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[Assistant] $1.50. Okay, anybody else have any, uh,


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[Assistant] A little more venturesome than that. You can't do worse than two dollars in this bet. What? Four dollars. Four dollars. Okay.


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[Assistant] Alright, so Bernoulli asked a bunch of people, and the average of what they say happens to have been four dollars. That's what they said on average.


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[Assistant] And so Bordelli said, well this is amazing, the expectation is infinite, and they're only willing to pay me a miserable four dollars for this.


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[Assistant] And uh, you know, they give up their money and they weren't going to get anything back. But let's ignore that temporarily and take it seriously. Bernoulli said the solution must be that people don't care about


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[Assistant] The dollar payoff, they care about the utility of the dollar payoff. So let's put in a utility function. So the utility of the dollar payoff would be u of 2 plus 1 quarter, u of 4 plus 1 eighth, u of 8 plus 1 quarter.


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[Assistant] The more dollars you get, maybe it increases utility, but by less and less. So you're not really gaining much.


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[Assistant] By getting these numbers way out here, they're not adding really much to utility. So you only care about these small numbers. I mean, you know, it's good to have more, but not much better to have more.


[User] Please transcribe this text

[Assistant] So he said, lo and behold, if I put in log natural as my utility function, which this looks like, that's the graph of log natural, okay, and I put this in, and now if you know a little bit of


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[Assistant] Okay, now, you see this is easy to solve, to compute, because log of 2 to the n is n times log of 2. So the log of 2's come out, and it's just the sum, it's log of 2 times the sum 1 over 2, times the sum of 1 over 2 to the n, uhm, times, uh, n. So it's n over 2 to the n. So it's, so this is equal to log of, uh, 2 times the sum


[User] Please transcribe this text

[Assistant] Okay, that's what it turns out that this thing is.


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[Assistant] Okay, so it's not just 1 over 2 to the n, which would have added up to 1, but n over 2 to the n. So the point is, because you have the log function here, this actually equals a log of 4.


[User] Please transcribe this text

[Assistant] So, anyway, I've worked out the arithmetic. It's very simple. You just have to, you all know how to sum 1 over 2 to the n. You probably don't know how to sum n over 2 to the n.


[User] Please transcribe this text

[Assistant] You never thought of doing it before, but the same trick gets you to be able to sum n over 2 to the n.


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[Assistant] It's obviously more than 1 over 2 to the n, in fact it's equal to 2.


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[Assistant] So this sum is equal to 2, and 2 times log of 2 is log of 2 squared, which is log of 4.


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[Assistant] So, by plugging in, instead of caring about the expectation, you care about the expected utility.


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[Assistant] You can explain why the average person was willing to pay $4 because the expected log of this is equal to the log of 4. This is equal to the log of 4. So Bernoulli thought he brilliantly explained


[User] Please transcribe this text

[Assistant] His paradox, so this is the other brother, Daniel, the dead brother, posed the problem, maybe solved the two for all I know, but the one who posed


[User] Please transcribe this text

[Assistant] The other brother, who still lives, came up with the, uh,


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[Assistant] Solution, maybe with his brother, that it's, uh,


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[Assistant] You know, people don't look at expected payoffs, they look at expected utility of payoffs, and the utility should have this con-k feature that more and more payoffs


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[Assistant] I mean, the payoff with expected utility of a con k function, you know, this log wouldn't have really solved the problem because suppose that Bernoulli had offered instead a bet not of 2 to the n but of 2 to the 2 to the n, you know, much more generous bet, okay, then even with logs you would have gotten an infinite number.


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[Assistant] So basically he should have said that you maximize, that people care about


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[Assistant] A concave function of payoff with a function is bounded unlike


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[Assistant] Log, which is not bounded. But anyway, let's leave that aside. Should be a concave function.


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[Assistant] Okay, so to put it another way, a concave function has the property, if you look at it, you know, let's say it looks like that, that if you have this payoff xA, and this payoff xD, and you've got, so this is the utility now,


[User] Please transcribe this text

[Assistant] Of either getting XA or getting XB, you're going to end up with this expected utility. Your utility is going to be half of UXA plus half of UXB. That's what we had down here. Half of this utility plus half of that utility. So nothing else can happen. Okay? But, if you give the person half the amounts of money, for sure, then he gets this utility, which is much bigger than that utility.


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[Assistant] Okay, because this is a concave function. The extra you gain by winning the bet,


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[Assistant] Compared to getting the average for sure, the extra you gain doesn't drive the utility up very much, because it's flattening out. Whereas losing the bet, even though you're losing the same number of dollars, because from here to here is the same as from here to here, the loss of the same number of dollars is more important to you than the gain of an equal amount of dollars.


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[Assistant] So the modern theory of risk aversion, which is to just assume risk aversion in modern economics means people care about expected utility


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[Assistant] I expect a discounted utility where the utility is concave.


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[Assistant] Okay, so whatever utility function we wrote in here, maybe it shouldn't be log, should be something else.


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[Assistant] Whatever that constant utility was, which was log of 4,


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[Assistant] So this random gamble gives this expected utility, which is equivalent to having that for sure. So here's the four.


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[Assistant] Okay, so log, so 4 for sure gives the utility, log of 4, that puts you here, which is the same as the expected utility of getting the random gamble. So that's what, uh,


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[Assistant] Okay, that's the modern theory of risk aversion. And it explains why people would rather have things for sure.


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[Assistant] It's now quantifiable, because if you can't have something for sure...


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[Assistant] If you can't have something for sure, then you know that it's more dangerous, but with this concrete utility function, you can find out exactly how much you're willing to pay to transform, you know, this risky gamble into a safe gamble. You give up this much expectation in order to get the payoff for sure.


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[Assistant] Okay, so we're gonna turn something into a vague theory into something quantifiable and get a surprising conclusion. Okay, so that's step one.


