So, so far where have we gotten
to?
We started summarizing what
general equilibrium was.
We saw that Irving Fisher of
Yale reinvented general
equilibrium in order to study
finance,
and we saw just by
reinterpreting the variables of
general equilibrium we could
start to say a lot of things
about finance,
and in particular we had the
idea of free markets,
an argument in favor of free
markets.
 
We had the idea of arbitrage
and no arbitrage so you could
deduce a lot of prices without
solving for the whole
equilibrium just by knowing what
other prices are.
And we also learned that the
price of many things is going to
have to do with the utility and
marginal utilities of people,
and that's going to have a lot
to do with what their impatience
is,
and whether they're rich people
or poor people,
redistributions of wealth,
who's got the money and how
impatient the people who have
the money are.
 
So those are the basic lessons
that we're going to now carry
into the course.
 
And so for several lectures now
I'm going to leave the abstract
theory of general equilibrium
and start teaching you some of
the basic vocabulary of finance
that you have to know and that
everybody in finance knows like
what is a mortgage,
what's an annuity and stuff
like that.
So before I go there,
though, I want to remind you of
what Shakespeare had done 300
years before Irving Fisher.
So Irving Fisher,
remember, he cleared up the
confusion of what interest was.
 
He said interest is
crystallized impatience.
It's not some horribly unjust
thing.
It's not, as Marx thought,
exploitation,
but Shakespeare had discovered
all this 300 years before.
Now, when I was your age or a
little bit younger than you in
high school we all had to read
the Merchant of Venice.
I have two Indian coauthors who
are vaguely my age,
maybe a little older,
but anyway they sort of grew up
in India and they had to learn
the Merchant of Venice,
and actually they learned it a
lot better than I did.
They both have memorized the
Merchant of Venice.
They can recite almost the
entire thing by heart.
But anyway, when I was in high
school it was completely typical
to study the Merchant of
Venice.
I wonder how many of you have
actually read it.
Who's read the Merchant of
Venice?
Whoa this is Yale, I'm shocked.
 
So a quarter of you have read
it.
Well, I recommend to the other
three-quarters that you do read
it.
 
Now, when it's taught nowadays,
especially at Yale,
it's taught as a love story and
a commentary on anti-Semitism.
Now, of course, it's both.
 
It is a love story and
commentary on anti-Semitism.
Shylock the lender is Jewish.
 
And remember what we heard
about the great religions?
They were all forbidding
lending at interest except for
Judaism which let you loan money
at interest to non-Jews.
So Shakespeare [correction:
Shylock],
who's the money lender,
is Jewish and lending it to
Christians and that plays a big
role in the play and what
happens to him,
and what people say about
Judaism is a big element of the
story.
But the way the play is read
now that's the whole story,
and I don't think it's the
whole story.
In fact, I think it's quite an
unimportant part of the story.
I think the heart of the story
is Shakespeare's commentary on
economics.
 
And so I'm going to try and
argue in the next ten minutes
that Shakespeare was not only a
great writer,
a great psychologist,
but a great economist.
And you're going to see that
almost all the elements of the
course are in this play,
and that if you read it the way
I think you should read it,
it should be obvious that it's
really about economics and not
about love.
So how do you know that?
 
Well, the very first line of
the play, Antonio walks in and
he says, "In sooth,
I know not why I am so
sad."
 
And there's an interlocutor,
a minor character,
whose name I've forgotten,
Salario or something,
says well it must be that
you're so nervous.
All your riches are on these
boats and they're at risk,
and so anyone who had so much
money at risk on boats would
naturally be nervous and
therefore maybe depressed.
And Antonio says,
no, no, no, no,
I'm not worried about the boats
because every boat is on a
different ocean and so I'm not
worried.
They are on a different ocean
and they're sailing at different
times.
 
I'm not worried about my boats.
 
And so then the interlocutor
says, well then you must be
worried about love.
 
And he says,
no, no, that's not it at all.
So what do we see at the very
beginning of the play?
It's business first,
love second;
and secondly,
he understands diversification.
Now, what is the plot of play?
 
Bassanio, who Harold Bloom--so
it happens that went to talk to
Harold Bloom--
I saw him in the Whitney
Humanities Center--
one of Yale's greatest scholars.
He's a polymath,
he knows about everything,
but including about Shakespeare
and he has a much advertised
photographic memory.
 
So I happened to run into him
at the Whitney Humanities
Center, actually in the men's
room of Whitney Humanities
Center.
 
While we were there I asked him
about the Merchant of
Venice and whether he
happened to remember the rate of
interest that Shylock ends of
charging Antonio,
and he said, "Dear boy,
I remember almost everything,
but that I've forgotten."
 
It was so unimportant to him
that he didn't even remember the
rate of interest.
 
But he said,
"I happen to be lecturing
about the Merchant of
Venice in my class this
afternoon,"
so I went and heard his lecture
on the Merchant of
Venice.
So Bassanio,
who's one of the heroes of the
play, according to Harold Bloom
is a complete loser.
He's the one who needs the
money to woo Portia,
who's this beautiful woman
living outside of Venice.
And so he's got to borrow a
huge amount of money.
So when he enters he's
described as a Venetian,
a scholar and a soldier.
 
Now, whenever Shakespeare says
a scholar and a soldier,
sometimes it's not a Venetian,
when he says that the guy's
always a great guy.
 
So this occurs repeatedly.
 
So anyway, Bassanio comes in as
a star.
He's a Venetian,
a scholar, a soldier.
What more can you want to be?
 
And so he needs the money to
woo Portia and he's got a
business plan to do it.
 
He's tried wooing her before
and it's come to nothing and
he's lost his money.
 
But he says,
"If you shoot an arrow and
you lose it,
shoot an arrow again the same
way and then follow the second
arrow more closely and you'll
figure out where the first arrow
goes."
So he's a man on a business
venture with a business plan.
So here's Bassanio,
and here's Shylock and Antonio.
Now, he needs 3,000 ducats and
he doesn't have any collateral
or anything.
 
And so he goes to Antonio,
who's an older man,
and according to Harold Bloom
there's some potential gay
relationship and maybe they're
lovers and maybe they're not
lovers.
 
That's half the lecture.
 
Anyway, so Shylock lends the
money,
3,000 ducats,
and it's so much money he has
to borrow it from another money
lender named Tubal who's even
richer than he is.
 
So they argue over what the
interest rate has got to be.
And so Antonio says--oh dear
I've forgotten to change this,
so this is out of order--so
they argue over what the
interest rate should be and
Antonio and Shylock make this
argument.
 
Antonio says,
it's disgusting that you want
to charge me interest.
 
I mean, good Christians never
charge interest.
I'm appalled at you.
 
It's because you're Jewish
you're charging me interest.
So he's throwing up epithets
and insults at Shylock,
but really he just wants a low
interest rate.
And so he says,
Antonio says,
"Shylock,
I would neither borrow nor lend
by taking or giving interest,
but to supply the ripe wants of
my friend I'll break a
custom."
So "ripe wants of my
friend" that's saying
because Bassanio was so
impatient to get his hands on
the money to find Portia,
she's going to get married if
he doesn't hurry up and marry
her himself.
Because of his impatience he's
willing to pay a high rate of
interest.
 
And Shylock says you're always
complaining about me that I
charge interest--I've left out a
whole bunch of stuff--but I'm
patient.
 
All of us are patient.
 
That's the badge of our tribe.
 
We're patient and so that's why
I'm willing to lend you the
money.
 
So here Shakespeare has laid
out, and it goes over five pages
patience and impatience.
 
