ECON 251: Financial Theory
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ECON 251 - Lecture 7 - Shakespeare's Merchant of Venice and Collateral, Present Value and the Vocabulary of Finance
Chapter 1. Introduction [00:00:00]
Professor John Geanakoplos: I think I’m going to start. 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.
Chapter 2. Contracts in Merchant of Venice [00:01:33]
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 theMerchant 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.
Chapter 3. The Doubling Rule [00:20:23]
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 pit 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 pit is something that is traded in the market and everybody at every hedge fund and every Wall Street bank knows what pit is at the beginning of each day.
Now, Fisher said, well don’t get too lost thinking about pit. Think about pt. Take out inflation. You have to make an expectation about what inflation is, but assuming you’re right you can figure out from these pit’s what pt 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 pits–there’s also more important stuff you can do with the pts. You should always keep those in mind. So let’s take the simplest case where pit 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, m1 through mT, 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, pi1 through piT. Because to get m2 dollars at time 2 just cost you pi2 times m2 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 m1 over (1 + i), m2 over (1 + i) squared, and mT 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 m1, m2, and mT 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.
Chapter 4. Coupon Bonds, Annuities, and Perpetuities [00:36:07]
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?
Professor John Geanakoplos: 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.
Professor John Geanakoplos: 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.
Professor John Geanakoplos: 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.
Chapter 5. Mortgage [00:54:24]
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.
Chapter 6. Applications of Financial Instruments [00:59:15]
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 <>.
Professor John Geanakoplos: 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 P1 X1 + P2 X2 + P60 X60, that’s on the left hand side, is less than or equal to P1 endowment 1 + P2 endowment 2 + P30 times endowment 30. You’re getting 115,000 of real goods every year for 30 years. P1 is 1 over 1.023. P2 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.
Professor John Geanakoplos: Yeah, and then what?
Student: I don’t know.
Professor John Geanakoplos: 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.
Professor John Geanakoplos: 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 <>.
Professor John Geanakoplos: So you’d short a 10 year annuity.
Professor John Geanakoplos: 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.
Professor John Geanakoplos: 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.
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