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[Assistant] We now think about people maximizing utility. Well of course we thought about that from the beginning. The very first class you had utility and diminishing margin utility. So actually this risk aversion with diminishing margin utility, fortunately for us,


[User] Please transcribe this text

[Assistant] Is exactly the same thing we've been thinking about all along anyway. Diminishing margin utility for consumption. So the very assumption of diminishing margin utility that we made from the beginning is also explaining risk aversion.


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[Assistant] So it's incredibly fortunate that we don't actually have to change any of our mathematics. And we've explained a new phenomenon. Okay, now the most simple utility function


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[Assistant] Is either the log one or the quadratic. So remember u of x equals, you know, a plus bx minus cx squared.


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[Assistant] Okay, now I'm, you know, adding a constant is never going to change anything, so I'm always going to write this as a x minus one half alpha x squared. Okay, that's going to be my, uh,


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[Assistant] Utility function, my quadratic utility. It could be like this or it could be like that. If you had a constant, you know, which doesn't depend on x, that's not changing what anybody does, so that's irrelevant. And if I divide it by a constant like b, that's not gonna change what everybody does, so I might as well assume the quadratic utility is x minus one half alpha x squared.


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[Assistant] Now why is that such a good, convenient thing for us to use?


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[Assistant] Where with probability gamma 1, you're gonna get x1. Probability gamma 2, you're gonna get x2. Probability gamma s, you're gonna get gamma s.


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[Assistant] Okay, so what's the expected utility? The analog of Bernoulli?


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[Assistant] That means u is going to equal summation s equals 1 to s of gamma f, uhm,


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[Assistant] xs minus one-half alpha xs squared.


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[Assistant] Okay, so that's all we're doing. We're just saying that people don't care about the payoff. They have to evaluate getting x1, x2, or xs.


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[Assistant] They're going to take, they're going to multiply the payoff by the expectation, but not look at the payoff itself, look at the utility of the payoff.


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[Assistant] Now quadratic is very simple, and the reason why we're gonna get such a beautiful theory out of it is because this number, you don't have to keep track of all the x's to express this utility. We're gonna be able to summarize it incredibly simply. This is going to equal some function f of the expectation of x, okay, and the variance of x.


[User] Please transcribe this text

[Assistant] So all we're going to have to worry about is the expectation of X and the variance of X. And so many, many very complicated things we can think about very simply.


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[Assistant] So more generally, if you put the log instead of the quadratic utility, we couldn't get this simplification, and so the theory would have to be more complicated. So the beautiful theory, the capital asset pricing model, comes out of using this simple quadratic utility. So why does it get so simplified? Well, if I just write it out, u is going to equal the summation of gamma s xs, so this


[User] Please transcribe this text

[Assistant] S equals 1 to S. I've let my probabilities be gamma. I don't know why I chose that. Minus 1 half alpha summation S equals 1 to S of gamma S XS squared. Okay?


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[Assistant] Now what is this 1f alpha gamma s xs squared? Well if I wrote xs minus the expectation of x, and I summed this squared, okay, that equals the variance of x. That's the def, oops, times gamma s. That by definition is the variance. But if I wrote this out, what would I get? I get summation s equals 1 to s, gamma


[User] Please transcribe this text

[Assistant] And then I'd have, you know, what would I have, minus 2 times summation gamma S XS expectation of X, okay, plus summation S equals 1.


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[Assistant] To s of gamma s expectation of x squared.


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[Assistant] What, uh, what's this? The second term.


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[Assistant] Minus 2 times that, the expectation of x is a constant.


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[Assistant] So I can take that out, minus 2, expectation of X, and take this out.


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[Assistant] And notice that summation gamma s xs, that's the expectation of x as well. So that's just minus 2 times the expectation of x squared. And so this, I can take the expectation of x squared out.


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[Assistant] The summation of the probabilities is 1, so therefore I just get


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[Assistant] Equal to summation s equals 1 to s gamma s x s squared minus expectation of x squared. So therefore this up here is equal to the expectation of x. Okay, so what have I got here? I've got this term. So I've got the variance of x equals


[User] Please transcribe this text

[Assistant] This summation xsx squared minus expectation of x squared. So this term equals the variance of x plus expectation of x squared.


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[Assistant] So therefore I've got this minus one half alpha expectation of X squared plus, uh, minus one half alpha variance of X.


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[Assistant] That's what the algebra gives me. So why is that again? Because the given quadratic utility up here, that thing,


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[Assistant] This term is obviously the expectation of x, but this term is just the variance of x.


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[Assistant] Okay, plus the expectation of x squared. So I subtract it, I keep the expectation of x minus the expectation of x squared. That's the first term.


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[Assistant] And then I've got minus the variance of x from that term times the 1 half alpha. Okay, so you see that depends on the expectation of x in a positive way, assuming alpha is a small number, and in a negative way on the variance of x. So just as I said, somewhere,


[User] Please transcribe this text

[Assistant] I said it was going to turn out like that and it did, right here. The utility is equal to the expectation of X and the variance of X in a positive way on the expectation of X and a negative way on the variance.


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[Assistant] We're ready to start the analysis. We've said, okay, so far, we've assumed people only care about the expectation.


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[Assistant] And then we said, well, we know they don't really care about the expectations. Hedge funds and everybody else, if they know what they're doing and they're trying to keep their investors happy, they're gonna hedge.


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[Assistant] We didn't say why they're gonna hedge, we just asserted they like to hedge so their investors don't get mad at them.


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[Assistant] But really what we had in mind is the investors have some utility function. They don't like risk.


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[Assistant] So the hedge fund is going to try and keep the payoff steady. But there's a trade-off, you know, you can't eliminate all risks.


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[Assistant] So how much is the hedge fund and the investor gonna suffer if all risk isn't eliminated? Now we have a way of quantifying it. People care about the utility and not about just the expected payoff. And so you add more risk to them. You replace a sure thing with a risky thing with the same expectation. They think it's worse. That much worse.


[User] Please transcribe this text

[Assistant] Okay, and so we said that of all the myriad of utility functions we could use, log, you know, some exponential e to the minus a x


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[Assistant] x to the r. There are lots of different utilities we could use. We're gonna deal with the quadratic because it has the simple property that in evaluating an entire risky proposition, people care about the expectation, which is what they cared about before.


