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[Assistant] So that's what we have to solve all of that. It looks way more complicated than before and so complicated that you can see why people might have gotten confused. But according to Fisher, it's going to turn out to be a very simple problem in the end once we look at it the right way.


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[Assistant] And this is original ownership by BE of Stock Alpha. This is original ownership.


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[Assistant] By A of stock beta, and this is the original ownership by B of stock beta. Original ownership of alpha.


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[Assistant] D is the dividend, that's the output that we can all, okay thank you, I should just write this down. This is the anticipated dividend.


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[Assistant] Which is the output, since that's the end of the world, of stock alpha in period two.


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[Assistant] And D beta of 2 is the anticipated dividend of stock beta in period 2.


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[Assistant] So it's one apple we're getting out of the alpha tree, two apples out of the beta tree.


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[Assistant] All the other stuff goes into helping us think about how many apples we're gonna get out of it in the end. So we cut to the bottom line, what are the apples we expect to get out of the tree? One from the apple tree, two from the beta tree. Someone else had their hand up.


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[Assistant] Okay, so we're returning to first principles here, very simple example, you know, the


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[Assistant] Uh, when there's ever a big confusion about something important, it's always good to go to first principles. There was a chess player when I was young named Mikhail Tal, who was a


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[Assistant] Uhm, so we're going back to the first principle. How would you define equilibrium here for financial equilibrium? Well, the first thing,


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[Assistant] You know, is this common sense? What are people doing? At time one, they're going to, what can they do? They can spend money, so let's say, so I'm going to look at the budget set.


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[Assistant] You know, and I can be A or B. I don't have to write it twice.


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[Assistant] Okay, so he's going to say to himself, let's say A is he and B is she, he's going to say to himself, uh, let's say I can, uhm, I will say, how much does it cost me to buy good? Well, the cost


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[Assistant] of apples is q1 times x1. That's how many apples I might end up with. Now how much does it cost me to buy shares? It's going to be pi alpha


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[Assistant] Times how many shares I end up with. Theta alpha plus pi beta.


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[Assistant] I'm buying alpha shares, and I'm buying beta shares, and this is how much I have to spend to get the holdings I want of each. Now where did I get the money to do that?


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[Assistant] I got the money to do that because I started with my endowment, you know, of good, which was E.


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[Assistant] I won, which in this case for A was 1.


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[Assistant] And I had shares to sell and apples to sell, so that's, uh, that's what I did.


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[Assistant] The number of apples I started with, that means I has bought apples. Because he ends up with more than he started with. So on that he must have been buying apples. If theta alpha


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[Assistant] Theta bar alpha is more than theta bar alpha. It means that alpha bought shares of stock alpha. Theta alpha is less than theta bar alpha. It means alpha sold shares of stock alpha.


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[Assistant] Alright, now in the second period what happens? Well in the second period we have q2 times x2. Now, nobody


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[Assistant] The shares are gonna be worthless in period two. So no one's gonna buy them. Why are the shares worthless? Remember that when you buy a share of stock, the dividend comes later. You don't get the dividend immediately. So someone buying stock in period two, it's too late to get the dividend. It's already gone to the owner who bought the shares in period one. So the buyer of a stock gets the dividend


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[Assistant] Period. You can buy the tree, but the world's coming to an end. That tree's not gonna do you any good. It doesn't produce any more apples.


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[Assistant] So nobody's gonna bother buying shares, I don't have to bother with them, the prices are zero.


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[Assistant] Okay, and so what income do people have in period two? Well, they've got the contemporaneous price times, you know, the apples that


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[Assistant] Somehow they find on the ground that they're, you know, or that their parents are going to leave them when they get old.


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[Assistant] So that's their endowment of apples. But what else do they have? They've got more apples than that. What else do they have? The dividends, okay? And so what are the dividends? Well, you bought theta alpha to begin with.


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[Assistant] So that's D alpha 2. So if you bought


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[Assistant] You know, the whole tree, then you've got all the dividends. And similarly with beta. Beta, beta, and d out d beta 2.


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[Assistant] Okay, so that's it. So the budget set is a little more complicated. It's, uh, so that's the budget set. Okay, so it's got two


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[Assistant] Equalities instead of one equality. Already things look a little more complicated. Now, so an equilibrium is going to have to be that, uh, i chooses x i1, x i2, theta, uh, alpha,


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[Assistant] You know, I could write theta I alpha, theta I theta.


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[Assistant] UI subject to this budget stat.


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[Assistant] Alright, so, and now in equilibrium we have to have that xA1 plus xA2.


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[Assistant] A alpha plus theta d alpha has to equal theta bar. A alpha plus theta bar


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[Assistant] The alpha, right, the stock market has to clear and theta A beta, plus theta B beta, that is equal to theta bar.


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[Assistant] Okay, but now what's the last equation? This is a little trickier. What's the last equation? XA2 plus XB2 equals


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[Assistant] It's gonna be the apples that they, you know, have on the ground, but also the ones that were picked off the trees. So these dividends. So it's going to be the total dividends, which are theta bar A alpha plus theta bar B alpha times that tree, B alpha two plus theta bar A beta plus theta bar B


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[Assistant] So just to say it in words, it's exactly what we had before, except we have to take into account


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[Assistant] In addition to the goods market clearing we have to take into account that the stock market has to clear and in the end demand for goods


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[Assistant] Has to equal the supply that people had in their endowments, but also what the companies are producing. These companies are producing output, you know, apples, and so that's part of what the consumption is going to be in the economy.


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[Assistant] Okay, are you with me here? It's a good time for questions, maybe? Yes.


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[Assistant] Okay, so in period two, you might wonder, pi alpha is the price of the stock at period one.


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[Assistant] Stock alpha in period one, pi beta is the price of the stock beta in period one. How come I didn't...


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[Assistant] You're buying it not for the dividends at the same moment in time. You don't get those dividends. The guy who already had it gets those dividends. When you buy the stock, you're buying it for the future dividends you can get. And I've assumed the world's gonna end after two periods, because nobody's utility


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[Assistant] You know, cares about period three. If you buy the stock in period two, it's too late


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[Assistant] For you to get anything, there are no dividends. Because period, you could only get them in period three and there won't be any dividends in period three and if there were you wouldn't care about them anyway. So the stock's worthless to you. So the price of the stock in period two of both stocks is gonna be zero. So there's no point in putting down what people are buying of the stocks or selling or anything, it's just not worth anything.


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[Assistant] Later, when you look at your income from having bought the stock, you'll have as your income the dividend from the stock plus the resale value of the stock. Because you could sell the stock next period. But I just know the resale value is going to be zero because you're in the last period of the economy. Okay, and I just want to keep it very simple this first time. So you, so it's, you could, yeaah. So we'll see, you know.


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[Assistant] Step by step, you'll be able to keep very complicated things in your head, but not right at first. So any other questions? Yes.


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[Assistant] Why do people, uhm, have endowments today and next year?


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[Assistant] Because you could think of the endowment, for example, as, here we've got apples, but usually the endowment is your labor. So you can work this year, what if, your most important endowment is your energy and your labor. So you've got it this year, next year if you're still alive, you're gonna, that's new labor that you have, it's a different good, so it's a second endowment that you have.


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[Assistant] So, you know, I don't want to get caught up in labor and all that and get...


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[Assistant] Involved with Marx again. So I'm just gonna talk about apples. You have an endowment of apples when you're young and next year somehow you're gonna have more apples.


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[Assistant] So you could think of the, you might have thought that the only apples next year come from what the firms are producing, but


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[Assistant] You know, I've allowed for the possibility that people have apples too, just like their labor next year. Okay, other questions?


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[Assistant] Well this is ea, this is ea, so I should have written this more carefully maybe, ea1 and ea2.


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[Assistant] So, as he was suggesting back there, I've assumed for person B that, you know, he's got an apple now, somehow the world, you know,


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[Assistant] We aren't modeling what happened to get us here. The guy's got an apple today. They both have an apple today. Somehow, A is also gonna have another apple tomorrow that, uh, you know, he's gonna find under his doorstep somehow that isn't being produced by the tree. And maybe think of it as labor that he's gonna have next period.


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[Assistant] That's it. Fisher says as soon as you write down the economy mathematically, all sorts of things are going to occur to you, which if you're talking in words about justice and injustice, you know, you're going to be lost.


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[Assistant] So what, what, what can we get right away out of this, uh, what can we get right away out of this?


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[Assistant] How would we define inflation? What is inflation?


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[Assistant] What's inflation in this economy? Assuming we've got the equilibrium which we're going


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[Assistant] So we're going to calculate it, but right now we don't know what the numbers, you know, we've got a bunch of equations and stuff, we don't know what


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[Assistant] XA1 and Q1 and Q2 are going to turn out to be, but we're going to find out very soon. But before we find out...


