kept going down from there. So since I can't remember the
numbers let's just call this B_1,
the remaining balance which happened to be,
you know, it was 99.11 the first time.
Let's call this B_1,
then I went to B_2, B_3 etcetera and then
B_30 is equal to 0, no remaining balance after that.
So we said, what should you
do–I'm going to do the calculation now a little bit
differently– I said after every payment of
8.88 you could always say to yourself,
"Do I want to continue or do I want to pay my
option?" Now, you notice that if I had
divided this by B_1, say, if you had a mortgage that
was a little bit smaller, barely over a 1 dollar for
example, that would divide everything by

The payments would all be
divided by B_1 and the remaining balances would all be
divided by B_1. So I could always scale this
thing up or down. There's nothing fancy about
100, nothing important about 100.
If the original loan was for
200 you just double all your payments and double all your
remaining balances. What could be more obvious than
that? So I want to think in those
terms of a mortgage that always has 1 dollar left.
So suppose at any stage you had
1 dollar left in your mortgage. Your remaining balance was 1.
So let's say at any node,
let's ask the question, what is the value of 1 dollar
of remaining balance? So if you start at 100 and you
haven't prepaid, here you've got B_2

Of course, whatever the value
of that is divided by B_2,
that's the value of 1 dollar. So I'm just going to figure out
the value of 1 dollar of remaining balance and I'm going
to call that W, let's say.
I'll call that W of some node S.
So where am I?
I'm in some node in this
interest rate tree, right?
Here's our interest rate tree,
and I'm anywhere just here, and I'm doing backward
induction so for all successor nodes I figured out what 1
dollar of remaining balance is. And let's say it's in period
1,2, 3,4, 5, so I'm in period 5, B_5.
So what is the remaining
balance at this node which I call S?
So it's some node right there
of–oh no, I've lost it.

So W_S is going to be
what? It's going to be the minimum of
1, you could just pay it if you wanted to, or you could wait.
1 over (1 r_S),
and then what would you have to do, you would have to make your
payment. Well, what's your payment?
The payment is this 8.88 but
divided by B_5 plus the remaining balance of 1
dollar. So (B_6 over
B_5) times the remainder times W_Sup.
Now, why is this right?
I hope it is right by the way.
I should have thought of this a
little before. So this is the remainder of 1
dollar left. So if I divide by B_5
here I'm not going to have a remaining balance of
B_6. I'm going to have a remaining
balance of B_6 over B_5.
So if I started with 1 dollar
of remaining balance then I know that in the next period I'm
going to have B_6 over B_5 dollars of
remaining balance left. It doesn't sound too
convincing, by the way.

Well, it's right,
and that happens with probability 1 half.
And then with the other
probability 1 half, plus I make the payment,
but I go down instead of up and so I have B_6 over
B_5 but I have W_Sdown,
and that's also times 1 half. So either I pay off my
remaining dollar or I end up with this many dollars.
Assuming I had a 1 dollar of
remaining balance I'm either going to pay it off,
the remaining balance, or I'm going to have this much
left next period and 1 dollar of remaining balance is going to be
that. So that's it.
So I know now by working this
backwards I can tell what 1 dollar at the beginning is
worth. And so it's exactly the same
calculation I did before except I'm talking about 1 dollar.
I'm always figuring out 1
dollar of remaining balance instead of the whole thing.
Present value of callable,
so here's present value of 1 dollar of principal.
And so remember the present
value of a callable mortgage was 98.8.
Here the present value of 1
dollar, figuring it out that way,
is .98, obviously it's divided by 100,
but the key is that now you can see just by looking at it where
the 1s are is where the guy decided to prepay.
So it's the same thing as
before, but you see before you couldn't tell very easily from
the numbers when I did the 100.

Sorry, that didn't quite make
it. Before when I did the present
value with the 100 all these numbers were 98s and 97s.
I mean, where has he prepaid?
It's hard to tell where the
prepayment is. If I do it all in terms of 1
dollar of remaining balance then just by looking at the screen I
can tell where the guy prepaid because there are 1s there.
So I know where he's prepaid.
Wherever the 1s are that means
he's prepaid. So I can tell very easily what
he did. All right, that's the only
purpose of doing the same calculation in a somewhat
trickier way. So if you think about it a
second you see I've just divided by–I've always reduced things
to if you had 1 dollar left.

All right, so this tells us
what to do, when the guy should prepay and when he shouldn't
prepay. So if you're now in the world
looking at what's happening you can find the historical record
of how people have prepaid. So let's just look at the
historical record, for example.
Here, if you can see this,
this is blown up as big as it goes.
So this is what you might see
as the historical record of percentage prepayments
annualized from '86 to '99, say.
So you notice that they're very
low here, and then they get to be very high,
and then they get low again, and then they get high again.
So why do you think that
happened? So what is this?
This is prepayments for a
particular mortgage, 8 percent.
You take all the people in the
country who started in 1986 with 8 percent mortgages.
There's a huge crowd of those
because that was about what the mortgage rate was that year.
So a huge collection of people
got these mortgages in '86 and you keep track of what
percentage of them prepaid, really every month,
but you write the annualized rate,
and then this is the record.

So why do you think it changed
so dramatically like that? What's the explanation?
Student: Stock market.
Prof: What?
Student: Stock market.
Prof: It looks like the
stock market, but I assure you the stock
market had almost nothing to do with it.
Why would prepayments be so
low, and then be so high, then be low,
then be high? What do you think was happening?
Student: Interest rate
change. Prof: Interest rate.
We just did that.
We just solved that.
That was the whole point of
what we were doing.