So then they get an argument,
again, about interest.
So I forgot a slide.
 
So the argument is Shylock
tells a story.
He says even in the Bible,
you say that it's un-Christian
to lend at interest,
but don't you know the story in
the Bible where Jacob was asked
to do,
perform a service in the field,
using his fields.
Somebody wanted to use his
fields for a while,
and so Jacob said okay you can
use my fields but I have to
charge you a fee and the fee's
going to be that however many
spotted lambs are born those are
the ones that I get.
And so it turned out that there
was a huge number of spotted
lambs,
and so although Jacob had lent
some of his sheep and his fields
to the person who wanted them he
got back vastly more than he
lent at the beginning.
And so Antonio answers,
well, this isn't interest this
is a risk.
 
Jacob got so much more because
he took a risk.
Who would have known how many
lambs were going to be born.
And so you don't really charge
interest, you're an investor.
So they haggle over this for a
while and they come to the
conclusion that he's going to
lend the money.
And so what is the interest
that they actually end up
charging, the thing that Harold
Bloom couldn't remember?
Well, 0.
 
"I'll lend you the money
and take no doit of usance for
my monies,"
not a single interest for my
money.
 
But they have to negotiate
something else,
something besides the rate of
interest.
They have to negotiate the
collateral.
And so they say,
"Go with me (blah,
blah, blah, blah),
and then if you don't pay let
the forfeit be an equal pound of
your fair flesh to be cut off
and taken in what part of your
body pleaseth me."
The other half of the--there
were two lectures by Harold
Bloom,
the second half of the first
lecture was what part of the
body is he really talking about,
and there seemed to be only two
possibilities,
the heart and another
possibility, and Harold Bloom
favored the second possibility.
 
But anyway, it's collateral
that they're putting up for the
loan.
 
So there's collateral.
 
So now what we found is that
Shakespeare has understood the
impatience theory of interest.
 
You've got an impatient
borrower and a patient lender,
and it's the tradeoff between
patience and impatience which is
going to decide what the rate of
interest is.
So that's already Irving
Fisher's biggest message.
And then the second thing he's
noticed,
which Irving Fisher didn't
notice at all,
and this is going to be a large
part of the rest of the course,
how do we know these people are
going to keep their promises.
Why is Antonio going to keep
his promises?
Well, it's because he's putting
up collateral.
And Antonio is stepping in for
Bassanio because his collateral
is worth more than Bassanio's.
 
Shylock wants his pound of
flesh, not Bassanio's pound of
flesh.
 
So all right,
so that's the beginning by the
way.
 
Just how does the play unfold?
 
It gets more interesting.
 
So what happens is after
getting his money Bassanio then
goes to woo fair Portia.
 
And how does he woo her?
 
Well, it turns out the way that
her fabulously wealthy father
has set up the marriage is,
there are three caskets,
a gold one,
a silver one and a lead one,
and he has to pick one,
and one of them contains her
picture,
and if you get the one with her
picture you get to marry her.
 
If you pick the wrong one,
and here's the shocking thing,
if you get the wrong one you
swear before you choose if you
choose wrong never to speak to
lady afterward in way of
marriage.
 
So not only don't you get
Portia, you don't get anybody.
So what is the purpose of this
absurd contract?
Well, the purpose is maybe to
make sure that people really
want to marry her.
 
Maybe the father set it up so
that only someone who really
wanted to marry her would bother
to enter this competition
because the risk is so high,
but another way of saying it is
it gives an excuse to
Shakespeare to talk about risk
and return and how people who
have a higher risk are going to
expect a higher return.
 
So they talk about risk and
return.
And Aragon basically says she's
really not that good looking to
justify such a high risk.
 
But anyway, all those other
guys picked the wrong casket and
Bassanio picks the lead one and
gets her, and so she becomes the
wife.
 
Now, of course,
she's delighted by this.
He's the one she wanted all
along, and so she says,
"Let me give you this
ring."
This is yet a third contract.
 
The first contract is the loan
of Shylock.
The second contract is the
choosing the caskets and a
contract that you won't marry
again if you choose wrong.
And now we have a third
contract, which is Portia
deciding that she gives a ring
to Bassanio and she says,
"Let this ring represent
your love."
And he says,
"When this ring parts from
this finger then parts life from
hence."
I'll never lose this ring.
 
I'll never give it up.
 
I love you so much.
 
So, of course,
the boats appear to sink.
Calamity appears.
 
"My ships have all
mis-credited [correction:
miscarried]."
 
Shylock wants his collateral.
 
So Portia now,
who turns out to be incredibly
wealthy--so we realize,
again, the play's not about
love.
 
She's beautiful,
but she's fabulously rich,
much richer than Shylock is,
much richer than Tubal was.
They had to scrounge around to
get the 3,000 ducats.
She hands 6,000 ducats,
and then 12,000 ducats,
then 36,000 ducats.
 
Says, look, offer Shylock all
this money.
Tell him, here,
I've got the money.
Tell him not to take his pound
of flesh.
So they hold the trial to
decide whether Shylock should
get his pound of flesh or not.
 
And so Shylock,
by this time,
is incredibly pissed off,
to say the least,
at Antonio and Bassanio.
 
And why is he so angry?
 
Because among other things his
daughter Jessica has run off
with a Christian named Lorenzo
and stolen his money.
And so he yells,
"My daughter,
my ducats."
 
And so she sold his wife's ring
for a monkey,
or his ring that was given to
him by his wife Leah.
And he says,
"I had it of Leah when I
was a bachelor.
 
I would not have given it for a
wilderness of monkeys."
So Shylock believes in keeping
his promise.
He would never have broken a
promise.
He never would have given away
the ring that was given to him
by his wife.
 
He absolutely wouldn't do it.
 
He believes in keeping promises
unlike everybody else in the
play, his daughter,
everybody as we'll see.
So Lancelot says,
"This making of Christians
will raise the price of
hogs."
This Jewish girl Jessica has
become a Christian so now she's
going to be able to eat pork so
it's going to increase the
demand for pork and therefore
raise the price of hogs.
So Shakespeare--the play is
full of economics.
It's all about teaching
economics.
So anyway, they go to the trial
and Shylock thinks the guy is a
complete fool.
 
He doesn't understand interest.
 
He doesn't understand the whole
point of a lending contract and
getting interest,
and basically he says--I've got
to skip over this a little
quickly.
He says, we've got a contract.
 
Your city, the greatest
commercial city in the world at
that time, can't possibly
survive if you don't uphold
contracts.
 
So, "If you deny me,
fie upon your law.
There's no force in the decrees
of Venice.
I stand for judgment."
 
I stand for keeping promises
and the law is supposed to
enforce promises.
 
I stand for law is what Shylock
says.
Now, at the trial,
who turns out to be the judge?
Well Portia has disguised
herself as the judge and she's
actually the judge.
 
And so she comes in and she has
this famous line,
"Who is the Merchant here
and which is the Jew?"
So, again, this confirms to me
it's about economics.
If it was about Judaism it
would be, who's the Christian
and who's the Jew?
 
She's saying,
who's the borrower,
and who's the lender.
 
That's how she comes in.
 
And so then she says,
"You've got to show
mercy."
 
And this is the most famous
line in the play,
"The quality of
mercy," blah,
blah.
 
You all remember it who've seen
it.
And he says,
and then Bassanio says look,
I've got 6,000 ducats.
 
I've got more than that,
take that.
And he says a contract is a
contract.
You've humiliated me,
all kinds of humiliations have
happened to me.
 
I've got feelings too.
 
I've been humiliated.
 