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[Assistant] Well, we're talking about inflation, when we talk about the consumer price index. Inflation, what are they talking about? Yep.


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[Assistant] Right, so that's the price of apples today, it's the price of apples next year. If the price of apples next year is bigger than this year, we've got inflation. If it's lower, we've got deflation. Okay, so already the model, you're talking about inflation. Okay, what else, uh, what's the next most obvious, uh,


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[Assistant] Key idea, key idea is arbitrage.


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[Assistant] Okay, so Fisher says people have


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[Assistant] You know, they have foresight. They're anticipating what the dividends are going to be.


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[Assistant] They, they, they, they can, you know, they understand that, you know, you can talk about how beautiful the tree is and how much you like the owners and how much they're watering it and all kinds of, and whether they have a good plan for irrigation and, you know, whether they did well in college and stuff like that. But in the end, all you care about the trees is what


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[Assistant] How many apples are going to produce? So knowing that, can we say something about pi alpha versus pi beta? In equilibrium, what's going to have to happen? There's going to be some connection. And what's the connection going to be? You've got two trees.


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[Assistant] So pi alpha is going to be pi beta times the, uhm, uh, let's see, pi alpha, okay, so alpha will be better as long as the


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[Assistant] Hope I haven't got, I've got that in the right, uh, order. If, and so in this case, pi alpha is gonna equal a half pi beta.


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[Assistant] Because alpha's producing half the dividend that beta's producing, so obviously it's going to turn out to have half the price.


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[Assistant] That's the fundamental principle, okay, we're doing it in the most trivial case, but the most fundamental principle of finance, that is, you've got two assets,


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[Assistant] And one of them, they're basically the same up to scale, then their prices have to be the same up to scale. Who's gonna bother to buy alpha if it costs three times, if it costs, you know, the same amount as beta when it only produces half as much? Okay, yeaah.


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[Assistant] That's a word that we haven't defined yet, so we're going to define it in the next class. So it's something like that, yup. Any other? Okay, so that's


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[Assistant] It's a very simple thing. Now suppose I added, suppose I added, okay, so here's a, uh, suppose I added, suppose after finding the equilibrium,


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[Assistant] Okay, now, it would have a price. I've added a third acid, gamma.


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[Assistant] Then what, so pi gamma, we have to solve for the equilibrium pi gamma, and is there some word that I could use? How could I?


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[Assistant] So, gamma is an asset that pays a dollar in period two. It's like a bond, promising a dollar in period two. The price of the bond would then have to be


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[Assistant] What? 1 over 1 plus i, where i is called the nominal interest rate.


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[Assistant] So we've got inflation is occurring in the model. If I added a bond, which I didn't bother to do, because it's just yet another thing I have to write down, I could have had a third asset, which pays a dollar, so the others


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[Assistant] Are paying off in, in apples. This one's paying a dollar and it's priced today. You know, if you pay 80 cents today, it's like saying


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[Assistant] I'm paying 80 cents today, I'm getting a whole dollar tomorrow, so it's like a 25% rate of interest. Okay, because


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[Assistant] Another way of saying it is that I put in pi, you know, so one over pi gamma is one plus i. I put only pi gamma in today, I get one out tomorrow, so I've gotten back not only the pi gamma I put in, but something extra, that's one plus the interest rate.


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[Assistant] Okay, so this world is going to have an interest rate in it, it's going to have inflation in it, with me so far. Alright, so now Fisher makes


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[Assistant] Okay, so let me add one more thing. I could come back to this. So I said that, I said that, uhm, I said that if you take


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[Assistant] Theta alpha less than theta bar alpha, it means you're selling the stock. So I'm going to allow people to go even further. Theta alpha less than zero.


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[Assistant] Okay, so let me just write that again. Theta alpha less than theta bar alpha means, uh, selling alpha.


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[Assistant] Well, the mathematics is telling you that over here, theta alpha is going to be negative. You've got, instead of getting extra dividends, you're going to be giving up dividends, because it's going to reduce your supply of money.


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[Assistant] So theta alpha less than zero is called selling short.


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[Assistant] Okay, so you're, you're, uh, if you can still hear me, you're selling short, this means.


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[Assistant] Okay, so you're selling something you don't even have. It's also called naked selling.


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[Assistant] Okay, so I'm going to, for now, allow for that.


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[Assistant] So we're not taking into account that anybody is defaulting. If you take theta, alpha, negative, it means your income in the future is going to be reduced because you're going to have to deliver the dividend. Because you're going to have negative dividends, which means effectively you take out of your endowment those dividends and hand them over. So it's as if you always keep your promise.


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[Assistant] So this model, so far, the official model, assumes no default. No collateral, we're not worrying about any of that stuff. Okay, and of course that's going to be a critical thing.


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[Assistant] Uh, that you also wanted. It was a trade of value for value. Everybody agreed the two things you traded were equally valuable. If you take theta alpha negative, by taking theta alpha negative, okay, that becomes a negative number here, so it allows you to spend more. You can buy more good.


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[Assistant] By taking theta alpha negative, that's negative, that means this can be more positive and still satisfy this constraint. So by selling a stock short, you're promising to do something in the future, you get more money now, you can eat more now, and then of course you have to consume less in the future because you have to pay back your promise.


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[Assistant] So, you're exchanging something valuable, you're, you know, you're getting money, something valuable, in exchange for a promise, which is worth nothing, until the future, when you deliver on your promise. When you buy the stock, you know, you're buying part of the tree, but the tree is doing nothing for you now, you're doing it because it's going to be valuable next period. You're actually not


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[Assistant] Okay, so, so far we're going, for the next few lectures we're going to ignore the fact that people get very nervous when they give something up that's valuable in exchange for something that's just a promise.


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[Assistant] What? Okay, so a critical thing has happened here. So we've kept the same mathematics except we surreptitiously added this huge assumption. Now Fischer said having done that, what can you


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[Assistant] This is the most important insight. He said this model, it looks so complicated. It looks like now we have vastly more equations. No wonder Marx and all those religious zealots were getting confused. You know, uh, we can simplify it all and be back to where we were before and yet talk about finance.


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[Assistant] So Fisher introduced the idea of present value prices.


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[Assistant] Okay, so he said, look, when you buy a stock, what are you really doing? This is the principle of arbitrage. He says when you buy a stock, you're saying to yourself, I'm giving up money today. Now money today is


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[Assistant] Consumption, because I would have used that money if I didn't buy stocks, I would have bought apples today and eaten them.


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[Assistant] So when I buy a stock, I'm giving up apples today, I'm getting the stock, which is then paying me dividends tomorrow, which whatever they are, I'm selling off, you know, I'm getting a profit out of the stock tomorrow, and then I'm ending up with apples tomorrow. Maybe I'm just eating the dividends straight off the tree.


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[Assistant] And no matter how I do it, whether it's through stock alpha, or through stock beta, or through a nominal bond, it's gotta be the case that all three ways, or all fifty other ways you could imagine doing it, have to give me the same trade-off.


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[Assistant] The amount of apples I effectively give up today in order to get apples tomorrow is going to be the same no matter which way I do it.


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[Assistant] Okay, so that's why this assumption is so important. What would they do? Not only would nobody buy alpha, but they'd start selling it. They'd say, well, alpha is so expensive,


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[Assistant] Uh, you know, let's say it's the same price as beta, I can sell alpha, with every alpha I sell, I can buy stock beta. And so, I haven't done anything today, but in the future, I've got stock beta, which is paying me too.


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[Assistant] I owe because I sold stock alpha short, I owe one, so I'll pay off the one I owe and I'll still be left with one. I'm making an arbitrage profit. And so I'm not gonna stop at selling one share of alpha, I'll sell two shares of alpha, then three shares of alpha, then a million shares of alpha, and everybody will be selling


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[Assistant] Uh, alpha short to buy beta and, and uh, you know, the market for alpha will never clear. So that's why the prices will have to adjust.


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[Assistant] And so it can't be, it has to be an equilibrium. The price of alpha is exactly the height, half the price of beta. Which is to say, in short, that if you solve for this equilibrium, you can solve for an equilibrium where p1 equals q1 is the price today of an apple today.


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[Assistant] Okay, so if you've got this equilibrium, by working your way through, you know, by figuring out what the price of alpha is,


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[Assistant] Uh, so the stock, for example, you want to figure out what the...


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[Assistant] What the stock is opposed to the stock of alpha, suppose the price turns out to be, uh, a half. Okay, then by paying a half today, you can buy stock alpha, which is going to pay you a whole dividend. Okay, so the price, therefore,


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[Assistant] Of an entire, or let's do beta. Suppose the price of beta is, uh,


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[Assistant] How much do you have to give up today in order to get


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[Assistant] By paying a quarter today, you're getting two dividends. So by paying a quarter today, you're getting two dividends. If you paid to get one dividend, you'd have to pay an ace today. So the price, uhm, P2 would be an ace in that case.