So you tell me,
what do you think happened in '93?
This is September '93.
I don't know if you can read
that. What do you think was going on
then? Student: Interest rates
got low. Prof: Interest rates got
low, exactly. So you may not remember this
because you were barely born. In the early '90s there was a
recession and then the government cut the interest
rates. In the '90s,
the early '90s there was a recession and the government
kept cutting interest rates further, and further and
further. There was this huge decline in
interest rates through the early '90s, and so what happened?
All these people who,
in '86, who had these 8 percent mortgages–the new interest
rates were lower and so they all prepaid.
You got this shocking amount of

So this graph,
which seems sort of surprising and looks like the stock market,
turns out to have nothing to do with the stock market.
It has to do with where the
interest rates are. Well, do you think interest
rates explain everything? No.
What else could you notice
about the–escape. What else have we learned here
by doing these calculations? Well, what we've learned so far
is that if the interest rates in the economy are at 6 percent,
that's where they started, remember we said they started
at 6 percent and there was 16 percent volatility.
Here I had 20 percent
volatility. It doesn't matter.
I mean, that's a plausible
amount of volatility, a little high,
but that volatility. The mortgage rate of 8 percent
is not going to give a value of 100.
It's going to cheat the bank if
the homeowners are acting rationally.
The bank could get 120 if the
people weren't acting rationally.
They were just never exercising
their option.

It they're exercising their
option optimally the thing was only worth 98.
Now, I told you at that time
the interest rates should have been around 7 and 1 half
percent, not 8 percent given this 6
percent interest rate in the economy.
The mortgage rate should be 7
and 1 half percent. So we deduced last time that
obviously not everybody's acting optimally.
Well, you can tell that looking
at this diagram. How do you know that not
everybody's acting optimally? Remember these are '86
mortgages, so everybody's taking them out at the same time within
a few months of each other, the same 8 percent mortgage.
How can you tell from this
graph that they're not exercising their option
optimally? It's completely obvious.
Just looking at it for one
second you can say, "Oh, these people can't be
exercising their option optimally,"
why is that? Yes?
Student: They should be
exercising all at the same time if they were acting rationally.
Prof: So as he says
we've just done the calculation with those 1s and 0s.
I told you when the right time
to exercise the option is, so, everybody's got the same

Every single person if all
they're trying to do is minimize the present value of their
payments they should all be prepaying at the same time.
Here you see that very few
people are prepaying, but it's getting up to almost
10 percent so probably this is a stupid time to prepay,
but the point is still 10 percent of them are prepaying.
And over here when presumably
you ought to prepay, in the entire year,
right, they have 12 chances during the year.
It takes them an entire year
and only 60 percent of them have figured out that they should

So you know they're not acting
optimally. So just from that graph that
would tell you, and you have further evidence
of that. That's evidence that they
aren't acting optimally. Furthermore you have evidence
that the banks don't expect them to be acting optimally because
the banks aren't charging them 8 or 9 percent interest,
which is what they would need to pay to get the thing worth
100, they're charging them 7 and 1
half percent interest which for the optimal pre-payer is worth
much less than 100 to the bank. So the banks wouldn't do that.
They would just go out of
business if they did something stupid like that.
They wouldn't do that unless
they thought that the homeowners weren't acting,
at least not all of them acting, optimally.
So suppose you had to predict
how people are going to act in the future and you wanted to
trade on that? What would you do?
How would you think about
predicting it? So this is the data that you

What would you do?
You have this data.
These are 8 percent things.
You also have 9 percent
mortgages issued the year before,
and then maybe a year before that there were 8 and 1 half
percent interest and you have that history,
and you've got all these different pools and all these
different histories. How would you think about
figuring out a prepayment–how would you predict prepayments?
Well, the way economists,
macro economists at least in the old days,
used to make predictions, they would say,
"Hum, the first quarter looks pretty good."
What are they predicting now?
Now, they're saying
unemployment is probably going to keep rising for the next
quarter or two well until the next year,
but at that point things are going to turn around and we
expect the economy to get stronger,
come out of its recession and unemployment should gradually
improve from its high which we expect will be 10 and 1 half
percent to something back down to 6 percent by the end of 2011.
That's more or less the
economists' prediction.

Now, can you make a prediction
like that about prepayments? Would it make sense to make a
prediction about that? Why is that an utterly stupid
kind of prediction? What is the essence of good
prediction? If you wanted to predict
something and you were going to lose a lot of money if your
prediction was wrong how would you refine your prediction
compared to what I just gave as a sample prediction?
Student: You have to
have a number of scenarios and >
to each one.
Prof: Exactly.
So what he said is if you're
even the slightest bit sophisticated you're not going
to make a bald non-contingent prediction.
Things are going to get worse
the next two quarters, then they're going to start
getting better, then things are going to get as
well as they're going to get after two years.
You'll solve the problem after
two years.

What happens if another war
breaks out in Iraq? What if Iran bombs Israel?
What if there's another crash
in commercial real estate? How could that prediction
possibly turn out to be true? It's a sure thing it's going to
be wrong. It's just impossible that's
going to be right because the guy making the prediction has
made no contingencies built in his prediction.
You know that guy's making a
prediction for free. Someone may be paying him to
hear him, but he's not going to be penalized if his prediction
is wrong. No one in their right mind
would make such a prediction. So the first thing you should
do in predicting prepayments is to realize that you've got a
tree of possible futures, and given this tree of possible
futures you're going to predict different prepayments depending
on where you go on the tree.

So you see, prediction is not a
simple one event–it's not a one shot thing.
Just as he so aptly put it,
it's a many scenario thing. You have to predict on many,
many scenarios what you think will happen and that makes your
prediction much better because, of course, if there is a war in
Iraq, and if there is a catastrophe
in Afghanistan, and if Iran does bomb Israel,
and if the commercial real estate market collapses things
are going to be a lot worse than this original guy's prediction.
So everybody knows that,
so why not make the prediction more sensible?
So, on Wall Street that's what
everybody's done for 20 years.