I want the contract and the
contract says that I should get
the pound of flesh.
 
And so Bassanio says,
"To do a great right do a
little wrong."
 
So let him default.
 
So what does Portia say?
 
What is the judgment?
 
She has to play the judge.
 
It seems like the whole city
depends on enforcing contracts
and here it seems like a
horrible thing to do.
You're going to have to kill
somebody.
So what judgment can she
possibly make?
She says, well,
the state has to enforce
contracts, of course.
 
Contracts have to be enforced,
but only good contracts should
be enforced.
 
And so what's wrong with the
contract?
Does she say we're going to
reduce what you owe from 3,000
to 1,500?
 
That's principal forgiveness.
 
Does she say you don't have
to--what does she say?
What she says is that what was
wrong is, the contract wasn't
right.
 
It wasn't the interest rate
that was wrong.
It wasn't the amount you owe
that was wrong.
It was the collateral that was
wrong.
So she says the right
collateral was a pound of flesh,
but not a drop of blood,
and so the state intervenes not
to change the interest rate,
not to change the principal,
but to change the collateral.
 
So all right,
that's going to turn out to be,
the leverage was wrong.
 
So then the play ends with
Bassanio asking the judge,
he's so pleased that things
have turned out right,
if he can reward the judge.
 
He doesn't know who the judge
is.
And he says,
I've got all these ducats that
I've just gotten,
why don't I give you some of
the ducats?
 
And the judge says,
well no, I don't want the
ducats, but I notice you've got
this ring on your finger,
why don't you give me that?
 
And he says,
well I can't do that,
my Portia, I've promised.
 
And the judge says,
well, give it to me anyway and
he gives her the ring.
 
And so the play finally ends
with her revealing herself and
he's incredibly embarrassed that
he's given her the ring.
So this is another contract
broken, another default.
And then he says,
but I'm never going to default
again.
 
And Antonio steps in and says
I'll guarantee again that he'll
never default again,
and of course,
we all know that he's going to
default.
So the whole play is just about
contracts and breaking
contracts.
 
And so at first it's about what
the rate of interest should be,
then it switches to,
should contracts always be
enforced,
and yes they should be
enforced, but the enforcement
should be the taking of
collateral and sometimes the
amount of collateral put up is
wrong.
 
So that's going to be the
conclusion of this course that
what went wrong in the last two
years or three years was a
horrible mistake about how much
collateral to be put up,
and the Fed instead of just
monitoring the interest rate,
which is what you're taught in
macroeconomics it's supposed to
do,
should be monitoring collateral
as well and maybe even most
importantly.
So with that introduction to
the rest of the course--
I don't know how convinced you
are about Shakespeare the
economist,
but anyway, let's now switch to
learning some of the basic words
of finance.
So I'm going to now follow
pretty closely what the notes
are.
 
All right, so let's imagine a
world where we've solved for the
equilibrium.
 
This could be the real world or
one of our models,
and there are many time
periods, not just two time
periods.
 
So let's suppose that there,
as there are as we're going to
see in great detail later,
suppose it's possible to pay
money today in order to get a
dollar next year,
or pay some amount of money
today in order to get a dollar
in two years,
or pay a different amount of
money today to get a dollar in
three years.
So pi_t is the amount
of money you pay today to get a
dollar at time t.
 
That's called a zero because
there's no coupon.
You just get something at the
end.
And so we'll see next class
we're going to start talking
about real markets and what the
prices of all those zeros are.
So anyway, that pi_t
is something that is traded in
the market and everybody at
every hedge fund and every Wall
Street bank knows what
pi_t is at the
beginning of each day.
 
Now, Fisher said,
well don't get too lost
thinking about pi_t.
 
Think about p_t.
 
Take out inflation.
 
You have to make an expectation
about what inflation is,
but assuming you're right you
can figure out from these
pi_t's what
p_t is,
the present value price:
how much would you pay today in
goods to get an apple at time t?
 
Not a dollar at time t,
but an apple at time t,
so it involves knowing what
inflation is and what the price
of apples is going to be at time
t.
So Fisher said,
there's a lot of stuff you can
do, but the
pi_ts--there's also
more important stuff you can do
with the p_ts.
You should always keep those in
mind.
So let's take the simplest case
where pi_t is a
constant interest rate.
 
There's a constant interest
rate i.
So if you ask what's a dollar
worth today in terms of how many
dollars you can get next year
it's 1 i.
What's a dollar worth today in
two years is (1 i) squared.
So putting it backwards,
a dollar in two years--the
price of it today must be 1 over
(1 i) squared.
So a dollar in t years,
the value today is 1 i over t
[correction: 1 over (1 i) to the
power t].
So this is just a
simplification.
So we'll see that lots of the
jargon of economics assumes that
there's this constant interest
rate that's determining all
these prices.
 
So the first thing to realize
is what Fisher calls the present
value price.
 
If there were some asset that
paid off money in the future,
m_1 through
m_T,
you don't have to solve the
whole equilibrium to figure out
what its price would be if you
knew the prices of these zeros,
pi_1 through
pi_T.
Because to get m_2
dollars at time 2 just cost you
pi_2 times
m_2 dollars today.
So you add up the cost of
buying all the cash flows or the
asset.
 
That has to be the price of the
asset today.
And if the prices of the zeros
are given just by the interest
rate discounting it then it's
just m_1 over (1 i),
m_2 over (1 i)
squared, and m_T over
(1 i) to the T.
 
I see there's a typo here--oh
no, no there's no typo--it's (1
i) to the T.
 
They're all these (1 i) to the
T.
So and now if the price weren't
that, if the price of this bond
were, let's say,
smaller than this,
what would you do?
 
You would buy the bond and at
the same time you'd sell
promises to deliver
m_1,
m_2,
and m_T in the
future.
 
If you sold those promises and
nobody doubted that you would
keep your promise--
so this is something
Shakespeare would have been
suspicious of--
but if you made those promises
and no one doubted you'd keep
them you could raise this much
money by selling all the
promises.
 
So you'd get this much money
and if the bond cost less than
that you could make all the
promises,
get all the money,
buy the bond,
have money left over,
and then you'd have to keep all
your promises,
but the bond itself would be
paying you money in the future
that you could use to keep all
your promises.
 
So it has to be that this is
the price of the bond provided
that everybody will allow you to
borrow and lend at those rates
of interest,
because if it weren't you would
either buy the bond and sell all
the promises,
or in the other case were the
prices higher you'd sell the
bond,
get all this money and then use
that money to buy all those
promises.
Then you could make the
payments of the bond because the
promises would come due to you.
 
So in that case you'd have to
believe people who made promises
to you.
 
So as long as nobody's doubting
the other people keeping their
promises it has to be that by no
arbitrage,
the price of a bond is just the
discounted cash flow.
That was Fisher's main
principle.
So we saw that last time.
 
We're just going to do it.
 
So we're now going to introduce
a few vocabularies.
So the first thing is the
doubling rule.
So I think at least half of you
probably know this,
but it's much better if you can
do things in your head than
having to calculate them all.
 
So the doubling rule says how
many years at i percent interest
does it take to double your
money.
So you can just solve this.
 
So (1 i) to the n means that if
you take the logs of both sides
and you know that log of 2 is
.69,
then you take the log of both
sides,
n log (1 i) has to be log of 2,
so n = .69 over log of (1 i).
So now log of (1 i) is
approximately i.
Why is that?
 
So this is Taylor's rule.
 
You don't actually have to know
this if you've never seen
Taylor's rule before.
 