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[Assistant] Okay, so by, by looking, piercing through the veil of the stock market, you can always figure out what you're effectively paying today in order to get an Apple next period. And that price, which we just computed, would be the same whether we looked at it from the point of view of going through stock beta or through stock alpha or through the nominal bond. It would always have to give us the same answer.


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[Assistant] Just add to the endowments of good.


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[Assistant] With one unit of each good. But A also owns all of stock alpha and half of stock theta.


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[Assistant] So all of stock alpha pays one dividend in the future. So really, A effectively has claim on two apples in the future, and another half of beta, which is another apple in the future. So really, A's initial endowment of goods is one-three.


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[Assistant] Okay, which pays one apple, so that's another one that's really his, and then in the future he's gonna get half of the...


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[Assistant] Well, one doesn't change today, but how many, what's his claim effectively on dividends in the future?


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[Assistant] One. Thank you. Somebody answered that. Uhm, so,


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[Assistant] We've now reduced the financial equilibrium to a general equilibrium. The same kind of economy we had before.


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[Assistant] It's just that we had to augment the endowments to take into account that people own stuff through the stock. Okay?


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[Assistant] So what's the equilibrium of this economy? This equilibrium has a, this has a simple general equilibrium. Okay, so what is it? How do we solve for equilibrium? Well, take P1 equal to 1.


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[Assistant] And we'll solve for P2. So let's just clear the first market. How do you clear it?


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[Assistant] You're with me here, it's a standard general equilibrium, the same kind we've done many times before. So, let's see if I can do it.


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[Assistant] So person one is going to spend a third of his money. And how much money does he have? He has P1 times, so he has 1 plus P2 times 3. That's A.


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[Assistant] Right, his endowment is 1, 3, this is his income, and he's spending a third of it on good one, and the price of good one, P1. Okay, here is just 1. And then B is gonna spend, he's a half-half Cobb Douglas guy. So this is 1, and this is 1. He's spending half of his money, and his income is 1 plus P2 times 1, divided by 1.


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[Assistant] Okay, and that has to equal the total endowment, which is 2. 1 plus 1. So did I go too fast? Yes.


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[Assistant] Well, that's probably wrong, so let's see if I, let's try it again. Maybe it's two-thirds. Okay, so let's try, uh, let's see what I was doing.


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[Assistant] I'm saying, here are the, I've taken the financial economy, which is very complicated, looks very hard to solve, and Fisher says, you know, of course when we add uncertainty and things like that, we're gonna have to do other tricks. But without uncertainty, with perfect foresight, and so on, and no uncertainty,


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[Assistant] Fisher says this is an easy problem to solve. You take the financial equilibrium with all its extra variables, and you realize if people are rational, they're gonna see through all that complicated stuff. They're gonna realize that alpha is just half as good as beta. And so they're gonna realize that by holding stock, they're making a certain tradeoff between


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[Assistant] I forgot what P2 was. Anyway, how much did you have to pay? Okay, if you pay pi alpha divided by d alpha 2, something like that, was P2. Because by, uhm,


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[Assistant] Okay, so if, if it cost you a certain amount of money and you get, you know, if it cost you a quarter, we said,


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[Assistant] And you get, uh, this is beta, so this is P2. So through either stock, like beta is the one I solved before, I said suppose beta is also equal to pi alpha, or


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[Assistant] By looking through the veil of the stocks, we realize that A actually owns three apples in the future. One because he owns all of stock alpha, and another one because he owns half of stock beta. So he's got the simple economy that we're used to solving that he did on the first problem set, so we can do it again and solve it. So I'm gonna, I'm gonna now solve for general equilibrium.


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[Assistant] I have to solve for P1 and P2 and all the XA1, XA2, XB1, XB2.


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[Assistant] But I can fix one of the prices to be one. So I'll fix P1 to be one.


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[Assistant] So that's why this should have been a two-thirds, as she pointed out. Thank you.


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[Assistant] Two-thirds of his money. What's his money? His endowment is one-three. So it's P one times one, which is one times one, plus three times P two.


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[Assistant] P2 times 1 divided by 1, that's how many apples today she wants, and that's what we have to clear to clear the apple market at time 1. Does this make sense now?


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[Assistant] Okay, remember how the, uh, Cobb Douglas worked? This is the, this is this trick I'm gonna use over and over again. With log utilities,


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[Assistant] Everybody will spend depending on the coefficient. So remember, this utility in the problem set, you know that this utility is just the same as if I put two thirds here.


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[Assistant] And one third here, right? Because I'm just multiplying, dividing this by three. So instead of one and a half,


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[Assistant] Log x2. So the sum of these things is 3, the sum of this plus this is 3 halves. I can multiply by 2 thirds. So if I multiply this by 2 thirds, I get 2 thirds here.


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[Assistant] And one third here, right, and I haven't changed the utility function. And now I know, this is a familiar pattern, A is always going to spend two thirds of his money on the first short. Okay?


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[Assistant] And B is gonna spend, and now we recognize it as a common Cobb Douglas thing, and we could say that A is gonna spend half his money on the first good and half his money on the second good.


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[Assistant] Okay, so what have I done? Fisher said, look, this model is so complicated, you're thinking in your head, people are deciding in period one, how much stock should I buy? How many bonds should I buy? How many apples should I eat? But really, if they're smart, they're not going to think that way. They're going to say to themselves, how many apples should I buy?


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[Assistant] Today, how many apples do I want to consume tomorrow? All these financial assets are just


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[Assistant] And what's the trade-off between the apples was that P1 and P2 is the trade-off between apples. You can look through the stocks and all that, but no matter which stock you think of buying, there's going to be the same trade-off between apples today and apples tomorrow, because of the no arbitrage.


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[Assistant] You know the price of alpha is going to have to be exactly half the price of theta.


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[Assistant] So if I, once I solve for this economy and get the price of alpha, I'll know how many apples today I have to trade off in order to get apples tomorrow.


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[Assistant] So that's why you can forget about the stocks, forget about the bonds. Everybody's thinking, I'm trading off apples today for apples next year.


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[Assistant] So it's as if everything happens today. It's as if there are present value prices today. We trade today at prices P1 and P2 for apples today and apples next year. Of course the apples won't appear till next year, but I can sell an apple today at price P1 and buy promises for apple next year at a price P2.


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[Assistant] And that's the trade-off I'm facing. If I face that trade-off, how much of my money am I gonna spend on apples today? I'm gonna spend...


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[Assistant] Two-thirds of my money on apples today, and the other third I'll spend on promises for apples next period.


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[Assistant] Okay, the price, often there'll be inflation, so Q2, the contemporaneous price next period might be higher than the contemporaneous price this period. But we don't care about that. What we care about is


[User] Please transcribe this text

[Assistant] How many apples you have to give up today in order to get apples tomorrow? So that's called the present, so P2 is the present value price. What do you have to give up today to get the apple next period? So we expect P2 to be less than P1.


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[Assistant] Precisely because, well we're gonna come to that, that's the next thing I was gonna talk about, cause everybody's putting more weight on consumption today than they are on consumption in the future. That's why the price P1 is gonna be bigger than the price P2. Yep.


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[Assistant] So we're solving for P1 and P2 and present value prices.


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[Assistant] Okay, so the crucial thing is he invented this term, present value price.


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[Assistant] Goods in the future. So when you buy the stocks today, you should think, how much am I having to pay today to get an apple in the future? You can deduce that from the price of stocks and how many dividends they're paying.


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[Assistant] So everybody must have figured out a P2. What does it cost today? How much money do I have to give up today to get an apple in the future? Well, I have to buy a stock and then sell the dividends and all that.


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[Assistant] Really what I should be thinking about is what's the price today I'm paying for one apple in the future. And that's P2. And so when you think about it that way, although it's an inter-temporal problem, it looks like a new model with time, Fisher said you can reduce it as if they're, think of it as if they're just the same problem you did before with two goods you're trading at the same time.


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[Assistant] That's not an obvious thing to have thought of. No one thought of it before him. Yep.


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[Assistant] Who, who just asked that question? Where are you? Okay, I'll remember you. That was very good, exactly. So in the future, you're getting the money for, so, what he pointed out is, I made another mistake here, when, in the future, the money you're spending on goods in the future,


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[Assistant] You're gonna get the dividends paid, of course you can sell the dividends for money, and the price is Q2, so a Q2 has to appear over here, just like there's a Q2 over there.


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[Assistant] Okay, so we're back to this standard general equilibrium problem. We can take a financial equilibrium and turn it into a general equilibrium. And so when we solve this, we're gonna have two-thirds plus P2 plus one-half plus one-half P2 equals 2. So, uhm, looks like three-halves, hope I haven't done this wrong, P2,


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[Assistant] 2p2, thank you, yeaah, plus 2p2, okay, so we have 2p2 plus a half plus a half p2, so we have 5 halves p2.