Now, they haven't done it for
30 years. It's just 20 years that they've
been doing that. So when I got to Kidder Peabody
in 1990 they were making these one scenario predictions.
It's a long story which I'll
tell maybe Sunday night. I ended up in charge of the
Research Department and so we made, you know,
other firms were doing this already, we made scenario
predictions, okay? So now what kind of scenario
predictions are you going to make?
When you make contingent
predictions there are an awful lot of them.
You can't even write them all
down, so what you have to do is you have to have a model.
So what kind of model should
you have? I'll tell you now what the
standard guys were doing on Wall Street at the time.
They were saying–here's
interest rate, sorry.
Here's the present value of a

Here's the present value of a
callable mortgage, present value of 1 dollar of
principal, so realistic prepayments.
So if we go over here we'll see
that people said, "Look, from this graph
it's clear," they would say,
"that when interest rates went down people prepay more so
why don't we have a function that looks like this?"
So, prepay, that's the
percentage of remaining balance that is paid off.
So what does that mean?
Remember, after you've made
your coupon payment you have a remaining balance,
B_5. You could pay all of it,
or none of it, or half of it.
So the prepay is what
percentage of the B_5–that's just after
you've paid, right? So, B_2 lets do that

just after you've paid 8.88 the remaining balance has now been
reduced to B_2. You could, in addition to the
8.88, pay off all of that B_2.
Typically some people who are
alert and think it's a good time to prepay will pay all of
B_2. Others will pay none of
B_2. So if you aggregate over the
whole collection of people the prepay percentages,
out of the sums of all their B_2s what percentage
of them are going to pay off.

So we look at the aggregate
prepayment. That's the old fashioned way.
And we say, "What
percentage of the remaining balance is paid off?"
So you'd make a function like
this. You'd say, "Well,
prepaid might equal 10 percent."
Why am I picking 10 percent?
So if you go back to this
picture you see that prepayments seem to be around 10 percent
when nothing's happening. So you say 10 percent plus
maybe you're going to get some more prepayments so you might
write–well, I just wrote down a function plus the min.
The min, say,
of .60 because it never seems to get over 60 percent if you
look at that you see it never gets over 60 percent really.
So the min of 60 and 15 times
the max of 0 and (M – r_S – sigma over 133).
That would be a kind of
prepayment function.

So what does this say?
What happens?
You're normally going to
pay–so this is this whole function here,
so I should write this as .1 plus, can you see that over
there, maybe not, so this plus .1.
So there's a baseline of 10
percent and if the interest rate is high,
so the interest rate is above the mortgage rate no one else is
going to prepay because this is going to be a negative number
and this will be 0. So you're just going to do
.1,10 percent. On the other hand,
as the interest rate gets low and falls far enough below the
mortgage rate people are going to say to themselves,
"Ah-ha! I have a big incentive to
prepay now. Maybe interest rates have gone
down so far I can no longer hope they're going to go back up
above the mortgage rate.

I should start prepaying
more." So more people are going to
prepay and this thing is going to go up.
I just multiply it by some
constant, but it'll never go up more than 60 percent.
That's what this function says.
And sigma, this is the
volatility–all right, so let's just leave that aside.
So there's a prepayment
function that seems to sort of capture what's going on.
It's usually around 10 percent
when there's no incentive. It never gets above 60 percent,
but as the incentive to prepay, as interest rates get lower and
the incentive to prepay increases,
more and more people prepay. That's kind of the idea.
All right, and then you would
fit fancier curves than that. You would look at M –
r_T and you would fit a curve that looks like this.
So if there's just a little bit
of incentive to prepay, the rates are a little bit
lower than the mortgage rate, nobody does it.
Then quickly a lot of people do
it and then they stop doing it.

So this is like 60 percent and
most of the time you're around 10 percent, and you try and fit
this curve. You're going to have millions
of parameters and since you have so much data you could fit
parameters. That was the old fashioned way
and that's how people would predict prepayments.
Now, that's not going to turn
out to be such a great way, but it certainly teaches you
something. So let's look at what happens
if you now–with those realistic prepayments you compute the
value of a mortgage. So this is the prepayment that
you'd get for the different rates and so you can see that as
the rates go down the total prepayment is going up.
And by the way,
it's more than 60 percent because you've got this 10
percent added to the 60 percent, so the most it could be is 70
percent, which it hits over here.
So you get 70 percent as the
maximum prepayments, and as interest rates get
higher no one prepays except the 10 percent of guys.
Now, by the way,
why are people prepaying over here even when the rates are so
high? It's because some people are
moving or they're getting divorced and they have to sell
their house.

So obviously you're going to
get some prepayments no matter what.
People have to prepay,
and why is it that people never prepay more than 60 percent
historically or 70 percent, because not everybody pays
attention. Now, I called them the dumb
guys last time, but as I said,
I probably fit into that category.
It's people who are distracted
and doing other things. They're just not paying
attention and so they don't realize.
They don't know what's going
on, so they don't realize they should be prepaying.
So as interest rates go down
more people prepay. As interest rates go up less
people prepay. And if you did some historical
thing and figured out the right parameters you'd get a
prepayment function. So how did I figure out this
was 15? How did I figure out this was
.6? Why should I divide this by 133?
What's sigma?
Once you get those parameters
historically you now have a well-determined behavior rule of
what people are going to prepay, and from that you can figure
out what the prices are of any mortgage by backward induction.
So how would you do it again by
backward induction? The same we always did it.
Over here, what would you do
over here? How would you change this rule?
Well, you would just be feeding
in the prepayment function.

So what would the prepayment
function be? Well, people wouldn't be doing
a minimum here, right?
They're not deciding whether or
not to prepay, they're just prepaying.
So let's get rid of that.
They're prepaying.
So this is the value of 1
dollars left of principal. So some of them are prepaying
and that's the function, so prepay, and that depends on
what node you're at. And here it says what
percentage of the remaining balance is being prepaid.
So that tells you,
that rule, who's prepaying, and then with the rest of the
money that's going on until next time 1 minus that same thing,
1 minus prepay times exactly what we had before.
So this part of 1 dollar got
prepaid immediately so that's the cash that went to the
mortgage holder.