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[Assistant] But they're punishing themselves for getting a bad variance. Okay, so.


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[Assistant] Because that's such a simple thing to say, we're gonna get a simple conclusion and a very surprising conclusion. Okay? So, let's now analyze a problem and see what would happen. Okay, so the problem I'm gonna choose to analyze is this one. I'm gonna say that there are three things that can happen in the economy.


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[Assistant] Uh, the first three, anyway, what are the probabilities?


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[Assistant] Okay, now there are many firms in the economy, A, B, and C.


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[Assistant] And let's say the first form, I don't want to invent the numbers here, so I might as well just write down the ones I picked.


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[Assistant] The first one is going to be 50, 175.


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[Assistant] Okay, so here we go. So by A, alpha and beta, I mean there are a million alpha agents and a million beta agents.


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[Assistant] And you know, they, you know, so everything can be scaled by a million, because I want a bigger con, like we always have. Okay, so there we have it.


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[Assistant] S equals 1, S equals 2, and S equals 3. So those are the three possible states.


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[Assistant] Just like we had before with this payoff, and those are the payoffs of all the assets.


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[Assistant] Alpha ohms, firm A, which is producing that output in the three states.


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[Assistant] It also owns 133.5 units of X0. So over here, alpha is owning 133.5 and beta is owning 65.5 of consumption at time 0. So this is time 0.


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[Assistant] Something is gonna happen. There's a lot of uncertainty between now and then. Some of the firms are gonna be paying off in some of the states and badly in other states and so on.


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[Assistant] Okay, and so the utility function for alpha, he cares about consumption at time zero and also in each of the three states.


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[Assistant] Okay, but now he's gonna have these quadratic utilities. He's gonna say to himself,


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[Assistant] Uhm, I care about my consumption and whatever I'm gonna, if I consumed, if I just hung on to my A in state one, this would be, okay, if I never traded, I just hung on to A, the utility function would be, you know, this quadratic thing of 133.5, so the, uhm,


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[Assistant] Minus 1 over 475 squared times a half.


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[Assistant] That would be his utility if he never traded. If he just stuck to A. He'd eat his own endowment, 133.5 at time zero.


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[Assistant] Oh, that isn't true. I wrote down the wrong utility at time zero. I said it's time zero utility is a half x zero. So it's a half 133.5.


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[Assistant] 100 in state 2, and 75 in state 3, and that's the utility he'd end up with.


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[Assistant] But that's not very good for him because he's running this gigantic risk. He's got this risk at time.


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[Assistant] You know, state one is a disaster for him if he just sticks to that. So he doesn't want to stick to that.


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[Assistant] So how should he evaluate the shares of firm A? How should he evaluate, you know, the shares of firm B, which he could get if he gave up some of A? Or how should he evaluate the firm, you know, shares of firm C? What should he do? Okay, so, beta, as a similar utility, beta's utility, u beta, is going to equal one, three quarters, x zero, plus the summation,


[User] Please transcribe this text

[Assistant] Minus one over eight hundred x x squared.


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[Assistant] They seem to prefer consuming in the future till now, but uhm,


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[Assistant] Okay, that was poor choice of numbers, but let's, you know, this number should be bigger than one, and this should be bigger than one, but anyway, uh, I put a half and three quarters.


[User] Please transcribe this text

[Assistant] Okay, so there's impatience built in, except it goes the wrong way. That was just poor choice of numbers. But the rest of it, uh, uh, uh, you know, expresses their risk aversion.


[User] Please transcribe this text

[Assistant] So alpha is looking at the expected payoff of what he gets to consume in the future, but he's punishing himself by the variance. So you look at this formula, you see it's not just the expectation, but he loses something because of the variance.


[User] Please transcribe this text

[Assistant] Shakespeare's idea of people not liking to be exposed to variance, to uncertainty, which we've quantified by calling variance. Okay, is everyone with me now? Yes, good, I'm glad you have a question.


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[Assistant] So that's the utility of alpha and that's the utility of beta. I could pick any people I wanted, I'd just pick those two people.


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[Assistant] Okay, now how do they differ? Which person is more afraid of risk than the other?


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[Assistant] Alpha is not going to punish himself too much, but he's going to punish himself, you know, somewhat more. So alpha is more risk averse.


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[Assistant] Okay, so I've taken two agents who are afraid of risk. One's more afraid than the other.


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[Assistant] And I put them in an economy where there are risky things that could happen. And so we now have, we now want to work out a more sophisticated version


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[Assistant] If we didn't, you know, if we were naive, you might say...


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[Assistant] It's a quarter, by the way, I hope I have those probabilities right. You'd say it's a quarter times fifty, plus a quarter times a hundred, plus a half times seventy-five. Is that what we would have said up until now?


[User] Please transcribe this text

[Assistant] Even up until now, we would have been more sophisticated than that. Discounted, okay, times discounted.


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[Assistant] Okay, that's what we've sort of figured up until now. That's the logical thing to do. Well, but we ignored risk aversion.


[User] Please transcribe this text

[Assistant] The stock market historically has had a lot higher returns than the bond market, you know. Even with the last stock market crash, of course it came back a lot, you know, average since 1926 the stock market's made something like 9% a year compared to 2.5% in the bond market.


[User] Please transcribe this text

[Assistant] So there's a huge disparity over such a long period of time. It can't just be it was luckier every year after year after year. Somehow people must have realized the stock market's riskier.


[User] Please transcribe this text

[Assistant] And so, as Shakespeare said, they wanted a higher return, meaning they were playing a lower price.


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[Assistant] But how much lower? How can you figure out how much lower? So in this example, in other words, what is the price of A? So this is the wrong price of A, apparently, because it doesn't recognize risk aversion.


[User] Please transcribe this text

[Assistant] Okay, so what would, okay, so that's where we are. So everybody,


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[Assistant] Uhm, that's taking expectation and discounting obviously can't be right because it doesn't recognize risk aversion. On the other hand, we always had a utility function in there from the beginning, even a quadratic one.