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[Assistant] That was lucky, you caught that. 5 halfs P2, and then over here we have 2 minus, uh, so 2 thirds is


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[Assistant] Four sixths and three sixths is nine sixths and twelve sixths minus nine sixths is, uh,


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[Assistant] Alright, so we've now solved for equilibrium. We know that P1 has got to be 1.


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[Assistant] P2 has got to be one third. We know that, uhm,


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[Assistant] A1, for example, if we wanted to solve for that, we'd just plug in a third here, so we'd have two thirds times one.


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[Assistant] Plus three, which is two-thirds times a four, which is eight-thirds, I guess.


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[Assistant] Okay, and, you know, xd1, we could have solved for that too if we wanted to. xd1 is going to be a half times four-thirds, which is two-thirds. Okay, a half plus, what was this? One-third, so, it's one plus one-third, which is four-thirds, times a half, which is two-thirds, is that right?


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[Assistant] Well that doesn't look right. So maybe I did this wrong. One plus, one plus one is two, so this is four-third. That works better. Okay, four-third.


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[Assistant] XA1 is four-thirds, XB1 is two-thirds, okay? So we have four-thirds and two-thirds, and so we could solve similarly for XA2 and XB2, which I won't bother to do.


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[Assistant] We can figure out what the prices are, the present value prices, and the present value consumption. But having done that, Fisher says, okay, we took a hard problem, we made it easy. Let's go back to the hard problem.


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[Assistant] Okay, so Fisher says the tradeoff between good one and good two is one to a third. So he defined, here's the nominal rate of interest, Fisher defined something called the real rate of interest.


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[Assistant] And he said that was a variable you should pay a lot of attention to. So the real rate of interest, he said, is, uhm, P1 divided by P2.


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[Assistant] Okay, so how did I get that? Just as someone in the front said,


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[Assistant] The good two is much less expensive. The present value of good two is much less than the present value of good one.


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[Assistant] I think an apple today is much more valuable than an apple tomorrow. So if you give up an apple today, you can get three apples next year. Just like if you put, so if you put an apple in the bank, it's like getting 200% interest on apple. So we call that the real rate, the apple rate of interest.


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[Assistant] You put an apple in the bank, you know, you give up an apple today, buy stocks, and when it comes out in the end, you've got 200% more apples than you started with. So it's the real rate of interest.


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[Assistant] Okay, so, that's his crucial variable. Now, let's go back to the original equilibrium. What is the stock price? Assume now that Q, assume Q1 equals 1. Okay, what is the stock price, pi alpha?


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[Assistant] Well, stock alpha pays one good tomorrow. So what is the price of phi alpha?


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[Assistant] What? Somebody said it, I couldn't hear it. A third. How did I get a third?


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[Assistant] How much do I have to pay today to get an apple in the future? I have to pay a third to get one apple in the future. P1 is 3 times P2, so to get one apple in the future is only a third of an apple today. So, the stock pays one apple in the future, so therefore


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[Assistant] How much do I have to pay today? I have to pay one third of an apple today, and since I took the price of apples to be one, it's going to be the price of one third. So what's the price beta? Two thirds.


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[Assistant] Okay, so Fisher said, look, we've solved now for all these financial things.


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[Assistant] How can you possibly tell whether somebody's gonna hold twice the alpha tree or just one beta tree? Either way is gonna get the same thing. So the theory can't possibly explain which one they're gonna do. Somehow they'll, they'll work it out and divide up the tree so that everybody ends up with the right number of apples in the end. And it also does not explain, does not explain inflation.


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[Assistant] You're not going to change the second equation. So q2 could be anything. And it, uh, you know, you can double it or triple it, it won't matter.


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[Assistant] Okay, so Q, and the same with Q1. It's just like Walroth said before, you can always normalize prices to be one. So the theory, you know, he had to add another theory of money and how many dollars were floating around in the economy to explain Q2. This theory won't explain it. It does not explain inflation, and it does not explain, uhm, who holds which stock, and so it does not explain the nominal rate of interest. It does not explain I.


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[Assistant] The nominal rate of interest, because one dollar, who knows what one dollar is going to be worth. It depends on how much inflation we have. But it does explain the real rate of interest. It does explain R.


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[Assistant] That's the crucial variable. So if you want to figure out what's the price today of a stock, so Fisher's famous equation is the price of a stock today


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[Assistant] Okay, so pi alpha divided by q1 for the real price.


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[Assistant] As somebody said, the price in terms of goods of the stock today is always going to equal the dividend in the future divided by 1 plus r.


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[Assistant] Okay, why is that? That's exactly what we already used. Okay, this is just a rewriting of the trick we did before. You take the stock tomorrow,


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[Assistant] Okay, and then you realize that 1 plus r is just p1 divided by p2, so replacing the p1s and p2s by q1 and 1 plus r, today's real stock price is just


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[Assistant] Just like we did. You know, once we knew P1 and P2, the present value, the present value of prices determine the interest rate because they're just as we said, P1 over P2, remember, is 1 plus r.


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[Assistant] So knowing P1 and P2, you're always normalizing P1 to be 1, that's the same, P2 is the same as 1 over 1 plus r. So if you know P2 or you know 1 over 1 plus r, you know what the value of the stocks are.


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[Assistant] That's his critical insight. Now, just to finish, insert, yes.


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[Assistant] Q1, which is the same as P1, you know, because it's today's price in both


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[Assistant] Okay, so let me just end on this one note.


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[Assistant] Fisher said, okay, we can take financial equilibrium without uncertainty, reduce it to general equilibrium, we know everything about general equilibrium, therefore we know everything about financial equilibrium. And we realize that the crucial variable in general equilibrium is relative prices. There is no just interest rate. The nominal interest rate, who the hell cares?


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[Assistant] The real interest rate is what we care about, and just in normal economics there's no just


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[Assistant] Relative price, there's no just real interest rate. It depends on people's utilities. You make them more patient.


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[Assistant] And the, you know, and that's gonna affect the real interest rate. You make them less patient, it's gonna affect the real interest rate. You give them more endowments today versus tomorrow, that's gonna affect the real interest rate. The relative price between today and tomorrow. That's the way you should think about finance, that's the way you should explain what's going on in the financial market. So in the problem set, you're just gonna do a problem like that, and then I'm gonna give more interpretations of this that Fisher gave. So I guess I'm out of time, so we'll stop here.


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[Assistant] Spending a couple classes these days learning basic facts and vocabulary about finance.


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[Assistant] And, uh, along the way we're trying to apply the simple lessons that Irving Fisher taught about


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[Assistant] Uh, turning a financial problem into a general equilibrium problem and making you


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[Assistant] In particular of the budget set, you know, that very simple budget set we wrote down at the very beginning turns out to be quite useful and people often can get quite confused. So the last issue we ended with I'm going to take up again. Suppose that you've got a very long lived institution like Yale, uhm,


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[Assistant] How should Yale think of how much to spend every year? What is Yale's budget set?


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[Assistant] It creates a fiction of an annual budget. And, you know, they talk about the deficit and having to bring the deficit under control and making cuts to, you know, close the deficit gap. But really, there is no such thing as a one year budget set.


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[Assistant] I mean, why one year? Why not one month? Why not one day? Nobody expects Yale to budget, balance its budget every day. Some expenditure comes in in the day, they have to hire an electrician to fix something unexpected. They're gonna spend more money than they take in, in, uh, student, uh, tuition that day.


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[Assistant] So, the fact that they're supposed to budget the balance every year is just a fiction. Irving Fisher taught us that Yale really has, if you can borrow and lend and you don't have to worry about lifts, there's one infinite lift budget set.


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[Assistant] It's an infinite horizon budget set where you just take the present value of all the expenditures. That's the...


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[Assistant] That's the left hand side and the present value of all the revenue, that's the right hand side and make sure that the left hand side is smaller than the right hand side over the whole course of Yale's life. So that simple principle


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[Assistant] has a tremendous implication which was overlooked to the


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[Assistant] Beno Schmitt, who was then the Yale president, released a white paper, he called it, documenting the fact that Yale hadn't, uh,


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[Assistant] Yale could be brought up to snuff and then gone afterwards as a normal running institution provided it spent a hundred million dollars a year for ten years. And that included fixing every college, you know, they're going to do more than one a year over a ten year period.


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[Assistant] But Yale's total budget, as I told you, was about a billion dollars at the time.


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[Assistant] And then here all of a sudden was this $100 million a year expense for 10 years. That's 10% of the Yale budget.


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[Assistant] A lot of the costs, you know, you can't change. I mean, you have to have the lights on, you have to heat the buildings. You can't really reduce those costs.


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[Assistant] And so, Ben Schmidt came to the conclusion that he'd have to reduce the cost he could change by 15% in order to balance the budget, to cut, you know, about $100 million a year out of the budget.