The rest of the cash got saved
until next time and here's what happens to it.
You have to make your coupon,
then you have a remaining balance, and then whatever is
going to happen is going to happen.
So you'll study this and you'll
figure out I'm sure. It takes a little bit of effort
to see that through, but with half an hour staring
at it you'll understand how this works and you'll read it in a
spreadsheet so you can figure out the value of a mortgage.
You get a value of a mortgage,
and now we can start doing experiments by changing the
parameters and see how the mortgage works.
Now, before I do that I want to
say that there's a better way to do this.
I mean, maybe these numbers are
estimated–what's a better way of doing it?
How did I do it at Ellington,
how did we–I mean at Kidder Peabody?
How did we predict prepayments?
What's another way at looking
at prepayments? Let me tell you something
that's missing. I used to ask people who wanted
to work at Kidder Peabody or Ellington the following little
simple puzzle, and most of the genius
mathematicians always got this answer wrong.
Of course we hired them anyway,
but they'd always get this wrong.
So the question is,
suppose you've got a group of people like this and you figure
out what the value of the mortgage is,
and interest rates have been constant all this time.
Let's suppose for one month
interest rates shoot down, interest rates collapse and
half the pool, 60 percent of the pool

So now you've only got 40
percent of the people left you had before, and then interest
rates return to exactly where they were to begin with.
Should the pool that's left be
worth 40 percent of the pool that you had just here,
or more than 40 percent, or less than 40 percent?
So remember,
you had 100 people here. You're the bank who's lent them
the money. You're valuing the mortgage
payments they're going to make to you,
you're getting a certain amount of money from them,
60 percent of them suddenly disappeared in 1 month leaving
40 left, but now interest rates are back
exactly where they were before. Is the value of the mortgage
starting here with the 40 percent pool worth 40 percent of
what it was originally, more than 40 percent or less
than 40 percent? What do you think?
Student: Is it worth
more than 40 percent because those people don't understand
interest rates and therefore they're not
> option properly and
> their mortgages?
Prof: Exactly.
So that's an incredibly
important point. It's called the opposite of
adverse selection.


Every one of these events is
selecting the people left not adversely,
not perversely, what's the opposite of
adversely, favorably to you,
so the guys who are left are all losers,
but that's who you want to deal with.
You don't want to trade with
the geniuses. You want to trade with the guy
who's not paying any attention. So the guys left are the people
who are never going to prepay or hardly ever going to prepay and
so it's much better. Now, this function doesn't
capture that at all, right?
It doesn't say anything.
It just says your prepayment's
depending on where you are. So whether you were here or
here you're going to get the same prepayment,
but we know that that's not going to be the case.
In fact, it's clear that over
here there must have been a much bigger incentive than there was
over there.

So the prepayments are the
same, but actually interest rates here were vastly lower
than interest rates there. So this is not such a good
function. So how would you improve?
What would you do to take into
account this adverse selection, or actually pro-verse
selection? What is the opposite of adverse?
Well, it doesn't matter.
What would you think to do?
Your whole livelihood depends
on it, millions, trillions of dollars at stake

You've got to model prepayments
correctly, so how would you think of doing this?
Just give me some sense of what
a hedge fund does or what anyone in this market would have to do.
Well, most of them did this.
So what would you do?
Student: Buy up old
mortgages, because the market is probably under estimating their
value. Prof: Well you would buy
it up when? Student: Right after…
Prof: Right here you'd
buy it up, right there, but what model would you use to
predict prepayments? Not this one,
so how would you imagine doing it.
You would imagine making a
model just like your intuition, so what does that mean doing?
Someone's asking you to run a
research department, make a model of forecasting

All the data you have is
aggregate data like that. You can't observe individual
homeowners in those days. They wouldn't give you the
information. I'll explain all that Sunday
night. So this is the kind of data you
have, what the whole group of people is doing every year,
but what would you do to build the model?
Adverse selection is very
important or pro-verse selection.
It's embarrassing I don't
remember the word, favorable selection,
a very important thing. So how would you capture that
in your model? Yep?
Student: Would you split
it into two groups and then model it separately?
Prof: So maybe another
thing you could do, what if you instead of having
this function that says what the aggregate's going to do all the
data's aggregate, so all you can do is test
against aggregate data. But suppose you said,
"The world, all we can see is the
aggregate, but the people really acting are individuals acting,
not the aggregate.

It's the sum of individual
activities, so what we should do now is have different kinds of
people." Oh gosh, sorry.
It was there already.
So let's go back to where we
were before, so realistic. What you ought to do is you
ought to say, well, 8 percent–remember we
had two kinds of people already. We've already got two kinds of
people, sorry. We've got these guys,
the guys who never call, so they're people.
That's a kind of person.
And suppose you go down here
and you have the people who are optimally prepaying?
Suppose you imagine that half
the people were optimally prepaying and half the people
never prepaid? Well, would that explain this
favorable selection? Absolutely it would explain it
because when you went through your little tree and you went
here, and here, and here,
and here, by the time you got down here all those people,
all the optimal pre-payers they're all prepaying.
So you start off with
half-optimal guys and half-asleep guys.
Once you get down here all the
optimal guys have disappeared and the pool that's left is all
asleep, so of course the pool is worth
much more here given the interest rate than it was over

In fact, if it goes back then
again to here where it was before–sorry that's same line.
If it goes back to here–have I
done this right? No, I've got to go back twice
here and then here. So once it goes back to here if
it goes here, here, here and here then the
pool is going to be much more valuable here than it started
there. There are half as many people,
but it's worth much more than half of what it was there.
So the way to do this is to
break–so then you're looking at the individuals.
You're saying one class of
people is very smart, or one class of people is very
alert, it's a much better word, one class of people is very