[User] Please transcribe this text

[Assistant] In 1951, this is the same guy who proved with Debreu the Pareto efficiency of equilibrium.


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[Assistant] He was my thesis advisor. He said, you know, we can do the same trick that Fisher did.


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[Assistant] So, only he, for some reason he never credited Fisher. I could never quite figure that out.


[User] Please transcribe this text

[Assistant] He had some obscure Danish guy he credited. But anyway, apply Fisher tricks.


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[Assistant] Look for GE, the general equilibrium trading output.


[User] Please transcribe this text

[Assistant] Okay, then go back to find out, then go back to deduce value affirmed.


[User] Please transcribe this text

[Assistant] Okay, now what good are we trading here? That was a conceptual advance. We call them arrow


[User] Please transcribe this text

[Assistant] And then De Bru got involved too. Errol De Bru, De Bru was the Yale assistant professor.


[User] Please transcribe this text

[Assistant] Well, Arrow is a Stanford Assistant Professor, so Arrow-DeBroux State Contingent Commodities.


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[Assistant] Okay, so Justice Fisher said, an apple today and an apple next year, even though they're identical apples, are different commodities with different prices because they come at different places in time. In fact, most people would prefer the apple today to the apple next year. So Arrow said, okay, an apple in the top state,


[User] Please transcribe this text

[Assistant] An apple in the top state is a different commodity from the same apple in the second state. Okay, so it should have a different price.


[User] Please transcribe this text

[Assistant] So we've got to start conventional equilibrium according to Arrow, where as long as you have these Arrow state contingent commodities that you can trade, trading today, you can buy, you can imagine today buying an Apple if state one occurs.


[User] Please transcribe this text

[Assistant] From the apple of state 1 occurs, and also an apple of state 3 occurs, which obviously is going to be more expensive, or it looks like it will be more expensive than the other apples because it's 50% likely to happen.


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[Assistant] And those are the prices we have to look for. And that's going to solve our problem because the prices of the aero securities are going to be different maybe from the probability.


[User] Please transcribe this text

[Assistant] And that's what will reflect the fact that when everybody's trying to hedge and not everybody can do it, you're going to have to change the trade-off.


[User] Please transcribe this text

[Assistant] So we've already seen this in our gambling thing, at least the prices. Remember with our bookies. The bookies were effectively willing


[User] Please transcribe this text

[Assistant] You know, a dollar for the Phillies one. So we've already had these.


[User] Please transcribe this text

[Assistant] Aero contracts, these aero securities implicitly in our equilibrium. And those 60-40 odds, those were the opinions of the bookie.


[User] Please transcribe this text

[Assistant] Betting odds depended on what the other bookies were willing to give. It didn't have to correspond to reality. There might not be a reality even.


[User] Please transcribe this text

[Assistant] So here there's a reality, 25, 25, 50, but that doesn't mean that the odds, the prices that are going to be quoted in the market will turn out to be that. We have to solve for equilibrium and see what they are.


[User] Please transcribe this text

[Assistant] Okay, so what's going to happen? Well, we can solve for equilibrium very easily. We've done this a million times before.


[User] Please transcribe this text

[Assistant] And I've chosen linear quadratic utilities, the kind we did on the very first day of class.


[User] Please transcribe this text

[Assistant] Because those are easiest to solve for equilibrium. You don't have to do get involved in the budget set or anything complicated. You just set margin utility equal to price. So we know for alpha, sorry, we know that the margin utility of alpha times zero divided by the price zero is going to have to equal


[User] Please transcribe this text

[Assistant] The margin utility of alpha at each state s times the price P s.


[User] Please transcribe this text

[Assistant] So the, the equilibrium, the arrow to brew equilibrium


[User] Please transcribe this text

[Assistant] Equilibrium is going to involve P0, P1, P2


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[Assistant] And P3, the prices of the arrow securities, the present value prices. P0 is what you pay today to get the apple today. P1 is what you pay today to get the apple a year from now in state 1. So these are the present value, present value, that's what Fisher would say. State contingent.


[User] Please transcribe this text

[Assistant] Now, you may ask whether there really are these aero securities floating in the economy, and we're going to come back to that question, but you could imagine all these aero-de-brew state contingent


[User] Please transcribe this text

[Assistant] Prices and, and commodities, those would be the prices we'd solve for equilibrium.


[User] Please transcribe this text

[Assistant] This over this, marginal utility of that, okay? So, what is this? And similarly for beta, marginal utility beta at zero over the price of zero equals marginal utility beta s over the price of s.


[User] Please transcribe this text

[Assistant] So what is this? For alpha, his marginal utility of consumption is a half.


[User] Please transcribe this text

[Assistant] At one of the prices is one. Let's take this price to be one. Okay, so beta


[User] Please transcribe this text

[Assistant] Her price is three-quarters, her marginal utility is three-quarters and the price is one. What's his marginal utility? It's in any state s. It's gamma s, gamma s times


[User] Please transcribe this text

[Assistant] I just differentiated this, I got 1 minus 2 over 400 times xs. And what's her marginal utility? It's gamma s and state s times 1 minus 1 over 400 times xs.


[User] Please transcribe this text

[Assistant] Okay, so I know in equilibrium that's going to imply that one half


[User] Please transcribe this text

[Assistant] Well now I have to screw around here.


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[Assistant] What? Over PS. Ah, glad that appeared. I was getting worried there. Thank you. Over PS. Okay, that helps a lot. So that implies that something like x


[User] Please transcribe this text

[Assistant] So this is what alpha is going to do and this is what beta is going to do. So this implies x alpha f equals what? So if I multiply through by 200


[User] Please transcribe this text

[Assistant] And I bring PS over Gamma S to the other side and I do a bunch of stuff. I'm gonna guess this is 200 minus 100, uh, PS over Gamma S.


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[Assistant] Then I multiply everything through by 200. So I get 100 ps over gamma s. And then I get the 200 here.