But an approximation of log of
(1 i)--
for any function F of X,
it's F of A F prime of A times
(X - A) 1 half F double prime of
A times (X - A) squared.
That's the standard Taylor's
rule thing.
So therefore log of (1 i) is,
you know log of 1 is 0,
so it's going to be 0 i because
the derivative of the log is 1
over X and if X = 1 that's 1
over 1.
So it's 0 i,
and then the second derivative
is minus 1 over X squared,
and if X = 1 that's minus 1.
So with the half here log (1 i)
is approximately 0 i - i squared
over 2.
 
So you can replace log of 1 i
with,
I mean .69 over log of 1 i with
.69 over i - i squared over 2,
so for very small interest
rates i squared is practically
nothing.
 
So .69 over i if the interest
rate is .023 percent and .69
divided by .023 is 30.
 
So it says that at 2.3 percent
interest you double your money
in 30 years.
 
Well, if i is 7 percent,
say, then i squared is starting
to get a little bit bigger.
 
So i - i squared over 2 is .07.
 
i squared .0049 over 2 that's
.0675.
So if I put in i = .07 it's .69
over .0675.
That's around 69 over 67.
 
There's a decimal thing,
so it's a little over 10,
say 10.2.
 
That's like .72 over .07,
so .72 is the doubling rule.
To get for interest rates
around 7 percent,
or 6 percent,
or 4 percent,
something like that,
you're going to divide not into
69 but into 72.
 
The interest rate is .07, right?
 
The interest rate is a percent,
so it's a decimal thing.
So .06 is like 72 over 6.
 
So at 6 percent it takes 12
years to double your money.
So the rule,
the basic rule is,
if you want to know how long it
takes at 6 percent interest to
double your money you just take
72 divided by 6 and it's 12
years.
 
If it's 8 percent interest 72
over 8 is about 9 years.
If it's 10 percent interest
it's a little over 7 years to
double your money.
 
And so that rule is incredibly
useful to keep in your head
because you can shock and amaze
people by how fast you can
compute things if you just
remember that rule.
So let's just check the rule,
by the way.
So suppose that you have 24
dollars in the bank,
and you have 6 percent
interest.
So here I took the 24 dollars.
 
You look at the top you see
that's the B1 number,
and I've just multiplied it by
the interest rate,
1.06, and so I keep doing that.
 
Here I've multiplied the thing
above it by 1.06 again.
So I keep investing the money
at 6 percent interest,
and over here I've invested the
money at 7 percent interest.
So anyway, after 12 years you
see that 24 dollars,
this is year 1,
so at year 12,24 dollars has
become 48.
 
So it's a very good
approximation.
And so 7 percent it's supposed
to take a little over 10 years.
So at 10 years you're not quite
there, but 11 years you're past
it.
 
So you can see how we're
starting with 24,
you can see how good the
doubling rule is.
So that's just something to
keep--so we can now do in class
lots of concrete examples
without having to take out our
calculators and stuff because we
can do them in class in your
head.
 
All right, so let's just do
that now.
Okay, so in fact why did I pick
24 dollars?
Well, this is a famous story
you hear in second grade.
The Indians sold Manhattan for
24 dollars in 1646,
so how bad a deal was that for
the Indians?
It looks incredibly stupid,
but actually interest
accumulates pretty fast.
 
So if you look at 6 percent
interest, so 360 years gets you
to 2006.
 
That's a sort of round number
at 6 percent.
So at 6 percent how long does
it take to double?
It takes 12 years to double.
 
So that means at 6 percent
interest you're doubling every
12 years.
 
So in 360 years you're going to
double 30 times.
So in your head you can figure
out that doubling 30 times is 2
to the thirtieth,
and of course 2 to the tenth is
something you should know.
 
It's 1,024.
 
So I'm sure you know that
number, right,
2 to the--anyway,
that's a good one to remember 2
to the tenth is about 1,000,
so 1,000 cubed is about a
billion, so basically 24 becomes
24 billion.
So at 6 percent interest they
sold Manhattan for 24 billion in
today's dollars.
 
So that's pitifully low,
but if you look at 7 percent
interest you can do the same
calculation.
So at 7 percent interest you
should do this in your head now.
So it's going to double every
72 over 7 years.
So there are 360 years,
about, 360 is a very round
number, so 360 divided by 72
over 7 that's 5 times 7,
it's 35.
 
So 2 to the thirty-fifth,
well it's like a billion times
2 to the fifth which is 32.
 
So 1 dollar becomes 32 billion,
but we started with 24 dollars
so it's 768 billion.
 
So now you're starting to get a
little bit closer to what the
value of Manhattan is.
 
I mean, the value of all the
real estate in the country,
all the houses in the country
used to be 20 trillion.
I'm not sure how far they've
gone down now.
Let's say they're 15 trillion.
 
So 15 trillion you add
commercial real estate,
maybe in the whole country
that's worth 25 trillion,
but that went down too so let's
say 20 trillion.
Now how much of the 20 trillion
could possibly be in New York
City?
 
I've actually got no idea,
but it can't be that much
more--the whole country is 20
trillion.
New York can't be worth more
than 1 or 2 trillion of the 20
trillion, so you're not that far
off.
So the deal's not that
spectacularly bad although it
sounds ridiculous.
 
Anyway, the point is you can do
this in your head.
So now the next thing that you
realize in this example is how
huge a difference a percent
makes.
So why is that so important?
 
Well, managers,
hedge funds,
we all charge a percent
interest.
So look at what's happening.
 
I mean, if you look at our
Indian investment of 24 and you
look--I don't know how many
years you want to look over,
but you can look over 36 years.
 
That's a sort of
typical--you're young and making
an investment.
 
When you get old what's the
difference?
This was the 6 percent growth
and this was the 7 percent
growth.
 
This is the difference and this
is the, I guess,
the percentage difference.
 
So it's 28 percent.
 
Of course I didn't label these,
but yeah so this is the
difference and this is the
percent difference.
So the percent difference I
just showed it to you.
Over 36 years it's a 28.6
percent difference.
So a typical--you're putting
money away right now.
You might be giving it to some
fund.
You might be investing
it--whatever fund you're
investing in,
they could be charging 1
percent interest.
 
And it seems,
what's 1 percent,
it's a tiny amount,
1 percent, but over 30 years
they're taking 28 percent of
your money, 36 years.
With the Indians over 360 years
we saw that it was an
astronomical amount that they
took.
They took almost all your
money, right?
So look at the percentage that
got taken, so 768 billion versus
24 billion, I mean,
it's astounding.
So giving 1 percent away to a
money manager is giving away a
fortune if you think you're
going to stick with the money
manager for a reasonable amount
of time.
So if you want the secret to
how hedge funds make money
that's the first way they make
it and the most important way.
They charge a fee that sounds
small, but it adds up over a few
years and it amounts to a huge
amount of money.
Now, you can make it much
smaller.
Why does it amount to so much
money?
Because the money that you put
in the fund you're keeping in
the fund, so it's growing and
growing and growing.
So they're taking 1 percent of
your 24 dollars today.
That sounds like nothing,
but the money is still there
and now 40 years later they're
taking 1 percent of a much
bigger number.
 
That's why that number gets to
be so large.
So that's the second thing.
 
So now, let's keep going.
 
So that's the basic thing.
 
So now let's go to define a few
terms that everybody should
know.
 
What's a coupon bond?
 
A coupon bond is the simplest
kind of bond,
the first one that was created,
and it pays a fixed coupon,
dollars, every period for T
periods.
The T's called the maturity.
 