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[Assistant] So, he announced one day that he was going to fire 15% of the faculty by attrition. You know, if they were junior faculty, wouldn't promote them.


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[Assistant] And if they were senior faculty, wait till they retired and not replace them.


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[Assistant] So this, needless to say, caused a tremendous commotion among the faculty, and as I told you, a committee was formed and


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[Assistant] And I had to present the report. So actually the report went pretty carefully through all the calculations made in the white paper, but the heart of the report was simply to real- to apply the lesson of Irving Fisher.


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[Assistant] So Irving Fisher would say, Yale's going to live forever. Let's suppose that we want, that Yale wants the same quality of education every year forever.


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[Assistant] So it should have the same wheel spending every year or forever after, after it, uh,


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[Assistant] It compensates for inflation. So at the moment we're assuming there's no inflation. So what does that mean?


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[Assistant] That means that you just look at the right hand side and you say, what's Yale's revenue? Well, whatever it was before,


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[Assistant] So what's the loss of revenue on the right hand side? It's 10 years of 100 million dollars a year, as you can see. Now you need an interest rate. What should the interest rate be? Well, uh,


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[Assistant] Should it be the nominal interest rate or the real interest rate? Well, we're supposing now there's no inflation, so it should be the real interest rate, because all the hundreds are, are, you know, have no inflation in them.


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[Assistant] So, what real interest rate should we use?


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[Assistant] The white paper used 5% because they thought that was the number that Yale could, could, uh,


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[Assistant] Real rate of return Yale gets, and so they discounted it 5%.


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[Assistant] Now, the real rate of interest, typically in the economy, is 3%.


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[Assistant] But let's suppose that we calculated this at 5%, the present value of, uh, 100 for 10 years.


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[Assistant] Is 772. Now how could you do that in your head? Well,


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[Assistant] You know, 5 into 72 is about 14, so in 10 years, you're going to get


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[Assistant] So, if you got 100 forever at 5% interest, that would be 2 billion. And we know for only 10 years, it's less than half the value.


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[Assistant] So, you know, considerably less than half the value, so it's not 1 billion, it's something less than that, it's 772.


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[Assistant] Did that in Excel and I calculated 7.72, but in your head you know that if it had gone on for 14 years, then the present value, or formula, or famous formula, is that you would take


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[Assistant] The coupon 100 divided by the interest rate .05 times 1 minus 1 plus r to the 10th.


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[Assistant] And so we know that if this were 14 instead of 10, you'd get a half here. So this is 20 times 100, which is 2 billion.


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[Assistant] So you could be sitting there in the audience hearing Ben-O-Schmidt talk and be computing in your head that we're talking about something under a billion, like, you know, three quarters of a billion.


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[Assistant] It should be spending 5% of $772 less every year. That's $38.6 million less every year.


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[Assistant] So that's a drastically smaller number than a hundred million dollars a year. It's crazy just to think that because you've got these expenditures for ten years and the no expenditures after that, that you should, you know, cut the budget by a hundred million and then let it go up after ten years. So uhm,


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[Assistant] You'd only need to cut it by 38 million. Now by the way, 5% is a pretty arbitrary number. Suppose you put 3% here.


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[Assistant] Well, 3% would give you a much higher present value, but then when you multiply it by 3% at the end for the annual reduction, it would give you a much smaller number. So that, anyway, this number, uh, which I


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[Assistant] But you multiply that by 3% and you get 25 or 26 million a year.


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[Assistant] So now the reduction is starting to sound like it's not such a frightening thing. So let's go stick with the


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[Assistant] Let's stick with the 5%, which is what the white paper, take all the assumptions of the white paper and take it literally and then notice that they never said anything about inflation. So actually, this, uh, this calculation of present value loss is, is, uh, it's, right, it's, it's, uhm,


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[Assistant] There's inflation, and say the inflation at that time was around 4%.


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[Assistant] In fact, all the Yale contracts that are still in place assume a 4% inflation, even though inflation is less than that now. But anyway, so that 100 million a year of dollars is actually less in present value terms.


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[Assistant] Because what should you discount by? If you look at the present value of 100 million dollars over 10 years and you take into account its dollars, you should be discounting by the real interest rate times the inflation.


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[Assistant] Okay, so by 9%, a little, uh, over 9, tiny bit over 9%. So if you discount that by 9%, you get, uh, 641 billion as the present value, a million as the present value loss to Yale. Okay, now,


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[Assistant] Given that there's inflation, how much should you be spending every year? Okay, you should be spending in wheel dollars.


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[Assistant] Reducing your spenditure, how much in real dollars? Well by 5, not by 9%, by 5% of the 641 million.


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[Assistant] So if you have 641, that's today's present value in, you know, there hasn't been any inflation yet. So that's the real loss.


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[Assistant] So if you ask what's the real expenditure reduction every year, it's 5% of 641. And that's 32 million.


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[Assistant] Okay, so 32 million is a far smaller number than 100 million and requires a small, far smaller drop in, in, you know, expenses. So our committee recommended that we cut the faculty by 6% instead of by 15%.


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[Assistant] And you know, 6%, there are a lot of people leaving every year, you can do 6% pretty quickly. So the upshot of this is that it's a simple application of present value, very elementary calculation. It came as somewhat of a revelation to our administrators, I'm afraid.


[User] Please transcribe this text

[Assistant] The day after the report, the provost of the university resigned.


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[Assistant] Two weeks later the dean of the university resigned, and two months later the president of the university resigned, and Rick Levin, the current president, took over, and he, uhm, cut the faculty by 6%.


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[Assistant] But by no more than that, and then of course the finances of Yale got much better, and he since added back that 6% plus a little bit more than that.


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[Assistant] So just to tell you about something good about the Yale administration, the provost who resigned that next day,


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[Assistant] Happens to be a friend of mine. I've had dinner with him every month for the last


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[Assistant] You know, 12 years since this happened and he's never once, uhm, criticized me or shown the slightest discomfort about the report that basically ended his, uh,


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[Assistant] He cares so much about Yale and was so determined to do the right thing.


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[Assistant] He just thought he made a miscalculation and stepped aside.


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[Assistant] Made the wrong calculation, not that he had some political agenda or something to cut such a huge part of the faculty. So it's the, the most honest and the most, uh, uhm, Yale loving, uh, administrator that, uhm, you could imagine.


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[Assistant] And uhm, I didn't know what the reaction would be of someone like him after giving the report. I was quite terrified actually that they'd be, they were still the president, the provost, and the dean, that they would be quite angry at our committee.


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[Assistant] Uhm, but they responded, uhm,


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[Assistant] Right, so the question was, why after I figured out the present value loss, like 772 million, why did I multiply that by 5% to figure out how much Yale should reduce its spending every year?


[User] Please transcribe this text

[Assistant] And the answer that was given down here is that I'm assuming that Yale is going to go on forever. So Yale can reduce its expenditures every year forever, and by doing that, make up for the same present value loss.


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[Assistant] Forever means perpetually, so it's a perpetuity. So how much do you have to reduce?


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[Assistant] What is the coupon reduction, the expenditure reduction every year at 5% interest?


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[Assistant] That just makes a present value decline of 772 million. Well, it's 5% of the principal. If you have 772 million in the bank, and


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[Assistant] Every year, 5% of that you throw away, you've thrown away the whole value at 5% interest of the $772 million.


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[Assistant] So by reducing your expenditures by 5% every year, you


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[Assistant] You, uh, defray the $772 million dollar law.


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[Assistant] So that's the critical, so the critical thing, the critical mistake that the administration made is they had a short run problem with a bunch of short run costs, but Yale's gonna live forever and Yale should


[User] Please transcribe this text

[Assistant] One-shot, uh, loss. You know, one-shot problem that Yale faced. So...


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[Assistant] They weren't thinking in Fisher's terms of taking the present value over the whole course of the lifetime of the institution. They were thinking, well, we've got to spend a hundred million dollars this year. We better cut costs by a hundred million dollars.


[User] Please transcribe this text

[Assistant] But that's obviously crazy, like suppose you had to spend an extra, you know, $100,000 in one day. Uh, does that mean you should, you know, lay off your faculty for one day so you can find the money to, uh, to pay that? And of course not. So you have to, you have to spread the loss over the lifetime of the university. Okay, is there any other questions about this?


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[Assistant] If I wanted to incur the whole cost over ten years, then I would have to reduce my expenditures by a hundred million dollars a year.


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[Assistant] If at the end of 10 years I want to be back even, then I would have to spend less.


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[Assistant] I would have to cut my expenditures by exactly the money I was pouring into the building.


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[Assistant] The maintenance costs go over a ten year period, and I want the expenditures to go over a ten year period. Right?