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[Assistant] So, and the xs goes to the other side and the ps over gamma s goes to the other side. So 200 minus 100 ps over gamma s. And this one is gonna be x beta s.


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[Assistant] Is going to equal, well I have to do the same trick here, except I'm going to be multiplying through by 400.


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[Assistant] And taking three quarters, which is three hundredths, will be three hundredths minus


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[Assistant] Minus 300 over PS over gamut.


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[Assistant] Okay, because if I multiply through by 400, put PS over gamma S on the other side, I have three quarters PS over gamma S.


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[Assistant] If I could figure out what the prices are, I know what everybody would demand in every state. Okay, so.


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[Assistant] Okay, so let me pause here. That was the first critical step. So what did I do? I said the long story, a lot of years went into this.


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[Assistant] I said people are risk averse. Shakespeare knew that. We want to quantify it. So we say people have concave utility functions. That quantifies risk aversion. We want to make a simple concave utility function. We pick quadratic.


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[Assistant] Okay, but of course we don't know what quadratic, different people could have different quadratic utility functions.


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[Assistant] Then we do the Fisher trick, and say that any equilibrium, as long as you can buy and sell every contingent commodity in the future, because all the arrow securities are there, it can always be reduced to general equilibrium, just like we did before. And so now, you have to feed the endowment into the agent's, or the payoff of the dividends into the agent's endowment. Okay, uhm, but here, so we haven't done that yet, and then we solve for supply equals demand.


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[Assistant] So all we have to do is we have to have X alpha S plus X beta S has to equal the endowment of alpha in S plus the endowment of beta in S. Okay, but this equals, alright, so we have to do that for every S. So this is, uh, 200 minus 100 PS over gamma S equals, uhm,


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[Assistant] Alright, so it equals whatever they are. So what is endowment of S?


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[Assistant] Endowment of alpha and s plus endowment beta events. We have to look at each state separately.


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[Assistant] And lo and behold, I picked the numbers so that if you add these all together, you get 500.


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[Assistant] And here you get 280 and 220 is also 500. And here you get 500 again. So lo and behold,


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[Assistant] There is no aggregate risk in the economy. Although the individual stocks are risky, the aggregate is totally unrisky. So I could put, no matter what S is, I could put in 500 here.


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[Assistant] It's gonna turn out that the total endowment of both people, because I've plugged the dividends into their personal endowments, added up to two people. It's 500. So it means that P.S.


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[Assistant] Uhm, minus 400 P S over gamma S equals 500.


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[Assistant] Because 500 from here is 100, put the 400 on the other side, divide by it, I get PS over gamma S equals a quarter for all S. Okay, so what did I find? I found that in this


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[Assistant] And so every, so it's the same. P s over gamma s is the same. Okay? Same in all states.


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[Assistant] Okay, so what would the price of A be here? What's the price of A in equilibrium?


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[Assistant] What's the price of A? I'm going to take, the price of A should be, price of A should be


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[Assistant] Well, P1 is just a quarter times


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[Assistant] Gamma 1, right? P1 over gamma 1 is a quarter.


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[Assistant] P2 is a quarter times gamma 2, and P3 is a quarter times gamma 3, so I just got this, multiplied by a quarter.


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[Assistant] So in fact all I did is I did what I had always done. I took the expected payoff and discounted it. The discount rate is a quarter. What's the price of the riskless asset? Pi of one, one, one is just going to be a quarter.


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[Assistant] Because it's a quarter times gamma one plus a quarter times gamma two plus a quarter times gamma three. Gamma one plus gamma two plus gamma three is one. So it's a quarter. So it implies the riskless interest rate is what?


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[Assistant] 300%, okay, and that's why the dis- so we're discounting by one over a quarter because the interest rate is 300%.


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[Assistant] Her, forgot who was her and who was him. Let's say her exposed her to a lot of risk.


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[Assistant] She's not gonna sit there stupidly just holding A. She's gonna trade it for B and C for different shares. In fact, she's gonna end up holding her consumption, this is her consumption, 200 minus 100 PS over gamut. This number doesn't depend on S. She's gonna consume the same thing in every state.


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[Assistant] And how can she do that? She can own equal shares of A, B, and C. She'll own a share of the whole economy. So in other words, by diversifying, by diversifying,


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[Assistant] Same thing. She doesn't just hold her A, she mixes B and C with it.


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[Assistant] So that you get the payoff of consumption that's exactly the same in every state. Because PS over GMS is independent of the state. She'll always consume the same thing. Everybody can hedge perfectly and there's no problem because there's no aggregate risk that anyone has to be stuck with. And therefore the price is just going to be the same as the probabilities, you know, discounted.


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[Assistant] And that's the theory we've worked with so far. So, so far you could say that everything we did with kosher, it's just that when we had these two different probabilities of things happening up or down,


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[Assistant] We thought that the aggregate economy would have the same endowments here as it did there and therefore the probabilities we used were the objective probabilities discounted. No reason to change them because nobody's going to be forced not to hedge. Everybody will hedge.


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[Assistant] I can't have said it as clearly as I ought to have. So would somebody like to say something? Yes.


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[Assistant] This is the new price. With the one quarter, this is the correct new price.


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[Assistant] So the theory so far hasn't changed in any interesting way. We just found the discount rate. It just looks like expected utility. But you shouldn't have expected it to change because the aggregate endowment was 500, the constant in every state. Nobody, there's no reason why we can't have everybody


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[Assistant] Any, before I do that, I'm gonna change the endowments so they're not all a constant. Any questions about where we're going? Yep.


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[Assistant] Yes, what I proved, okay, so thank you for the question. Okay, so I went a little quickly. I said that what we proved by solving for the general equilibrium


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[Assistant] Is that the price in every state was just going to be a quarter times the probability.


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[Assistant] Okay, that's what we showed had to happen in, in equilibrium. Now what's the consequence of that? The consequences are twofold. Number one, the price of all the assets is the same expectation we naively would have taken before where we used the discount rate a quarter. Okay, so that, and that's the first implication.