So it's defined by the coupon
which is the fixed payment it
makes every year until period T
which is the maturity of the
bond,
and then it also,
at the end of period T,
pays a principal which is
usually how the bond is
denominated--
the face value of the bond--it
pays the principal or face
value.
 
That's usually 100 or 1,000.
 
So a coupon might be 6,6,
6,6, 106.
That would be a 6 percent
coupon bond.
You can also define the coupon
by the percentage of the face
that it pays every year as a
coupon,
so little c is the percentage,
so .06 times 100 is 6,
6,6, 6,6, 6.
 
I could use big C as 6 dollars.
 
So it's defined by its
percentage, by the face and by
the maturity.
 
So the first obvious thing to
say is if the interest rate is 6
percent and the bond is paying a
6 percent coupon then it has to
be worth its face--
so let's always assume the face
is 100.
 
So why is that?
 
It doesn't seem totally obvious
because the formula is you take
100 times c (that's the first
payment) divided by 1 i,
then 100 times c divided by (1
i) squared etcetera.
It's not so obvious that's
going to turn out to be equal to
100.
 
But so you just have to think
for a second why that should be.
And the way to think of this is
if you had 100 dollars in the
bank at 6 percent interest you
get 106 dollars the next year.
Take the 6 and spend it,
you'd still have 100 dollars in
the bank.
 
That would give you 6 dollars
again the next year.
You could take that 6 dollars
and spend it,
you'd still have 100 dollars in
the bank.
You keep doing that until the
last year when you've got 106
dollars.
 
So at 6 percent interest
putting the money in the bank
and spending the coupons would
give you exactly the same cash
flow as the bond's giving you.
 
So therefore whether you put
the money at the bank at 6
percent interest or buy the bond
you're getting the same cash
flow,
so it has to be by no arbitrage
that the initial outlay was the
same,
so it has to be 100 dollars.
 
Well, that's obvious.
 
Now you can prove it many
different ways.
Now you can also imagine
keeping a bond forever paying 6
percent interest.
 
Then you get--a 100 dollars at
6 percent interest would give
you 6 dollars forever.
 
So if there was 6 percent
interest and you were getting 12
dollars forever,
how much would that be worth at
6 percent interest,
12 dollars forever,
6 percent interest,
you get 12 dollars every year
forever,
what's that?
How many dollars is that worth
originally?
If the interest rate that all
banks are giving,
and the whole world's agreeing
6 percent is the rate of
interest and someone is offering
to give you 12 dollars every
year forever,
how much money in present value
terms is he giving you?
 
Student: 200.
 
Prof: 200, right?
 
Because 200 at 6 percent would
give you 12 dollars every year,
so these are the most basic
formulas to keep in mind.
So those you may be hearing
these things for the first time
so it takes a second to adjust
to it,
but there's no cleverness
involved in figuring these out.
Now, so we've got the doubling
rule, we've got coupon bonds,
so that's simple.
 
Now, somewhat subtler thing is
an annuity.
So an annuity pays you a fixed
amount for a fixed number of
years.
 
So it doesn't pay the principal
at the end, so it pays that C.
That's supposed to be a capital
C.
It pays C, C,
C, C for a fixed number of
years.
 
So it's a T period annuity.
 
Now annuities also can be
changed in two important ways.
They can be indexed to
inflation.
That's a much better annuity
because now you're protected
against inflation.
 
It also could be timed to last
the rest of your life.
So we're going to come to this
when we talk about Social
Security.
 
The most important annuity by
far in the whole economy is the
Social Security annuity.
 
Once you retire and you're in
Social Security they figure out
what your coupon is going to be
every year.
I'll tell you the formula in a
couple of classes.
So depending on how much you've
contributed they calculate what
your coupon is every year.
 
So from the day you turn 65 for
the rest of your life you get
the same C inflation corrected.
 
So we're going to have to talk
about why they decided on that
contract.
 
But anyway, that's an annuity.
 
So it depends on the length of
life.
So these annuities are famous
in history.
Jane Austen in Sense and
Sensibility said it was a
disaster because whenever you
give someone an annuity they
live forever.
 
And she said that,
some character says her mother
gave the servants in the houses
annuities after their husbands
died and she figured that they
were so old--
she gave the annuities.
 
They were the servants of her
mother's and she gave them the
annuity after their husbands
died,
and since they were so old she
figured she'd pay them a few
years and that'd be the end of
it,
and they just went on and on
and on,
and she got tired of giving
them all the money.
But anyway, so obviously when
you're giving a life annuity you
have to calculate how long the
person's going to live,
and so we're going to come back
to that,
the selection of who takes
annuities.
Do they know that they're going
to live longer or not?
Anyway, that market's all
screwed up and we're going to
come to that later,
but it's a famous market,
the annuity market.
 
Now, how can you figure out the
value of an annuity?
So this is a very simple thing
to do once you've come this far.
So this is the next thing to
remember.
So remember,
an annuity's paying C,
C, C, C up to period T.
 
Here are the periods T.
 
So how much should this be
worth, the present value,
what is the present value today
at time 0?
Well, we know that if it
actually went forever,
C, C like that it would
be--forever, it would go C over
the interest rate i.
 
Annuities are often inflation
corrected so I wrote r for the
real rate of interest.
 
So you could call it C over r
for the real rate of interest,
whatever.
 
Let's say it's nominal.
 
Let's keep to i even though I
haven't used that notation
there, so C over i.
 
If you get C dollars forever
it's called a perpetuity.
So a perpetuity we already know
how to value.
We said C over i, right?
 
At 6 percent if you're getting
12 dollars every year,
it's worth 200 dollars.
 
Now, what if it gets cutoff at
T?
It sounds like there's going to
be a very complicated formula to
calculate, but actually it's a
very simple formula.
Why is that?
 
Because the T period--so here's
the perpetuity and the T period
annuity equals the perpetuity
minus a perpetuity starting at
time T--
minus perpetuity contracted at
time T--
right?
So why is that?
 
Here we have a perpetuity.
 
At time 0 you say to someone,
the state, the government says,
we'll pay you and your
descendants C dollars forever.
So we know what that's worth,
C over i.
Now we say suppose the state
tells you we're going to pay you
C until time T?
 
What's that worth?
 
Well, it's worth this,
the whole thing,
minus this part of it,
but looked at from this point
of view here the whole part of
it is just a perpetuity again.
So it's just the perpetuity
which is C over i - C over i,
another perpetuity here,
but as of time T because that's
like the 0 time--
the money's coming,
the next period,
forever.
Just like at time 0 the money
came starting at period 1
forever,
so at time T starting 1 period
later forever,
so therefore it's this divided
by (1 i) to the T.
 
So it's just C over i times (1
- (1 over (1 i) to the T)).
So this is the next thing you
have to memorize,
unfortunately,
but there are only a few things
you have to memorize.
 
So this is a very famous
formula for the value of an
annuity.
 
So let's just do an example.
 
Suppose somebody--maybe I can
just do the same examples,
so you've got the proof of
that, right?
This is no surprise?
 
Remember the whole perpetuity
is obviously,
it's 6 percent.
 
Let's just think of something
at 6 percent.
So let's do the 6 percent
annuity.
At 6 percent interest a 12
dollar perpetuity is worth 200
dollars.
 
That's what we said before.
 
So what is a 36 year?
 
So at 6 percent interest a 12
dollar perpetuity is worth 200
dollars.
 
So at 6 percent interest what
is a 12 dollar 30 year annuity
worth?
 
It's worth what?
 
How much is that worth?
 
So if it went on forever it
would be worth $200 dollars.
If we cut it off after 30
years--30 years is a bad time to
cut it off.
 