One class of people is very
un-alert and as you go through the tree the alert people are
going to disappear faster than the non-alert people and that's
why you're going to have a favorable selection of people
who's left in the pool. Well, of course,
there are no extremes of perfectly rational or perfectly
asleep in the economy so what you can do is you can make
people in between. How do you make them in between?
Well, suppose that,
for example, I only did one thing.
Suppose it's costly to prepay?
Some people just say to
themselves, "I'm going to have to take a whole day off of
work. I'm not going to write my paper.
I might lose some business that
I was going to do that day. A whole bunch of stuff I'm
losing, so I'm going to subtract that.
I'm not going to prepay.
I'm not going to even think
about doing it unless I can get at least a certain benefit from
having done it." So you can add a cost of
prepaying and people aren't going to prepay unless the gain
that they have by prepaying exceeds the cost of doing the

So to take the simplest case
let's suppose the very act of– never mind the thinking and all
that– the very act of prepaying,
going to the bank literally costs you money.
So if you have a value,
if the thing is 100 and you can prepay,
you know, if you do your calculations and don't prepay
today it's worth 98 and if you prepay today the remaining
balance is 94 you're saving 4 dollars,
but if the cost of prepayment is 5 you're still not going to
do it. So you get a guy with a high
cost of prepaying, an infinite cost of prepaying,
he's going to look like he's totally un-alert.
A guy with zero cost of paying
is going to look like he's totally alert.
So you can have gradations of
rationality, and you can have different dimensions.
So you can have cost of
prepaying and you can have alertness.
What's the percentage of time
you're actually paying attention that month?
What fraction of the months do
you actually pay attention, and you can have a distribution
of people, different costs and different alertnesses.
So that's the model that I

It's a simplified form of it.
It gives you an idea.
So here's this burnout effect
that I showed that if you take the same coupons,
but an older one rather than a–an older one that's burned
out will always prepay slower, so the pink one is always less
than the blue one because it went through an opportunity to
prepay. So here you start with a pool
of guys on the right, and then after a while,
after time has gone down a lot of them have prepaid.
So here's alertness and cost.
So you describe a person by
what his cost of prepaying is and how alert he is.
The more alert he is and the
lower the cost of prepaying the closer to rational he is.
The less alert he is,
the higher the cost of prepaying the closer to the
totally dumb guy he is.

And so you could have a whole
normally distributed distribution of people and over
time those groups are going to be reduced because a lot of them
are prepaying, but they won't be reduce
symmetrically. The low cost high alertness
guys are going to disappear much faster and the pool's going to
get more and more favorable to you.
And so anyway,
all you have to do is parameterize the cost,
what the distribution of people in the population,
what the standard deviation and expectation of cost is and of
alertness is, and that tells you what this
distribution looks like. So you're fitting four numbers
and you've got thousands of pools and hundreds and hundreds
of months, and fitting four parameters you
can end up fitting all the data.

So look at what happens here.
So here's the same data.
So I just tell you I know that
in a population, given what I've calculated in
the '90s there, I know what fraction of the
people have this cost and that alertness,
what fraction of the people are so close to dumb that their
costs are astronomical and their alertness is tiny,
what fraction of the people have almost no cost and a very
high alertness, so I'm only estimating four
parameters because I'm assuming it's normally distributed.
Given that fixed pool of people
I apply that to the beginning of every single mortgage and I just
crank out what would those guys do.
In the tree if they knew what
the volatilities were when would they decide to prepay,
and then I have to follow a scenario out in the future and I
say, "Well, along this path
which guy would prepay and which guy wouldn't prepay and what
would the total prepayments look along that path?"
And so this has generated the
pink line from the model with no knowledge of the world except I
fit those parameters and look how close it is to what actually

So it turns out that it was
incredibly easy to predict, contingently predict what
prepayments were going to be and therefore to be able to value
mortgages. And this was a secret that not
many people, you know, a bunch of people
understood, but not that many understood,
and so for years we were trading at our hedge fund,
first at Kidder and then at Ellington with this ability to
contingently forecast prepayments at a very high rate.
And why was it so stable,
the prediction, and so reliable?
It's because the class of
people stayed pretty much the same and every year there'd be
the same kinds of people with the same kinds of behavior.
Some were very alert.
Some were very not alert,
but the distribution of types was more or less the same and
you could predict with pretty good accuracy what was going to
happen from year to year.

Of course, then after 2003 or
so the class of people started to radically change and many
more people who never got mortgages before got them and it
became much harder to predict what they were going to do.
But so in the old days it was
pretty easy to predict. And why was it so easy to
predict? Because it was an agent based
model, agent based. So, by the way,
I added this volatility here, so these guys who just ran
regressions they had to have a volatility or something
parameter. So you see as volatility goes
up the prepayments are slower. Well, they just had to notice
that and build it right into their function.
I didn't even have to think of
that or burnout. None of those things did I have
to think about because if you're a guy optimizing here and
volatility goes up, so you reset the tree so that
the interest rates can change faster.
The option is worth more so
you're going to wait longer.

You're not going to just
exercise it right away because you've got a chance that prices
will really go up so you can wait a little longer,
afford to wait longer. So prepayments will slow down.
So all I'm saying,
all of this is just to say that if you have the right–
so it's agent based, it's contingent predictions,
those two things together enable you to make quite
reliable predictions about the future if you're in a stable
environment. And so what seems like a
bewildering amount of stuff turns out to be pretty easy to
explain. So now what happens?
So do you have any questions
here or should I–yes? Student: You said you
assume that those two parameters are normally distributed.
Did you select among some sort
of variance? Prof: Some sort of what?
Student: Variance.
Prof: I had to figure
out what the mean and the variance is.
There's mean and variance of
cost and mean and variance of alertness to get that
distribution, right?
So how do I know what the
population–so let me just put the picture up again.
So who are the hyper rational
guys? They are the people with the
really high alertness up there and the really low cost,
so they're the guys back there.