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[Assistant] The second implication is that from the formula for consumption, we notice that she consumes the same amount in all three states.


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[Assistant] So then I took a little bit of a leap and I interpreted that conclusion that her consumption doesn't depend on the state. What's the interpretation of that?


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[Assistant] She has obviously somewhere behind the scenes given up some of her A to get B and C and held them in a mixture, uh, so as to get the same consumption in every state. What must the mixture be? Obviously she held


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[Assistant] So she diversified, she didn't just stick with her A, she substituted


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[Assistant] A little bit of A, a little bit of B, and a little bit of C, a different boat on every ocean, and now she runs no risk at all. So she diversified, but in the language we used last time, I could call diversification hedging if I wanted to, she just sort of sold aero securities in the right proportions to turn her A into something that was completely riskless.


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[Assistant] Okay, so whether you call it diversifying or call it hedging, she's achieved the same end of totally balancing her consumption, and everybody, he did the same thing, and they both could do it because the aggregate consumption was a constant.


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[Assistant] In real life? Okay, the closest we've come to an arrow security in real life is a CDF.


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[Assistant] And this is part of the reason why these economists, Larry Summers, my classmate,


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[Assistant] And Ruben, who, you know, who, uh, was the Secretary of the Treasury and ran Citicorp and who was a Yale Law School student and a Harvard undergraduate and who I sat on many committees with, uhm, he, they were seduced by the, okay, so what's a CDS? A CDS pays a dollar if some, uh, bond defaults by a dollar.


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[Assistant] Okay, so that isn't an arrow security because an arrow security is a much more detailed thing. An arrow security says, I'll pay a dollar in state one. An arrow security says, you get an apple in state one. But state one, remember, is not described by a single firm. State one


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[Assistant] The states of nature are total descriptions of everything that could happen in the economy.


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[Assistant] If it stops snowing in Siberia, if, uh, you know,


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[Assistant] Khamenei loses power in Iran if, uh, there's a...


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[Assistant] Favourable election outcome in Afghanistan. And, uh, if, uh, Obama wins re-election, and if, uh, the U.S. solves the energy problem, then I'll give you a dollar. Okay? So the Aerosecurity lists incredible number of contingent things, every contingency possible, and says, in that case, I'll give you a dollar.


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[Assistant] A CDS says, if this thing happens, I'll give you a dollar, whether or not Obama wins the election, whether or not America discovers a new source of energy, whether or not Afghanistan turns around. You know, just as long as the bond defaults, I'll give you a dollar. So the CDS is an event contingent. It's an event contingent. A lot of event contingent security, that's the CDS.


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[Assistant] And an arrow security is a much more finely specified thing. It's a state contingent. You say everything that happens in the economy. So we'll never get to arrow securities. But CDS looks like we're on the way to them. And these guys blundered by thinking since CDSs are on the way to arrow securities,


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[Assistant] We should have as many CDSs and let people trade as much of them as we can.


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[Assistant] But we're gonna get to that in the last lecture about how all that was a, you know, how all this theory, what's wrong with all the theory. But, okay, so any other questions before we...


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[Assistant] Make the change that he suggested up there. Let's now change the economy just a little bit. Let's eliminate C. Okay, so, so this just disappears. Okay, so obviously now the total endowment is very contingent. It's 200, it's 280, and it's, what is it, 440.


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[Assistant] Okay, so now what's that equilibrium going to be? What do you think is going to happen? So we're going to, we want to quantify this. We want to give a beautiful simple theory that's quantifiable. But what do you anticipate happening to P1, P2, and P3?


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[Assistant] Alpha, she's going to say, look, my A is risky. I don't want to hold my risky thing. I want to start hedging and trading these aero securities so I get the same constant in every state. Of course, Beta, Beta, who owns B, he's going to do the same thing. So they're both going to be trying to trade aero securities. What's going to happen, do you think?


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[Assistant] Yeah, there's no way that they can exactly, there's no way that each of them can be perfectly hedged. So no matter what they do, they're gonna be exposed to more risk and state, you know, state three is gonna be a great state, state one is gonna be a terrible state.


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[Assistant] So what do you think that means about the prices? So there's no way to hedge, everybody can't be hedged.


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[Assistant] And so in fact what will happen is nobody will be hedged. Okay, although alpha


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[Assistant] Uhm, so beta will end up bearing more of the risk than alpha. And what do you think will happen to the prices of the arrow security relative to the probability?


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[Assistant] There won't be a constant ratio of a quarter as we had before, but can you be more specific?


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[Assistant] Exactly. So that's what's gonna turn out. Because everybody, the world is short of commodities in state one, there just aren't many apples. That's the disaster. That's when we, we can't solve the energy crisis. We're totally screwed. Okay?


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[Assistant] People, everybody wants to consume more in that state. Everyone's gonna try and hedge against that state. They're all gonna be trying to buy aero securities in that state, which means that because there aren't as many to buy, there's just not enough apples to go around, the price of aero securities in state one is gonna be high relative to state three where everybody's, you know, you know, there's plenty to go around there.


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[Assistant] So she is gonna sell some of her A and get some B, okay, to diversify, but B's got so good in state three that all of a sudden she's not gonna be, you know, so worried about state three anymore, but state one she's still gonna be worried about, and there's nothing to be done about that, so the price is gonna have to be very expensive in state one.


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[Assistant] Okay, so alright, that's all blah, blah, blah. Let's solve for equilibrium and see what happens. We can solve immediately.


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[Assistant] Still depends on P0, P1, P2, and P3, but now we have to be a little bit more careful. In state 1, so demand in every state is 600 minus 400 P1 over gamma 1 equals endowment of alpha plus endowment of beta.


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[Assistant] My handwriting, it looked like a 1 anyway. What's the aggregate endowment in state 1?


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[Assistant] The aggregate endowment in state 1 is 200. This is a 1 now. Okay, that's 200. Okay, so that means P1 over gamma 1.


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[Assistant] So, you're not discounting the first state at all. You're looking at the probability of it.


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[Assistant] So what if I go to P2 over gamma 2? The demand is going to be the same.