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[Assistant] So if I want the expenditures to be reduced evenly over a 10 year period, I'd have to do 100 million dollars a year. If I wanted to eat away the cost all in one year, I guess I didn't understand your question, you're saying if I wanted to reduce expenditures entirely in one year and then return next year to my usual pattern of expenditure, then I'd have to cut the 1 billion dollar budget by 772 million.


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[Assistant] So that would, you know, involve basically firing the whole faculty and saying take a year off, you're on furlough. Sort of what they're doing in California.


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[Assistant] I could do that if I wanted. I could rearrange the hundred million anywhere I wanted to. So you would cut less than a hundred million dollars this year out of the budget and a little bit more every year after that.


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[Assistant] One way to do the cost over within a 10 year time frame is reduce.


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[Assistant] No one, you know, why do that? And that makes the, you know, that's kind of what bad politicians do. They say, well we're not gonna, you know,


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[Assistant] Not our fault, we'll just make those guys in year ten get totally crushed and sooner


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[Assistant] Right, okay, so there's a good question. So there, there, so actually some people, you know, the administration, that was their best response. They said, well, even though the white paper said this was a one-shot thing, and after we do the


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[Assistant] 10 year plan, Yale is back in good shape, and of course every year Yale has allowed


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[Assistant] You know, maintenance expenses, that's part of the budget, is maintenance. And so after we get the buildings back and ship, you know, tip-top shape, we're going to keep them in tip-top shape by doing these, uh, you know, normal expenditures every year. So we should never have another period where we have to do something drastic like that.


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[Assistant] That's what the white paper said. But after the report, they responded just like you said. They said, well, we didn't really mean that. Maybe in 25 years, we're gonna have to do another remodeling, uh, effort.


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[Assistant] And so, well, if you needed to do that, then, uh, it wasn't just, you know, a one-time deferred maintenance. It means that you've drastically underestimated the cost of keeping up the buildings over,


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[Assistant] You know, over Yale's whole future. And then you would have to...


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[Assistant] Reductions in expenditures for the first 10 years, and then, you know, in year 35, you'd have to have more reductions, and then in year 70, you'd have more reductions like that. So you'd have to take the present value of all those things, and then figure out how much to reduce expenditures on an even basis, and so it would be much more than 32 million, it would, you know, be 60 or something million, or 50 million.


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[Assistant] So you're exactly right, but that's not what they said in the white paper, you know? So I took them literally what they meant. And uhm,


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[Assistant] So what happened after that? You know, Yale has done much more building expenditures than that. But that's because Yale's endowment went up to 23 billion.


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[Assistant] You know, so from 3 billion, which it was at the time, it's now 23 billion. So Yale's launched an incredible program of building construction. Basically they've done two things. They've hired a huge number of, uh, built a huge number of


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[Assistant] The faculty still not that much bigger, you know, went down 6%, it's now back a little bigger than it was before.


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[Assistant] So, you know, the plan to expand the college and expand the faculty hasn't happened yet. So,


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[Assistant] Yale faces another choice now. You know, the endowment went to 23 billion.


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[Assistant] And then this past year they managed to lose down to 17 billion, you know, 30% got lost. So we're down to...


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[Assistant] To 17 billion now. So again, we have the same question. We just lost 6 billion dollars.


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[Assistant] How much should we reduce, uh, expenditures every year? So, what's your answer to that?


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[Assistant] Not thirty percent. What, uh, yeaah. So how much would you reduce expenditure?


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[Assistant] Well, why wouldn't it be 30%? Because Yale spends a lot of money, it doesn't get


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[Assistant] From the endowment, right? It gets money from tuition, for example.


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[Assistant] So, how much would you, yeaah. So what should Yale do? What do you think Yale is going to have to take out of the budget now?


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[Assistant] Yeaah, and let's say it's still 5% real interest, then what would you do?


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[Assistant] Okay, 300 million. So Yale's got to somehow cut 300 million out of its budget.


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[Assistant] So it's not gonna do it in one year, but over the course of the next few years it's gonna have to cut 300 million. Now the budget is well over 2 billion.


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[Assistant] So that's 15% of the budget, though, Yale's gonna have to cut. So this is a serious thing. How do you cut 15% of the Yale budget?


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[Assistant] Well, I hope they don't do that. I think they learned their lesson, so I doubt if they'll do that. But they're, it's, uh, you know.


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[Assistant] There's gonna be a bunch of stuff like that. So anyway, we have another budget problem, by the way. So these kinds of budget problems are happening all over the country. I gave a talk at Albany University, and they're gonna, they're gonna abolish their graduate economics program. Uhm, SUNY Albany, that's, you know, so they're doing all, you know, these are serious


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[Assistant] Problems, losing that much money. In any case, 6 billion translates to, right, the difference here is 6 billion.


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[Assistant] And you multiply that by .05, just as we said, and that equals 300 million a year. So, Yale won't do it right away, but over the course of, you know, a few years, Yale's gonna have to reduce it.


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[Assistant] So they're gonna obviously choose to do a lot less building and some of all those, those


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[Assistant] Right, okay, very good question. So on the front, he's saying, you know, this presumes that we know for sure that the endowment lost six billion, it'll never recover it.


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[Assistant] Uhm, maybe that's a temporary drop in the stock market, it'll go back up.


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[Assistant] And basically the principle he's applying is he's saying you can't make these drastic reductions in annual expenditures, you know, firing people and then two years later realizing you've got a lot of money trying to hire them back because you won't be able to hire them back.


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[Assistant] So clearly Yale has to have a more complicated rule about how it gradually adjusts its spending when there's a change in the endowment.


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[Assistant] And by the way, you know, this number could go down as well as it goes up. So we're going to come back to Yale's investments and, you know, what they're like. A lot of Yale's investments are called private equity investments that are very hard to value. So for all we know, this 17 is a lot worse than that.


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[Assistant] But we'll be finding out, uh, in the next year or two. It's not like a hedge fund where you have to


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[Assistant] You know, value all your assets by what the market will be willing to pay. A lot of these assets, there is no market, so they just sort of make up what the number is.


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[Assistant] So we have to, we're going to come back and discuss this. A very interesting question. Okay.


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[Assistant] Alright, so one last thing about this present value calculations, one last obvious thing.


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[Assistant] It's hard to keep in your mind the difference between real and nominal. Okay, so let's just do a very simple thing. The mortgage, mortgages


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[Assistant] Our traditionally nominal pa- no- Traditionally nominal fixed payment.


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[Assistant] How did I do that so quickly in my head? Well, because I know if the interest rate is 2.3% and you're gonna pay it forever, you'd pay $2,300 a year. We know at 30 years, a 2.3%, 2.3% doubles almost in 30 years.


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[Assistant] 69, that's getting pretty close to 72, so maybe it takes 31 years or something to double. So after 30 years, the remaining part of the mortgage is only


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[Assistant] You know, the remainder, the remainder is worth half the mortgage. You've lost half of the value by only getting it for 30 years.


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[Assistant] So instead of paying 2,300, you have to pay 4,600, you know? So the coupon over .023 times 1 minus 1.023 to the 30th, okay, that's 1 minus a half about. So if this coupon equals 100,000, the payment is going to be


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[Assistant] Since this is a half, the payment isn't gonna be, you know, $2,300. It has to be twice that, $4,600. Okay, so it's $4,600 per year.


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[Assistant] So if there's no inflation, that means you're making the same real payment every year. Now what happens if there's inflation? What if inflation goes up? What if inflation goes up? Okay, now what's gonna happen to what you have to pay? Well, this is just


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[Assistant] Well, if inflation is, you know, another 2.3% or something, then you would pay


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[Assistant] Then the nominal interest rate, 1 plus i, is going to equal the real interest rate.


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[Assistant] Times the nominal interest rate, times the rate of inflation. Okay, so let's say this is


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[Assistant] Little bit more than four-six, almost, you know, little bit more than four-six.


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[Assistant] So you know that the interest rate the mortgage companies are now gonna charge is gonna be, uh, 1.046.


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[Assistant] So the 4.6 is going to be the mortgage interest rate and so you can figure out by the same calculation


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[Assistant] What's the coupon going to be? So what's the coupon going to be?


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[Assistant] You know, instead of doubling every 30 years, that 4.6% is going to double approximately every 15 years. So this is going to be


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[Assistant] Doubling twice, this is a quarter, so this will be three quarters here, and so if you multiply


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[Assistant] You know, everything, 2,300 a year at times.


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[Assistant] 2,300 a year times four-thirds


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[Assistant] Oh, I forgot to change this to 4 sixths. So the interest rate is 4 sixths. Okay, so you multiply 4, so if you multiply, so this to the other side is 4,600 times 4 thirds. Okay, and that's, you know, 6,000 about.


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[Assistant] Okay, so the annual payment is going to go up to six thousand instead of


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[Assistant] 4,600. It was 4,600. The interest rate went up because there was inflation. So of course, they're gonna ask you for more money.