Let's cut if off after 24 years.
 
So you have 6 percent interest.
 
You get 12 dollars,
not for every year in the
future, but for 24 years,
how much is it worth?
Well, it can't be worth 200
dollars because it would be
worth 200 dollars if it went
forever.
So it's worth less than 200
dollars, but how much less?
So my uncle died,
left my sister an annuity.
She just had no idea what the
annuity was worth.
So do you have any idea what
it's worth?
Yep?
 
Student: 150 bucks.
 
Prof: It is,
and how did you get that?
Student: The doubling
rule at 6 percent interest it
takes 12 years to double your
money so in 24 years you're
going to quadruple your money.
 
So then you just put that 4
into the equation.
Prof: So it's 1 - 1
quarter, so that's 3 quarters,
and 3 quarters times the 200 we
got before is 150,
exactly.
 
Exactly right what he said.
 
So does everybody get that?
 
Let's try another one.
 
So let's suppose I pay 8
dollars.
Let's see, how long is this
going to be?
Let me say that again just in
case you didn't follow that
because I'm going to give a
slightly harder one this time.
So he's saying how do you
figure out the value of an
annuity?
 
Something by next class you'll
be able to do in your head.
It's going to be,
take the cash flow that you get
over here.
 
If it went forever it'd be so
easy to figure out what the
value was.
 
If it's 12 dollars at 6 percent
interest that's like having 200
dollars in the bank because then
at 6 percent interest you're
going to get 12 dollars every
year forever.
So we know that 12 dollars a
year forever is clearly 200
dollars.
 
That's C over i,
12 over .06 is 200 dollars,
but it's going to get cut off
in year 24.
So we're going to lose all this
future stuff,
but the future's not worth very
much.
Why isn't it worth very much?
 
Because by the time we get here
we've already discounted by a
lot.
 
So a dollar starting here is
actually only 1 quarter of a
dollar starting back here,
because in 24 years at 6
percent interest you've doubled
twice,
so it's worth 1 quarter.
 
So you just take 1 quarter of
the same annuity.
So it's 1 - 1 quarter of the
same annuity.
So the one that ends in 24
years is like 3 quarters of the
value of the perpetuity,
3 quarters of 200 is 150,
that's how we did it.
 
So let's reverse the thing.
 
Suppose we know the present
value's 100.
You're now the company,
and you're trying to figure out
how much to pay.
 
So what is the C going to be?
 
Let's say it's 8 percent
interest, and I'll just do the
same example in the notes.
 
In 30 years a typical thing,
so it's not going to work out
exactly evenly,
so 8 percent interest for 30
years.
 
So we know it's worth 100.
 
So let's get rid of all the
irrelevant stuff so you don't
have the board cluttered.
 
We've got something that's
worth 100.
There's the formula down here.
 
So the thing is worth 100.
 
You know the interest rate is 8
percent now, and it's a 30 year
annuity.
 
So if somebody tells you the
interest rate's 8 percent these
days, you're going to get a 30
year annuity,
you've got 100 dollars to
invest.
You go to the annuities
company, the insurance company,
you tell them,
"I want an annuity,"
how much do they give you every
year?
Well, you just have to figure
out what C is.
So you put C over .08,
so what does that tell you?
What would that be?
 
What would they be paying you
if it was a perpetuity?
If it was a perpetuity what
would they be paying you?
They'd be paying 8 dollars a
year, right?
So but they're only going to
pay you for 30 years,
so how much are they going to
pay you, (1 I) to the T.
So this is 1.08 to the
thirtieth, and what's 1.08 to
the thirtieth?
 
Well, 1.08 to the thirtieth by
our rule is what?
It's equal to 1.08 to the
twenty-seventh--
in 9 years at 8 percent
interest it'll double,
so after 27 years it's going to
double 3 times,
1.08 to the third power.
 
So after 27 years it's going to
double 3 times,
so that's eight,
right, 2 to the third power is
8 and then at 8 percent over 3
years it's 108 goes to about
116,
goes to about 124,
but you know it's going to grow
a little faster because 1.08
times 1.08 is a little bit more
than 1.16 so it's going to grow
to like 1.25 instead of 1.24,
so that's 10.
So this is just 1 over 10.
 
So the whole thing is 9 tenths.
 
So basically you get almost all
the value.
After 30 years at 8 percent
interest it's such a high
interest rate that after 30
years you're getting 9 tenths of
the value of the annuity
[correction: perpetuity].
So you're going to have to get
paid 10 ninths,
the 8,000 dollars.
 
It would have been 8,000
dollars if it were a
perpetuity--you have to pay the
guy a little bit more.
You have to get a little bit
more because you're only getting
it for 30 years.
 
But because the interest rate's
so high the stuff after 30 years
isn't very important.
 
You have to be given an extra
tenth.
So it's 10 ninths times 8,000
so it's 8,888 is what your
annuity's going to be every
year.
So just to summarize it,
to say it all again,
we know how to compute
perpetuities with ease,
and so if you want 100
[thousand]
dollars and a coupon forever,
and the interest rate's 8
percent this is 8,000 forever.
 
If you only get it for a
shorter amount of time,
obviously you have to get more.
 
How much more?
 
Well, it depends.
 
Each year for only 30 years it
depends on how much you're
giving up at that end.
 
And at 8 percent you're only
giving up a tenth of the whole
value.
 
So you have to be compensated
for that each year by getting 10
ninths of what you would have
gotten before,
so we're up to 8,888.
 
So those are the words that
everybody has to know.
So now let's just do a couple
more simple computations here
just to give you an idea of how
Fisher helped here.
So I'm going to do a few
mortgage things.
I haven't defined mortgage yet.
 
Why didn't I do that?
 
So a mortgage is just a 30 year
annuity.
So one more thing--a mortgage
is an annuity.
A fixed mortgage,
a fixed rate mortgage is
defined by a principal.
 
So when we talk about the
crisis this word principal will
come up all the time.
 
So that's the face value,
a principal,
a mortgage coupon rate,
and a maturity.
This is a fixed rate mortgage.
 
So the most common kind of
maturity is 30 years,
30 years is the most common and
then sometimes there are 15 year
mortgages,
and then there's a whole host
of other mortgages we're going
to come to later where there's
floating interest rates.
 
So the 30 year mortgage,
how much do you have to pay?
Well, if it were on an annual
payment and it were an 8 percent
mortgage for 30 years on a
100,000 dollars,
so if it was a 100,000 dollar
principle at 8 percent coupon
for 30 years we just calculated
that you would have to pay,
the payment would be 8,888
dollars a year,
because 8,888 dollars a year
discounted at 8 percent is going
to give you 100,000.
 
So that's how the mortgage
works.
So whenever you hear about a
mortgage you always hear the
mortgage rate,
that's the coupon rate.
The maturity is usually 30
years.
Then you'd have to be told how
much the mortgage is for.
Then you can figure out what
does the guy have to pay every
year.
 
You just figure out the annuity
payment that at this interest
rate makes his payments have
present value at this interest
rate equal to the principal.
 
So we just saw it was 8,888 a
year.
And there's one more little
twist with mortgages.
So that's not literally true
what I said.
Mortgages have monthly payments.
 
There are monthly payments and
so the monthly rate--at monthly
rate equal the coupon divided by
12.
So if it's an 8 percent
mortgage then it means that
they're taking 2 thirds--
so in this case we'd have 8
percent over 12 which equals 2
thirds of a percent.
So the mortgage would be .67
percent, and so then you do the
monthly calculation.
 