They're the hyper–or maybe it
was the guys, you know, one of these corners
with very high alertness and very low cost.
I forgot which way the scale
works. It might be going down.
So anyway, the guys with very
high alertness and very low costs are the hyper rational
people. At the other corner you've got
the guys who have very low alertness and very high costs.
They're the people who you're
going to make a lot of money on if you're the bank.
So how do I know how many
people are of each type? Well, I don't.
I have to fit this distribution.
But you see I have so much data.
I've got this kind of curve.
This kind of curve I've got for
every starting year for the whole history and there's so
many different interest rates and so many different–
so I'm applying that same population at the beginning of
every single curve and then seeing what happens to my
prediction versus what really happened.
So I've got thousands,
and thousands, and thousands of data points
and only four parameters to fit.

So I pick the four parameters
to fit the data as much as possible.
If I assumed everybody was
perfectly alert instead of that curve that I showed you,
I put a huge crowd here of perfectly rational people then I
would have found that I would have gotten prepayments at 100
percent up there and at 0 all the way over here and so it
wouldn't have fit that curve. So that's how I knew that there
couldn't be that many perfectly rational people.
Student: How can you
know for sure that there are only two patterns?
Prof: You mean how do I
know cost and alertness, maybe there's some other
factors? Yes, well there probably are
other factors.

So what would you
commonsensically think are the factors?
What keeps people from
prepaying? I think the most obvious one is
it's a huge hassle and they're not paying attention.
So those are the first two that
I thought of. Could you think of another one?
Student: Maybe their age.
Prof: Their age, exactly.
So maybe demography has an
effect on it. So maybe, for example,
you get more sophisticated the older you get.
So that was another factor we
put in. So I'm not telling you all the
factors, but these were the two main factors.
Another factor was growing
sophistication. We called it the smart factor.
That's another factor.
So over time you get more
sophisticated. So anyway, the point is with a
few of these factors you got a pretty good fit,
and it was pretty reliable, and you could predict what was
going to happen contingently. So now if you want to trade
mortgages what are some of the interesting things that happen?
The first interesting thing to
notice is that what do you think happens as the interest rate
goes down? So the first thing to notice
is–so I'll just ask you two questions.
Let's go on the other side.
I'm running out of room.
Suppose that you have the
mortgage value, what you get in the tree?
So in this tree that we've
built, here's the tree, it's going like that,
and at every node we're predicting–
for each class of people we're predicting where his 1s are.
So that class is prepaying.
The other class is not as smart
so they're not prepaying here, but maybe when things get
really low they'll start prepaying here.
So each class of people,
each cost, alertness type has its own tree.
They're the same tree,
but it's own behavior on the tree, and then I add them all

So what happens with the
starting interest rate? So here we had .06 and this
value was 98 or something, right?
Now, suppose the interest rate
went down to .05. I drew this picture of interest
and mortgage value. What do you think happens?
So the interest starts–this is
'98,6 percent is there. As the interest rate goes down
what do you think happens to the value of the mortgage?
If you're a bank and you've
fixed–the mortgage rate is 8 percent.
That's a fixed mortgage rate,
but now you've moved in the tree from here to here.
Do you think your mortgage is
going to go up in value or down in value?
Student: It's going up.
Prof: It's going to go
up because the interest rates are lower and the present value
of the payments is getting higher.
So if the interest rate goes
down the mortgage is going to go up like that,

But will it keep going up like
this and this? If it were a bond it would go
up like that, right?
A bond, a 1 year bond which
owed 1 over 1 r would keep going up and up the value before it
got negative, say.
It would go up.
As r got negative it would go
way up like that. So does the mortgage keep going
up like that? As the interest rate goes down
is the value of the mortgage going to get higher and higher
and higher? Suppose the guy's optimal,
what's going to happen? This is 100 here.
What'll happen?
Student: He's going to

Prof: He's going to
eventually figure out that he should prepay so it'll go like
this. If he's perfectly optimal he'll
never let it go above 100. So it's going to go something
like this. As the interest rate gets
higher you get crushed, and as the interest rate gets
lower you don't get the full upside because he's prepaying at
100. He's never letting it go above
100, right? So if he's not so optimal maybe
your value will go up, but not so astronomically high.
So this idea that the mortgage
curve, instead of being like this goes like that,
this is what was called negative convexity.
Now, the next thing to know is
suppose that the guys are partly irrational so it's going above
100. So it's starting to go like
this. Then what do you think?
As the interest gets really low
what's going to happen? All right, you just said it, so.
If the guy was rational,
perfectly rational it would go like that.
He'd never let it go above 100,
but now suppose guys are not totally rational?
What's going to happen is
they're going to, sort of–as rates get a little
bit low they're going to overlook the fact that they
should prepay.

So now it's advantageous to you.
Things are worth more than 100,
but if rates get incredibly low even the dumbest guy,
the highest cost guy is going to realize he has an advantage
to prepay and so things are going to go back down like that.
So the value's going to be
quite complicated. So this is the mortgage value
as a function of interest rates. Just common sense will tell you
this. In a typical bond as the
interest rate gets lower the present value gets higher.
You should expect a curve like
that, but because of the option if it were rationally exercised
the curve would never get above 100.
It would have to go like that.
But now if people are
irrational you can take advantage of them and get more
than 100 out of them. But if the situation gets so
favorable to you it becomes blindingly obvious,
eventually to them, that they're getting screwed,
and eventually they act and bring it all the way back to 100

So this value of the mortgage
looks like that. So that's a very tricky thing.
I'll even write, very tricky.
So if you don't know what
you're doing you could easily get yourself hurt holding
mortgages. You could suddenly find
yourself losing money holding mortgages.
So that's my next subject here.
I want to talk about hedging.
So we know something now about
valuing mortgages. Now I want to talk about
hedging, and what hedge funds do, and what everyone on Wall
Street should be doing which is hedging.
So if you hold a mortgage
you're going to hold it because maybe you can lend 100 to a
bunch of people but actually get a value that's more than 100.
So it looks like you're here,
but if interest rates change a little bit suddenly this huge
value you thought you had might collapse back down to 100,
or the interest rates might go up and it might collapse to way
below 100. So you look like you're well
off, but there are scenarios where you could lose money and
you want to protect yourself against that.
So how do you go about doing it?
What does hedging mean?
And I want to put it in the
context, the old context of the World Series which we started
with before.