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[Assistant] Every year, because if you pay the same amount every year, and this is the real payment,


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[Assistant] In terms of inflation corrected dollars, you're paying less and less every year.


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[Assistant] So clearly, if you started with no inflation, and a number like this, no inflation,


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[Assistant] Okay, so no inflation. And now you've got inflation, but the same real interest rate and the present value of your expenditures.


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[Assistant] The real present value, right? The mortgage company's gonna want, the lender's gonna want to get the same amount back in real terms as it got before because the inflation hasn't changed.


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[Assistant] The real world. So Irving Fisher would say the inflation is just a veil. Everybody's gonna want the same real interest rate and so the mortgage is gonna have to return the same real thing it did before.


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[Assistant] The present value in real terms of the real payments is still going to be a hundred thousand.


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[Assistant] Real payments go down over time and have the same present value they had before. It's gotta be that they're higher at the beginning and in real terms lower at the end. So sure enough, 6,000 is a much higher number at the beginning than 4,600.


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[Assistant] The mortgage companies are going to ask for higher annual payments.


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[Assistant] So the 6,000 a year instead of 4,600 a year, but now if you inflation correct that, the 6,000 every year is going to be a less, you know, in terms of real goods, less and less every year.


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[Assistant] But the present discounted value of this thing has to be the same as where you started. So the effect is the young, the young borrowers are gonna be spending a lot more in real goods when they're young and a lot less when they're old. So inflation has an unfortunate impact


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[Assistant] The repayments happen earlier, so the young who have less money are having to pay a huge amount. And when they get old, you know, the inflation's so high that that same $6,000 is practically nothing. So when they're 50 and 60, they're paying practically the peanuts to them. But when they were young, it was really a hardship. So there's a big problem with nominal mortgages.


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[Assistant] Which is that in inflationary times it kills the housing market. Fortunately we're not in an inflationary time. Okay, any questions about that?


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[Assistant] Alright, so that's the basic lesson of taking the present value and again you've always got to sort out the nominal from the real and look through the veil and don't get all mixed up by the fact that there's inflation. It's the real thing that you want to concentrate on as much as you can.


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[Assistant] Alright, so that's it for the obvious lesson of present value. Now I want to introduce another word which is very famous in finance.


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[Assistant] Okay, so yield, the next topic, or yield to maturity, is a way of trying to compute one number that summarizes how good a bond is


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[Assistant] Or how good, how well a hedge fund has done.


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[Assistant] So I think the more interesting case and the less obvious one.


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[Assistant] Is to start with a hedge fund.


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[Assistant] How, how do you measure how well a hedge fund's doing or how well it's done in the past? Never mind, yea, how well it's done in the past? We're gonna spend a lot of the course talking about this in various ways, but the first way to do it involves yield to maturity. So, let's


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[Assistant] See why the problem is a little complicated. So I imagine that there's three investors in this hedge fund.


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[Assistant] Okay, so every year some of the investors are going to decide what to do.


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[Assistant] The hedge fund before this guy put in his money had nothing. It's just beginning. He's put in his hundred dollars. So the hedge fund's got a hundred dollars.


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[Assistant] Okay, so that's it. So I'm imagining, you know, these all happen.


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[Assistant] They usually happen quarterly or annually or something. They don't happen every day. There's a fixed moment at which everyone deposits their money. So let's say they happen annually.


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[Assistant] The guy puts in a hundred dollars at the beginning of the year. For the rest of the year, nobody can do anything. They can't take money out, they can't put money in. So the hedge fund, let's say, uhm, manages to get


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[Assistant] Put the $100 to work and finds a 7% return. Okay, so it's now got


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[Assistant] The hedge fund altogether has got $107.


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[Assistant] So let's just go to the hedge fund all together, uh, has a hundred, had a hundred


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[Assistant] The hedge fund had 100 after these guys, cause only one guy put in money, and now the hedge fund's got 107 dollars. Well that 107 dollars is all the first guy's money, cause nobody else has put anything in, he still owns it. Okay, so so far the hedge fund got a 7% return.


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[Assistant] You know, our first investor, thanks to himself, well, you know, they did fine, 7%, not great, but you know, uhm,


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[Assistant] I'm okay, I'm not gonna do anything, won't take any money out or put any in. A rich second investor puts in a thousand. And another guy puts in two hundred.


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[Assistant] Okay, so now what's happened to the, uhm,


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[Assistant] To the N.A.V. of the fund. Well, the first guy, he's, you know, at the moment they put in the money, the first guy still has 107, and his, the first guy still owns 107 of the dollars.


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[Assistant] The second guy's now got a thousand in the fund, and the third guy's got two hundred in the fund. And the, uh, the hedge fund now has, uh,


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[Assistant] 1200 plus 107, that's 1307, that's how much money is in the fund. Okay, so that's at the beginning of year two.


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[Assistant] If you're still following this, if you're not following it, interrupt me, sorry, I mean, so. What happens in year three, in the beginning of the next year,


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[Assistant] Year 3, well, let's say our guy, uh, so the hedge fund makes money, and this time it made 3%, a crappier, sorry, a less good return, only 3% cut it off.


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[Assistant] 1307, ended the year at 1307, now by the end of this next year, it's made 3% on that, so that's, you know, it's up to 1346. Okay, so uhm,


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[Assistant] He's now made, you know, everyone made 3%. So his 107 turned into 110. The second guy's got 1030 in the fund, and the third guy's got 206 in the fund. Okay, and the total fund is, you know, 1346. Now let's suppose that our


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[Assistant] Guy, this is the beginning of year three, our first guy says three percent, that's a terrible return. I'm taking my money out. I've had it.


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[Assistant] It's $110, you know, that's what I have, I'm taking it out. Okay, so, and no one else does anything. So at the end of the year, you know, now he's down to zero, and everybody else is where they were.


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[Assistant] Okay, and the hedge fund thing has gone down a little bit. Well now next year, the thing does even worse. It makes a 0% return. Okay, so everybody's money's just the same, except that the second guy decides, you know, this is really getting lousy.


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[Assistant] Uh, and he takes half his money out. So this is taking half his money out. What was his money?


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[Assistant] He was, you know, 11-12, and half of that is 5-56, so he takes half of it out, leaving half.


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[Assistant] Oh, sorry, so this guy takes half his money out at the beginning of the year, the fund does badly, then the fund, uh, I skipped the 8%, there was an 8% return, sorry. So, oh, what an idiot. Anyway, so the, uh, the,


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[Assistant] The next year, the fund returned not 0%, returned 8%, okay? So the first guy, after the 3% return, this guy took his money out, the other guys left it in, and then the fund had an 8% return, okay? So they're, they're, you know, it's a little bit better, but this guy decides to take half his money out.


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[Assistant] But then finally, the last year, the fund gives a 50% return, which is fantastic. So all these guys, you know, everybody does well. And now let's say they all decide to take their money out. So now there's nothing left in the fund. The funds, and you know, they withdrew the total 934.


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[Assistant] Okay, so what I've done here, just to summarize, is every year, people are putting in money or taking out of money at the beginning of the year.


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[Assistant] How is the fund due? Done. How would you summarize in one number how the fund has done over its one, two, three, four, five years of earning return? That's the question.


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[Assistant] Okay, so this is a standard, you know, this happens, you know, every day, this is obviously what happens every day with hedge funds. So how do you, how do hedge funds report how they've done historically? So do you have any suggestions? What would you do to summarize how the hedge fund is done? If you had to pick one number, what would it be?


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[Assistant] So one thing you could do is you could say, let's say, what, he said, literally take, multiply all these returns. What does that mean? That means if you put a dollar in the fund at the beginning, you'd earn 7%. If you left it there and never took it out, you'd get another 3%. Then you'd get 8% on top of that, then 0%, then 50%.


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[Assistant] Of course this is a multiplicative thing, so he says you'd get 1.07 times 1.03 times 1.08 times 1 times 1.5. Multiplying all that would give you the number of dollars you'd have at the end of five years, given that you put a dollar in in the beginning and never took it out.


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[Assistant] That's the constant rate of return that would have given you the same amount of money at the end that you would get by having left a dollar from the beginning in the fund all the way to the end. That clear to everybody? That seems like a logical thing to do.


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[Assistant] Okay, that's what money hedge funds do do, in fact. That's the number they tell you. Now, why might not that be a great number? Did everyone follow what his suggestion was? So, by the way, his number's gonna come out to be... Well, I don't know. Anyway, it's gonna be some number. We could do that.


[User] Please transcribe this text

[Assistant] Oh, because I'm trying to multiply these, I'm adding instead. Okay, so equals, I'll just have to do it one by one. That, uhm, times the next one.


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[Assistant] Okay, so that's what a dollar would have gotten you if you put it in and kept it to the end, and now he's saying take this to the fifth power.