So you have to figure out the C.
 
So it's the summation over 1 2
thirds percent so that's 1.067
in other words,
1.067 to the t,
t = 1 to 360 has to = 100,000.
 
That's how much you'd have to
pay every month.
And so it wouldn't be 8,888 a
year it'd be slightly more than
8,888 divided by 12 every month.
 
This is the last of these
definitions for today's class.
So everybody who's on Wall
Street knows what a perpetuity
is, and they know how to compute
its value at a given interest
rate.
 
They know what an annuity is
and how to compute its present
value.
 
A mortgage is almost the same
thing as an annuity,
the only twist is that the
mortgage is computed monthly
instead of annually,
using a monthly interest rate,
but when they say the monthly
interest rate,
instead of telling you the
monthly interest rate they tell
you the monthly interest rate
times 12.
It's just a convention.
 
All right, so those are all the
things you have to memorize.
Now, let's try to use the way
we can think and do some
practical problems here.
 
So here's one of the simplest
and most important ones is let's
say you're a Yale professor.
 
I gave this example on the very
first day of class.
You're a Yale professor.
 
When I first gave the example a
few years ago when I wrote the
notes the average Yale
professor,
so this was quite a few years
ago, eight or nine years ago,
was making 115,000 a year.
 
And as it happens the average
Yale salary now is 150,000 a
year.
 
But anyway, when I wrote it,
it was 115,000 a year.
So let's suppose this year's
professors are making 115,000
and let's say your salary will
go up at 3 percent inflation
every year,
and that's inflation that's
equal to the general inflation.
 
So your salary is keeping pace
with general inflation and no
more.
 
So professors,
we're not doing that well.
So the salary is 115,000.
 
Now it would be 150,000,
but anyway it was 115,000 on
average and let's say you're
just going to be kept up with
inflation.
 
You're going to be told every
year you have a 3 percent raise,
but that's just going to keep
you up with inflation.
Now you know you're going to
work for 30 years,
let's say, and retire for 30
years.
That's a little ambitious about
how long you're going to live,
but let's just suppose that's
what you think.
You're going to live for 30
years after that.
So how much should you spend
every year?
Well, you can't answer
that--and let's say you want
level real consumption.
 
So you want to consume the same
amount every year for the rest
of your life,
which is going to 60 years,
30 at Yale, 30 retired.
 
So how much do you spend every
year in consumption?
Well, you can't answer that
until you know the interest
rate.
 
So let's say the interest rate,
the nominal interest rate
equals, let's say,
5.3 percent about,
a little bit more than 5.3
percent.
So if the nominal interest rate
is 5.3 percent,
inflation you know is going to
be 3 percent and you've got
115,000 coming going up with
inflation every year,
how are you ever going to
figure out how much to spend
starting next year when your job
starts?
It looks like a hopeless thing.
 
You'd have to say,
well, if I get 115,000 next
year,
I consume some of it,
I put the rest in the bank,
it makes interest,
it grows at 5.3 percent,
but then inflation is 3
percent, so I take that into
account and I figure out how
much to spend the year after
that,
but then I'm going to get
115,000 more of that so I'll
save something,
maybe more from my next thing
and then I'll deposit that at
another 5.3 percent,
and I have to take into account
inflation at 3 percent.
It sounds like it's going to
get very complicated.
How are you going to figure
this out?
But in fact it's very simple,
and Irving Fisher pointed the
way to do it and we can now do
it in our heads.
So Fisher said,
don't figure out all this year
by year stuff and don't get
mixed up with the rate of
inflation.
 
You don't care about inflation.
 
You're going to look through
the inflation and only care
about the real consumption.
 
So the fact is you care about
the real rate of interest.
So the real rate of interest 1
r = 1 i over the rate of
inflation which equals 1.053
over 1.03 which is about 1.023,
right?
 
We're doing things in our head
now so we have to be a little
bit approximating,
so, right?
If I divide this by this these
numbers are so close to 1 that
I'm basically just subtracting
the bottom from the--the
denominator from the numerator.
 
If I multiply 1 g times 1 i,
all right,
1 g times 1 r that's going to
equal 1 g r rg and if this is
.02 and this is .03 then the
multiplication is .0006 so this
is practically irrelevant.
 
So multiplying numbers like
this, or dividing them,
is just like adding these
things.
So it's just like taking this
term and subtracting that.
So when you get a number near 1
divided by another number near 1
you just take the difference
from 1 in the numerator minus
the difference from 1 in the
denominator.
It's pretty close to doing
actually the division.
So this is about 2.3 percent
interest.
So the real interest rate is
about 2.3 percent.
So Fisher would say,
ah ha, use the real rate of
interest.
 
You're getting 115,000 real
payments every year for 30 years
at a real interest rate of 2.3
percent, but we know what that
is.
 
So what is 115,000 of real
dollars every year at a real
rate of .023 percent?
 
Well, remember what our formula
was.
It's the cash you're getting
every year--
if it were a perpetuity,
you got it every year forever,
it would just be the cash
you're getting divided by the
interest.
 
So 115,000, that's over a tenth
of a million every year forever
at 2.3 percent interest that
would be worth--
that's 5 million so far,
but you're not getting it every
year.
 
You're getting it for 30 years.
 
So 1 - [1 over]
1.023 to the thirtieth,
so what's that equal to?
 
It's no longer 5 million
because that would be getting
the money forever,
1.023 to the thirtieth is what?
Student: It's like
>
Prof: So we said that
2.3 percent interest,
2.3 into 72 is--2.3 times 30 is
69.
So because it's close to 0 the
69 rule almost works.
So anyway, an approximation
would say 2.3 into 72 is just a
little bit over 30,
so it doubles every 30 years
and you've got it for 30 years,
so it's just going to be 2.
This number's about 2,
right, 2.3 percent into 72 is
approximately 30,
so every 30 years at this
interest rate it doubles.
 
So therefore you've got 115,000
over .023 times 1 - 1 half.
You've lost half the value by
not getting it forever.
So that's 2.5 million.
 
So Fisher says,
look, remember the budget set.
In GE we studied budget sets.
 
We put P_1
X_1 P_2
X_2 P_60
X_60,
that's on the left hand side,
is less than or equal to
P_1 endowment 1
P_2 endowment 2
P_30 times endowment
30.
You're getting 115,000 of real
goods every year for 30 years.
P_1 is 1 over 1.023.
 
P_2 is 1 over 1.023
squared etcetera.
So this revenue on the right is
just the annuity of 30 years of
115,000 of real goods.
 
So it's worth 5 million reduced
because it's not a perpetuity,
it's only an annuity for 30
years and so it's worth 2.5
million.
 
So we've got the right hand
side.
That's this, 2.5 million.
 
That's how much the present
value is.
So that's what a professor at
Yale can look forward to his
entire,
her entire career if she
started 10 or 20 years ago would
be,
she'll make 2 and 1 half
million in present value terms.
If she'd gone to Wall Street,
in five years,
if she were a Yale
undergraduate and went to Wall
Street,
in five or ten years she'd be
making more than that every
year.
So well, not everybody,
but anyway.
So how much should she spend
every single year of her 60,
or life?
 
Well, so we have to figure out
this number C,
the coupon, right?
 
We have to figure out how much
can she spend every year of her
life, what C can she spend at
2.3 percent interest where now I
have a 60 here?
 
So it's an annuity of 60 years
of constant consumption at this
interest rate,
so how much is it worth?
Well, I have to just figure out
1.023 to the sixtieth,
1.023 to the thirtieth was
2,1.023 to the sixtieth is 4,
so this is 1 fourth.
 