So it's easier to understand
there, and so many of you will have
thought about this before so you'll be able to answer it,
but if I put it in the mortgage context it would seem just too
difficult. I don't know why I did that.
So the World Series–I'm going
to lower it in a second. So suppose that the Yankees
have a 60 percent chance, I said beating the Dodgers,
I thought the Dodgers would be in the World Series,
a 60 percent chance of winning any game against the Phillies in
the World Series. And you are a bookie and your
fellow bookies all understand that it's 60 percent.
So some naive Philly fan comes
to you and says I want to bet 100 dollars that the Phillies
win the World Series. Should you take the bet or not?
Yes you should take the bet
because 60 percent of the time you're going to win 100
dollars–no. Yes you should take the bet.
If he bet on one game you would
make, with 60 percent probability you'd win 100 and
with 40 percent probability you'd lose 100.
So that means on average your
expectation is equal to 20.

So if he's willing to bet 100
dollars on the Phillies winning the first game of the series
with you, you know that your expected
chance of winning is 20 dollars. You're expecting to win 20
dollars from the guy. Now, suppose he's willing to
make the same bet, 100 dollars for the entire
series? What's your chance of winning
and what's your expected profit from him?
Is it less than 20,20,
or more than 20? Student: More than 20.
Prof: More than 20.
It's going to turn out to be,
so a 7 game series, it's going to turn out to be 42
which we're going to figure out in a second.
But what's your risk?
What's your risk?
In either case you might lose
100 dollars. The Phillies,
they're probably going to lose, but there's a chance something
goes crazy and some unknown guy hits five home runs in the first
four games or something, and some other unknown guy hits
another four home runs and you lose the World Series.
You could lose 100 dollars,
and maybe the guy's not betting 100 dollars but 100 thousand
dollars or a hundred million dollars.
You know you've got a favorable
bet, but you don't want to run the risk of losing even though
there's not that high a chance you're going to lose.
What can you do about it?
Well, you know that there are
these other bookies out there who every game are willing to
bet at odds 60/40 either direction on the Phillies or the
Yankees because they just all know–
they're just like you.

You all know that the odds are
60 percent for the Yankees winning every game.
So suppose this naive guy,
the Phillies fan, comes up to you and bets 100
dollars on the World Series that the Phillies will win.
You don't want to run the risk
of losing 100 dollars. You know there are these other
bookies who are willing to take bets a game at a time 60/40
odds. What should you be doing?
What would you do?
Student: Bet on the
Phillies winning because they give you better odds so you're
guaranteed your profit.

Prof: So what would you
do? So this guy's come to you,
and you're not going to be able to give the–
we're going to find out exactly what you should do in one
second, but let's just see how far you
can get by reason without calculation.
So this guy's come to you and
said, "I'm betting 100 dollars on the Phillies winning
the World Series." This is the night before the
first game. Every bookie is standing by
ready to take bets at 30 to 20 odds.
What would you do?
Student: You'd bet with
the bookie that the Phillies would win because…
Prof: That what?
Student: That the
Phillies would win. Prof: Yeah, how much?
Student: 100 dollars.
Prof: You'd bet the
whole 100 dollars? Student: Well,
you get better odds, so.
Prof: But would you bet
the whole 100 dollars on the first game?
The guy's only bet 100 dollars
on the whole series.

Student: You'd bet 80
> dollars.
Prof: So it's not so
obvious what to do, right, but he's got exactly the
right idea. You can hedge your bet.
So here we are.
I shouldn't have put that down.
Don't tell me I turned it off.
That would just kill me.
God, I meant to hit mute.
I think I hit off.
Oh, how dumb?
So you would bet on the–while
that warms up. I can see it.
All right, so what happens is
you'll have a tree which looks like this and like this,
like this and like this, like this and like this and
let's say we go out a few games like this.
Now, this is a 1,2,
3 game series.

All right, so I've done it.
Here's the start of World
Series. This is the World Series
spreadsheet you had before. Now, here's the start.
Here's game 1,2, 3,4, 5,6, 7.
So if the Yankees win the
series they get 100 dollars. You get 100 dollars, sorry.
Oh, what an idiot.
So every time you end up above
the start, win more than you lose, you get 100 dollars.
On the other hand,
if you lose more than you win you lose 100 dollars,
and so ctrl, copy.
Here is losing 100 dollars.
So now this tree,
remember from doing it before, is just by backward induction.
If you look at this thing up
there it says you get, 60 percent I think was the
number we figured out over here, so right?
So 60 percent is the
probability of the Yankees winning a game.
So you take any node like this
one you're always taking 60 percent of the value up here
plus 40 percent of the value here.
So if you do that you find out
that the value to you is 42 dollars, just what we said.
So let's put that in the middle
of the screen.

So the value is 42 dollars.
Now, if the Yankees win the
first game you're in much better shape.
So winning the first game means
you moved up to this node here. All of a sudden you went from
42 dollars to 64 dollars. And if the Yankees lost the
first game you would have gone down to that value which is like
9 dollars. Your expected winnings when the
Yankees are down a game, you know, they're still a
better team so actually it's more likely even after losing
the first game that the Yankees would still win the series.
So you see the risk that you're
running and you can calculate this.
So what should you do in the
very first game? This tells you that your
expected winnings is 42.