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[Assistant] Uhm, this 0.2, enter. Okay, so 12.2%. Alright, now why isn't that the right number? Why might you think there should be another number?


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[Assistant] Well, is it grossly inflated, but why is it grossly inflated by that?


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[Assistant] That's true, but you've taken that into account. So the years we didn't do that well, like the 0% return, that got, that brought the average down. So why is that?


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[Assistant] Uh, a problem, exactly. Yeaah, it's averaging the good years with the bad years, so uhm, yeaah.


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[Assistant] So, I didn't do extreme enough numbers, but let's, let's, uh, the crucial thing to take into account is that, uhm,


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[Assistant] Suppose you have a fund that starts off, many funds, like my fund, started off with very little money, but we did it at the right time because we, you know, we knew that was a good time. You're gonna see when the leverage cycle we talked about was at the bottom of the leverage cycle, just like this past year. You know, we're up there.


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[Assistant] So at the bottom of the leverage cycle, you're gonna have a great year. Of course, we hardly had any money because the fund was, uh, just starting. And so we made 50% the first year. Okay? But then everybody said, oh, these guys must be geniuses. And they poured a huge amount of money into our fund. And let's say the next year we did, uh,


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[Assistant] 10%. Actually, we had another great year the second year.


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[Assistant] But let's say that we did ten percent the second year.


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[Assistant] Young man in the front is saying, you know, you made 50% on pennies and then you made 10% on a gigantic amount of money. It isn't right to take, you know, the average of 50% and 10% because almost all the money that you managed you made 10% on, not 50% on. That's his point. Okay? So how would you deal with his point? How could you figure out a way of computing the right


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[Assistant] Return to compensate for the fact that some years you have a lot more money at stake than you have in other years. Take a weighted average. Well how would you take the weighted average? Sounds a little complicated.


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[Assistant] I'm going to now give you the answer, which is a little bit like that. She's saying do some dollar weighted thing. And so, you know, you, you, uh, somehow weight


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[Assistant] The numbers by how much money there was invested for that year. So because there was a lot more money invested in year, you know, three than there was in year one, that number should somehow have a bigger weight. And, uh,


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[Assistant] You know, it's not exactly clear how the weights are going to get in there, but it's obvious that that's something like that you ought to do. So here's the mathematical solution that, uhm, the internal rate of return, or yield, uh, gives. So it says, let's look, every year, let's look at what happened in the fund. So, you know, for, for, you know, if you,


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[Assistant] We shouldn't care about which investor put in which dollar. We care about how the fund managed dollars. The names of the investors don't make any difference. How did the fund manage its money? So the first year, the fund got a hundred dollars.


[User] Please transcribe this text

[Assistant] So, as for producing money, it produced a negative hundred, you know? Money went into the fund.


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[Assistant] That was one guy invested. We don't care who it was. The total that went into the fund was 100. The second year, 1200 went into the fund. Those are the second two guys.


[User] Please transcribe this text

[Assistant] The third year, 110 came out. That was the first guy. The fourth year, 556 came out. That was the second guy.


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[Assistant] The beginning of the fifth year, the third guy took out $155, and then the last year, the second and third guys took out everything that was left. So from the point of view of money creation and use, this is every year, what went into the fund and what came out of the fund, the net inflow out. Okay, so there are a bunch of numbers.


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[Assistant] The question is, so the one number summary is, what rate of return, it's called the internal rate of return


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[Assistant] Which if you discounted all these numbers, what would Fisher say if you discounted all these numbers? If you had to use just one interest rate, Fisher would say, what's the present value of these cash flows? Well, if the interest rate was zero, you just add up all the numbers, and you're going to get a big positive.


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[Assistant] If the interest rate is infinity, then that means that the, the, you know, after the first year you're discounting everything to zero and you're gonna get negative a hundred.


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[Assistant] So you could say what interest rate could you discount all the cash flows at to produce zero? That would be like saying at that interest rate


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[Assistant] All the fund has done is rearranged its money. It's taken money in and put money out but at that same constant interest rate the present value is zero. So it's basically, it's allowed you to trade money across periods at this


[User] Please transcribe this text

[Assistant] Internal rate of return, the interest rate which makes the present value zero. So if you guessed, if you guessed, uh,


[User] Please transcribe this text

[Assistant] So 10% turns out to be the right number, so if you discounted things by 10%, you would take, you see what the formula is, you take the inflow, and you discount it by 1.10 to the first power, and this you discount by 1.0, 1.10 to the second power, and this is what, you know, the inflow,


[User] Please transcribe this text

[Assistant] The net outflow, I guess, which is 110, you discount by the third, you know, 1.10 to the third power, etc. You keep discounting by that.


[User] Please transcribe this text

[Assistant] Okay, and you get all the discounted net flows, you add them all up, and you get practically nothing. Uh, that's adding them up, and this is taking the square, and I use Solver to figure out what the right discount weight was to make all the zeroes. So that's the


[User] Please transcribe this text

[Assistant] That's the simple way of averaging, you know, dollar weighting everything.


[User] Please transcribe this text

[Assistant] You, you, you, you don't care about who the people are. You don't care about, uh, whether one guy's putting money in and another guy's taking it out at the same time. You just care about the net. And if you've got a bunch of net numbers, that's the net outflow.


[User] Please transcribe this text

[Assistant] And you're trying to say what's the one, what's the single rate of return. The idea is to say at what interest rate


[User] Please transcribe this text

[Assistant] If you, if there were a bank paying an interest rate, and all those things discounted gave you zero, that would be like saying the fund is functioning just like a bank. No matter when people put the money in or take it out, they're always getting this rate of return, which is 10%.


[User] Please transcribe this text

[Assistant] Okay, so they're getting a constant 10% rate of return no matter when the money goes in or out, and that's kind of


[User] Please transcribe this text

[Assistant] Gives you a measure of how well the fund is doing.


[User] Please transcribe this text

[Assistant] 10%, which is a different number from 12%.


[User] Please transcribe this text

[Assistant] Before we got the geometric average of 12%. Now, very typically, this is the case. That this internal rate of return is lower than the, you know, dollar return from putting the money in at the beginning.


[User] Please transcribe this text

[Assistant] Assuming the fund doesn't just collapse and go to zero at the end. So why is that? For funds that have survived.


[User] Please transcribe this text

[Assistant] Typically, that number, 12% is higher than 10%. Why would that be? What does that tell you about the world? Oh, someone who hasn't, uh, well, go ahead.


[User] Please transcribe this text

[Assistant] It tells you that, and why, and how is it that the hedge funds, okay, could tell you that. It tells you that the hedge funds are doing better when they have a smaller amount of money than when they have a larger amount of money. Exactly. You've concluded that it's easier to make money when you have a small amount. What's another possible explanation? Yep.


[User] Please transcribe this text

[Assistant] Right, I think that's the reason. The reason is that people pour money into hedge funds just after they've done incredibly well, and they keep pouring money in, and then eventually there's a blow up, and so when the blow up comes, the hedge funds have zillions of dollars, and they're losing a lot of money, losing a lot of money, and then you know, then everybody pulls their money out,


[User] Please transcribe this text

[Assistant] And that's when the cycle is going up and all of a sudden the hedge funds have these huge returns again, but they hardly have any money. So, okay, any other questions about this?


[User] Please transcribe this text

[Assistant] Alright, so internal rate of return is the way to, uh, is a way


[User] Please transcribe this text

[Assistant] We're gonna see that has many shortcomings, but it's a way at getting at the idea, as several of you have said, that you can't just take the geometric average, which is what every hedge fund, like ours, always


[User] Please transcribe this text

[Assistant] What's your geometric average of all your returns?


[User] Please transcribe this text

[Assistant] That's not a good reflection of how the hedge fund's done necessarily because some years the fund had a lot of money to work with.


[User] Please transcribe this text

[Assistant] And so the average dollar didn't do that well. And now there's a question about how should you measure the average dollar? And I've given the internal rate of return. There are actually other formulas you could give. This one isn't, uh, this is the most famous one. Okay? So uhm, alright. Now let's


[User] Please transcribe this text

[Assistant] See how this internal rate of return is used all the time on Wall Street.


[User] Please transcribe this text

[Assistant] By the way, if you're not following what I'm saying, you should, uh, please interrupt me. Okay, so what if you took a bond, a simple coupon bond,


[User] Please transcribe this text

[Assistant] Okay, well a simple coupon bond, what is a simple coupon bond? The yield to maturity of a simple coupon bond.


[User] Please transcribe this text

[Assistant] Lecture called yield. What is a simple coupon bond? A simple coupon bond pays the same coupon every year and then pays the principal in the coupon at its maturity. So the yield to maturity is going to be the price, which is like a negative payment, okay, such that if you


[User] Please transcribe this text

[Assistant] So if you knew the price, suppose you, sorry, suppose you knew the price of this bond, if you knew the price of the bond,