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[Assistant] It's the price that's gonna change to make up for the fact that the supply is much different. Namely, namely what? 280.


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[Assistant] And then P3 over gamma 3.


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[Assistant] Okay, equal, now the outcome is, uh, 440. So if I subtract 440 from this, I get 160 divided by 400.


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[Assistant] What's that? Two-fifths, thank you. P3, P3 over gamma 3 equals two-fifths. Okay, so the prices turned out to be quite different. Now, the reason why


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[Assistant] But, more interesting is that the prices are no longer proportional to the probabilities. Just as he said over there, the price in state 1 is going to be much higher relative to the probability, namely 100% of it.


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[Assistant] That's the conclusion. So now what do we do for our price? What's the price of A?


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[Assistant] What's the price of A? What do I plug in here? That. Okay, so that equals one-quarter times fifty plus, uhm, four-fifths times a quarter, four-fifths times a quarter, four-fifths times a quarter, one-fifths times a hundred plus P3 was, uh, two-fifths times a half plus one-fifths


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[Assistant] Times 75. Okay, which equals something.


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[Assistant] Twenty, thirty-five and twelve and a half, forty-seven and a half.


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[Assistant] Why did I what? Okay, so what is P1? P1 is equal to gamma 1. And gamma 1 is a quarter. So that's how I got a quarter here. So that's 1 times a quarter. Say P2 was two-fifth, was three-fifth. Now what was P2? Maybe I did it wrong anyway. P2 was four-fifth times a quarter.


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[Assistant] Times a quarter, okay, which is equal to one-fifth, and P3 was two-fifths times a half.


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[Assistant] Okay, which is equal to one-fifth. That's how I got the prices, okay? So.


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[Assistant] Alright, so you see that things change, and we've captured the idea that people can't hedge.


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[Assistant] Fully by making the price of the air security in the state where the economy is worse off much, uh,


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[Assistant] So we haven't gotten close to the punchline, sorry. I got... Okay, so we're about, yeaah.


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[Assistant] Okay, so the prices, the two things happen to the prices.


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[Assistant] And that's why, relative to the probability, the price is much higher than the third state. You agree with that, right?


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[Assistant] Okay, but there's a second effect, which is that all these numbers, one, four-fifths, and two-fifths, they're bigger than the quarter, quarter, quarter we had before, but that's obvious, that's because we wiped out the future, you know, half the endowment in the future disappeared, so naturally people are willing to pay more for the future because they're poorer there.


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[Assistant] That's what we, in the first day of class, we said that the interest rate, or you know, the third week, the interest rate, according to Fisher, would go down if you got poorer in the future.


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[Assistant] So that's part of the reason that's happened. By the way, what is the riskless rate of interest?


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[Assistant] So P1 plus P2 plus P3 equals what now? It's equal to, uhm, a quarter plus a fifth plus a fifth. So a quarter plus a fifth plus a fifth. These are the prices. A quarter plus a fifth plus a fifth, and that's equal to


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[Assistant] Okay, so that's 5 over 7 over 10.


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[Assistant] So therefore the interest rate, 1 plus r equals 10 over 7.


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[Assistant] So r equals three-sevenths, okay, which is like forty percent. Okay, so the interest rate went from three-hundred percent to forty percent.


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[Assistant] But that's because we lost all this future consumption. But that's not what I'm concentrating on. Fisher would have already known that. What I'm concentrating on is the fact that the prices are no longer proportional to the probabilities. You're discounting every probability but adjusting the probability because people are much more worried about the first state than the third state.


[User] Please transcribe this text

[Assistant] Not A, they're more worried about the first state. The firms are A and B. The states are 1, 2, and 3. So they're much more worried about the first state where the payoff is 200 than they are about the third state where the payoff total dividends in the economy are 440.


[User] Please transcribe this text

[Assistant] Oh boy. That sounded so unconvincing. So let me just say, like I want to say the punchline. So I've got three more minutes to go. There are two punchlines which are


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[Assistant] Okay, I haven't gotten to the stunning conclusion. So far I've said stuff which is, uh, you know, which Arrow and De Bruy had already figured out. But now I want to go to the thing that Tobin and Markowitz figured out. Okay, which is one more step we haven't noticed yet. Arrow has already figured out that because not everybody could hedge, that means that the


[User] Please transcribe this text

[Assistant] The price of a narrow security is not exactly equal to the probability. It's relatively high if the economy is poor, like in state one, and relatively lower if the economy is rich, like in state three. Okay, that's common sense.


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[Assistant] Now what's not common sense is the extraordinary conclusion I'm about to show you. Let's look at what the consumption is, the final consumption of this, uhm, these two people.


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[Assistant] Okay, so if we look at the final consumption of these two people, what's her final consumption?


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[Assistant] Okay, and the last step was 200 minus 100 times, uhm, two-fifths. Okay, two-fifths is 20, so this is 180. Okay, that's hers.


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[Assistant] Consumption in the future, I'll put a twill, I haven't talked about x zero yet, is uh, 400 minus 300 times 1, okay, which equals 100.


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[Assistant] And here it's 400 minus 300 times four-fifth, okay, which is equal to, hell, four-fifth of 300 is 160, okay, and here it's 400 minus six, no, 400 minus 300 times two-fifth, which is equal to, uhm, one-twenty, two-two-eighty. Is this right?


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[Assistant] 80 is 120, thank you. Okay. So these are all right now?


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[Assistant] 180 should be okay. This is, this is 40, so this should be 160. Thank you. That's it. Okay, great.


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[Assistant] Okay, so now, what's so shocking about those numbers that I finally got them right? Thank you. What's shocking is, uh, this


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[Assistant] So let's say you take a quarter of this, okay, so the difference is, let's take a quarter of that. That's 50, 70. Okay, that's 50, 70 and 110. Okay, so a quarter of this plus


[User] Please transcribe this text

[Assistant] If you add to that 150, you're gonna get all these numbers. So this person, alpha A, I claim, just holds 150 as a riskless bond, pays 150, 150, plus 50, 70, and 110. That's, uh, no.