Of course .6 times 64 that's
38.4 .4 times 9 is 3.6. That is 42 dollars.
So that's 42 because it's the
average of this and this, and 64 is the average of .6 of
this and .4 of that. So what should you do?
Well, on average you're going
to make 42 dollars. What's the essence of hedging?
You want to guarantee that you
make 42 dollars no matter what happens.
No matter who wins the series
you want to end up with 42 extra dollars assuming the interest
rate is 0 from the beginning to the end of the series.
So how can you arrange that?
What can you do?
Well, so that's the mystery.
I'll give you one second to try
and think it through. You should get this.
What would you do here?
Are there no baseball bookies
in the–yep? Student: Didn't we just
bring this up before like with our hedge funds?
Can we put something else aside
that you view at a percentage rate that you think you can
trust and then you can trust the rest of it to whatever the real
probabilities are? Prof: Well,
you can bet with another bookie at 60 to 40 odds.
If the Yankees win the first
game you're just doing great.

If the Yankees lose the first
game you're looking to be in a little bit of trouble.
So the point is you're not
going to get the payoffs until the very end either plus 100 or
minus 100, but already by the first game
you're either doing better than you were before or worse than
you were before. You're already,
in effect, suffering some risk at the very beginning.
So this is one of the great
ideas of finance. You shouldn't hedge the final
outcome. You should hedge next day's
outcome. If you're marking to market
that's what you'd have to do. Marking to market you'd have to
say my position now–my bet is worth 64 dollars.
The Yankees lost the first
game, the bet would be worth 9 dollars.
So what does it mean to protect
yourself? Not just protect yourself
against what's happening at the end,
that's really what you want to do, but in order to do that you
should protect yourself every day against what could happen.
So every day you should end up
with 42 here and 42 there because, after all,
that's what you're trying to lock in.
No matter who wins the first
game you should still say I'm 42 dollars ahead because I got
myself in this position.

So how could you do that?
Well, let's bet at 3 to 2 odds,
right, 60/40 is 3 to 2 odds. Let's make a bet with another
bookie at 22 and 33 here. So 22–I put it in the wrong
place. This is the 33 and this is 22,
but plus 33 and minus 22. So what are you doing here?
Notice that this is 2 times 11,
this is 3 times 11. This is 60/40 odds.
I'm betting on the Phillies.
If the Phillies win one game I
collect 33 dollars. That's what I should do that he
said. He said, "Go to the bookie
across the street and bet on one game, not the whole series.
Bet on one game with that
bookie across the street, 33 dollars versus 22
dollars." Let's say you can only bet 1
game at a time with the other bookies,
actually, maybe you were saying all along bet on the whole
series, but let's say you can only bet
one game at a time with the other bookies.
You'd bet 33 dollars on the
Phillies in the first game.

That naive Philly fan has put
up 100 dollars on the series. You're, in the first game,
going to put 33 dollars. You've taken his bet so you're
hoping the Yankees win, but that's bad to be in a
position where you have to hope. You don't want to do that.
So you take his bet on the
Phillies because he's given you 100 to 100 odds.
That's even odds even though
you know the Yankees have a 60 percent change of winning.
You go to the bookie across the
street and you bet at 60/40 odds on the Phillies,
but you don't bet the whole 100.
You only bet 33 dollars of it.
So if you win you get 33
dollars. If you lose you only have to
pay the guy 22. So what's going to happen?
After the first day this
position is going to be worth 42 and this position is also going
to be worth 42, exactly where you started.
So because a win in the first
game is going to put you so far ahead in your bet with the first
naive Philly better, and a loss in the first game is
going to put you so far behind, you hedge that possibility by
going 33/22 in favor of the Phillies.
You take a big bet on the
Yankees and then you make a smaller bet on the Phillies that
cancels out part of the big bet on the Yankees,
but you've made the two at different odds and so on net
you're still going to be 42 dollars ahead.
Let's just pause for a second
and see if you got that.

So by doing this you can't
possibly lose any money. And now you're going to repeat
this bet down here and here. So in the next–you see where
do things go next? Here you're down 8 dollars.
If you lost again you'd be down
32 dollars. Now things would really be bad.
After the Yankees lost two
games in a row your original bet would look terrible,
but things aren't so bad because you bet on the Phillies

You already made 33 dollars.
So how much money do you think
you should be betting on the Phillies down here?
Well, you want to lock in 42
dollars at every node no matter what happens.
This 42 dollars,
by making the right offsetting bet you can keep 42 everywhere,
here until the very end, and so no matter what happened
you can always end up with 42 dollars.
That's the essence of hedging.
So let's just say it again what
the idea is. It's a great idea and we don't
have time to go through all the details, but the great idea is
this. You've made some gigantic bet
with somebody. Why do you bet with anybody?
Because you think you know more
than they do. The whole essence of trading
and finance is you think you understand the world better than
somebody else. So understand it means you
think something's going to turn out one way that the other guy
doesn't really know is going to happen.
So you're making a bet on
whether you're right or wrong.

So when you say you know you
don't know for sure. You just have a better idea
than he does, so you want to use your idea
without running the risk. So how can you do it?
If your idea is really correct
there may be a way so that you can eliminate the luck.
So here if you really know the
odds are 60/40, your class of bookies knows the
odds are 60/40, and some other guys who doesn't
know thinks the odds are 50/50 and is willing to bet against
you, you can lock in your 42 dollars
for sure. You don't just take a bet and
hope you win. You can take a bet and then
hedge it to lock in your profit for sure,
step by step, and that's what we have to
explain how that dynamic hedging works.
So I have to stop.


As found on YouTube

Looking to see what kind of mortgage you can get? Click here to see

Leave a reply

Your email address will not be published. Required fields are marked *