So we have 3 quarters here.
 
So this is C over .023 times 3
quarters.
That's what we have.
 
So we multiply 4 thirds by 2.5
million.
You get 10 million--this is
.023 divided by 3 times 10
million = C.
 
So that's like 76,000
something, 3 into .023 is 76 and
then you have to figure out what
decimal place you're at and you
know it's going to be less than
115,000.
So it's got to be some number,
some reasonable percentage of
115,000 so it works out to
76,000.
So that's it.
 
You can do that in your head.
 
I mean, not today but after
looking at it by next time
you'll be able to do that in
your head.
So this professor can figure
out what you should do.
It seems like a hard problem.
 
It's life.
 
You've got to figure out what
to do every year and now you
know how to do it very easily,
so any questions about this?
I want to do one more little
example.
All right, so let's do a harder
example.
It's easier computationally,
but harder conceptually.
When I just got tenure at Yale,
actually I had tenure for a few
years;
you'll see why this is relevant.
The President,
Benno Schmidt,
of Yale said,
"A horrible thing has
happened.
 
Generations of Yale presidents
before me have not realized that
the buildings were not getting
the proper care.
There's deferred maintenance.
 
Generation after generation did
no fixing up of the buildings.
I'm the first president who's
going to act responsibly and fix
up the buildings."
 
And he said,
"I'm going to fix up the
buildings,
and I can tell you that I've
hired these planners,
and they've come and done an
exhaustive study and we have to
spend 100 million a year for 10
years,
each year for 10 years,
to fix up the buildings
properly."
So that plan,
by the way, is the thing that
got turned into fixing one
college a year.
"So 100 million for 10
years that's what we need.
These presidents before me have
overlooked it.
They've spent as if we don't
have to keep up the buildings.
I've recognized the problem.
 
I'm going to correct it.
 
This is a huge expense they
didn't take into account.
We have to reduce the
budget."
So how much do you reduce the
budget by?
How many cuts should he have
made in that first year?
How would you have figured out
what to do?
So what he did is he
recommended firing 15 percent of
the faculty which didn't go over
very well.
And the faculty,
it was an amazing thing,
there's no structure at Yale.
 
The president runs Yale.
 
There's no senate.
 
There's no labor union.
 
There's no nothing.
 
It's just the president.
 
Suppose the president
announces, "I'm going to
get rid of 15 percent of
faculty,"
what is the faculty supposed to
do?
There's no mechanism.
 
So what happened is,
the old deans who are no longer
deans, they were just old,
almost all of them were men,
again, I guess,
old guys.
They got together and said,
"Well, we have no power.
We have no position,
but we used to be deans at
Yale.
 
It's up to us to do something.
 
We're going to appoint the
committee who's going to examine
the logic of the president's
decision.
So we're going to appoint six
people that we're going to pick
out of the blue and they're
going make a report to the
faculty and tell us what to
do."
So I was one of the six and the
other five guys were pretty
nervous.
 
Well, we all were nervous about
actually getting up in front of
the president and the provost
and the dean and saying that it
was all wrong and he shouldn't
do this,
but we had tenure so we could
get up and say whatever the hell
we wanted to.
 
So what did I say?
 
What would you have said if you
were me?
Now the whole budget of Yale
was--1 billion equals annual
budget, and a lot of things you
can't cut.
So notice 1 hundred million a
year is 10 percent of the annual
budget.
 
So he basically said,
"Well, we've got 100
million a year.
 
We ought to cut out 10 percent
of the budget,
and since there are some things
we can't get rid of,
we've got to keep making fixed
payments and the faculty is
something--
of course I'm not going to fire
the tenured faculty,
I'm going to fire people who
aren't tenured and when faculty
retire I just won't hire anyone
to replace them.
 
That's how I'm going to get rid
of the faculty."
That's how he got to 15 percent.
 
So what would you have done?
 
What would you have said if you
were me?
"Don't do it,"
but what else would you have
said.
 
What calculation could you do?
 
So you know now what to say yet
you can't think of what to say.
So what would you say?
 
I'll come to you in a second.
 
What's a reasonable number?
 
How would you think of a
reasonable number?
Let's take his facts as correct.
 
In fact they didn't turn out to
be that far off.
The people he hired were pretty
good at assessing how much stuff
needed to be done.
 
Yes?
 
Student: Well,
I guess the first thing you do
is figure out what a 10 year 100
million dollar annuity would be
worth.
 
Prof: Yeah,
and then what?
Student: I don't know.
 
Prof: So that's a good
start.
He says the first thing he'd do
is he'd think about what a 10
year annuity at 100 million
dollars a year is worth.
So why would he want to do that?
 
So that's good.
 
That's what he should do.
 
That's how I started,
but what's the relevance of
that?
 
Yep?
 
Student: Then if he got
that lump sum now he could
afford it without needing to
fire everybody.
Prof: So he could say
alumni,
please do something about,
you know,
it's 1 billion dollars,
not quite, something less than
1 billion dollars.
 
We'll figure it out in a minute.
 
So alumni please hand over 3
quarters of 1 billion dollars
and I won't have to fire my
faculty.
He could try that.
 
What if the alumni didn't come
through?
I know you're going to say
something, but I want to give a
couple more people a chance back
there.
Yep?
 
Student: It might be
cheaper to short an annuity,
and that way it definitely
turns out to be less and
>
 
Prof: So you'd short a
10 year annuity.
Student: Yes.
 
Prof: What were you
going to say?
Student: We're going to
have to pay for the colleges for
like 10 years or 12 years,
however many years you're
renovating them,
but the faculty are like
perpetuities.
 
You have to pay for them the
entire time.
So if you calculate the C over
i for a professor it's probably
a lot more than he was
estimating,
and see how many of those it
would take to compensate for the
annuity.
 
Prof: Now we're on the
right track, exactly.
So I'm going to go a step
further.
I went a step further.
 
I said, "Yale is
forever."
So what he's telling us is that
we need to catch up to where we
should be to make up for all
that lost maintenance.
He's not saying,
by the way, that the presidents
who built the colleges in the
1920s and stuff weren't paying
attention to the physical plan.
 
He was talking about the few
generations before him.
So once we make up for those
losses and then return to a
steady state after spending the
100 million a year for 10 years
we'll be back to Yale in a
steady state.
Yale's going to go on forever.
 
So the point is,
why should the next 10
years-generation pay for
something that's going to make
Yale better for the whole
infinite future of Yale.
So I said, "How much would
every generation,
not just today,
but forever in the future have
to consume less in order to make
up for this one shot problem,
this deferred maintenance that
a couple generations of Yale
presidents didn't put in?"
 
So in other words,
I would figure out the present
value of the 10 year $100 mil
annuity,
and then I would set that equal
to what coupon perpetuity gives
you the same present value.
 
So how can you figure that out?
 
So in other words,
if you lose 100 million for ten
years that's equivalent to how
much less for every year,
so it turns out to be quite a
big difference.
So it depends on what the
interest rate is.
Now, it happens that Yale has
an interest rate.
Yale always uses this 5 percent
rule.
So if you take R = 5 percent
that's supposedly the money
Yale, after inflation,
is confident that it can get on
its endowment.
 
Usually it thinks it can get
more.
You'd figure out what the
annuity value is of 100 million.
So I'm overtime.
 
So anyway, we're going to have
to--so the punch line is it
comes to 32 million a year not
100 million a year.
We'll do this calculation next
time.
And you don't need to fire 15
percent of the faculty to get 32
million a year.
 
So we'll start next time.
 
 
 