[User] Please transcribe this text

[Assistant] 100, 120, and 160, that's the right number, and that's equal to 50, 50 of the bond plus


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[Assistant] Uhm, a quarter of this thing, so 50 plus 50 is, okay, a quarter of this is 50, 70 and 110, right? So if you hold 50 of the bond,


[User] Please transcribe this text

[Assistant] 3 quarters of the aggregate endowment plus minus 50 of the bond. So 3 quarters of the aggregate endowment, 3 quarters of this thing.


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[Assistant] So what they've done in equilibrium is everybody, despite having a million stocks to choose from and thousands of states and all that stuff, what everybody does is hold the riskless bond, puts money in the bank, and holds the whole stock market. So the first theorem we're going to prove next time is called the mutual fund theorem.


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[Assistant] Which is that everybody diversifies. Everybody diversifies.


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[Assistant] Uh, the, the aggregate economy...


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[Assistant] Okay, so that theorem of Shakespeare of diversifying, what does it amount to do? We have a very concrete thing. You hold 10%, this person's holding 50, uh, 25% of every stock in the whole economy, plus, uh, lending, putting some money, $50 in the bank.


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[Assistant] The other person is doing three quarters of every stock in the whole economy plus lending the money to the first person. So that's the first of the two amazing results and I'll start next time by explaining it. Okay, so you're going to see it's going to be very concrete.


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[Assistant] We've taken a long view so far in looking at overall shape, size, and capacity of the string section.


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[Assistant] Now it's time to focus in on the details and build our understanding of strings from the ground up. The best starting point is in looking at the features of their construction, which all these instruments share.


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[Assistant] This body's shape and size differs from instrument to instrument depending on many factors. The main design principles are the size and pitch of the strings, the intended volume and projection, the tone, and the aesthetics.


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[Assistant] The older family of veils had deeper bodies that were actually more accurate in size for the length of strings.


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[Assistant] But that depth of body also swallowed the upper harmonics.


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[Assistant] The unique voice of the violin comes from a shallower body.


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[Assistant] favoring higher overtones that contribute to a stronger, brighter tone.


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[Assistant] This trait is common to violas and cellos as well, and even the double bass has a degree of redesign that's influenced by this principle.


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[Assistant] The body is shaped in two lobes which contribute to the resonance of the higher and lower tones.


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[Assistant] These lobes are divided by a narrower waist, allowing greater freedom of motion for bowing the top and bottom strings. There are also two F-holes that allow the sound to escape, and the characteristic decorative points which really don't make much difference to the tone whatsoever.


[User] Please transcribe this text

[Assistant] The top and bottom pieces, called the soundboard and the back, are carefully carved and varnished to favor a certain type of sound, then glued to the separating walls called the ribs. The exact proportions of these elements can make all the difference between a violin that's worth ten dollars and one that costs a million.


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[Assistant] The other major structural feature of orchestral string instruments is the neck, rising out of the smaller end of the body.


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[Assistant] The neck is topped with a fingerboard over which four strings are strung, starting with a peg box at the top and ending with a tail piece at the bottom.


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[Assistant] The peg box is the basic tuning mechanism for the instrument, using tension alone to secure the pegs into the peg holes, though double basses have mechanical tuning pegs.


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[Assistant] The strings wind around these pegs, then rest on the nut, a small ridge that elevates the string slightly above the level of the fingerboard.


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[Assistant] At the other end of the instrument, the strings rest across a bridge, which helps spread them slightly and transmit their vibration down to the sounding board.


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[Assistant] The bridge also forms a curve that allows the bow to play one string at a time.


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[Assistant] Finally, the strings end in the tailpiece, which secures them to the bottom edge of the body.


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[Assistant] Tail pieces often have little tuners in them to help the players make minute adjustments to the tuning of the strings.


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[Assistant] A shoulder rest is also commonly used for comfort and to elevate the instrument an inch or two.


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[Assistant] All these design features added together equal a violin, viola, cello, or double bass.


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[Assistant] But that's just the beginning. How string instruments are played is just as important as how they're built. Let's examine that in the next few videos.


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[Assistant] Tire Town School, Episode 2, New Friends.


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[Assistant] Honk honk, Poppy tooted her horn. Eddie laughed hard.


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[Assistant] Eddie, said Poppy, meet Dipper and Jeb. I'm an ice cream truck, said Dipper.


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[Assistant] Hey, Eddie, said Jeb. Want to carry this dirt? No, thanks, said Eddie. I don't like to carry dirt. Everyone looks surprised. What do you like to do? asked Dipper.


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[Assistant] Eddie grinned. I like to race.


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[Assistant] Modesty and conscientiousness receive their reward only in novels and life. They are exploited and then shoved aside.


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[Assistant] He who wants to hold on to that is losing.


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[Assistant] The first person you think of in the morning and the last person you think at night is either the cause of your happiness or the cause of your pain.


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[Assistant] I feel like I am among people who are going to live forever. At least that is how they act.


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[Assistant] They are so obsessed with money that they have forgotten about life.


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[Assistant] The best way to lose a woman was to show her a kind of life that one could offer her for only a few days.


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[Assistant] What you can't get always seems better than what you have. This includes the romance and idiocy of human life.


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[Assistant] Only when you finally part with a person, you begin to be truly interested in everything that concerns him. This is one of the paradoxes of love.


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[Assistant] It is a mistake to assume that all people have the same ability to feel.


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[Assistant] A person can become close to you in three days, and the one who lives near you for years may never know what your favorite color is.


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[Assistant] It is very queer that the unhappiness of the world is so often brought on by small men.


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[Assistant] We are forlorn like children and experienced like old men. We are crude and sorrowful and superficial. We are lost.


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[Assistant] Keep things at arm's length. If you let anything come too near, you want to hold on to it. And there is nothing a man can hold on to.


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[Assistant] We have so much to say, and we shall never say it.


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[Assistant] It was a melancholy secret that reality can arouse desires but never satisfy them.


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[Assistant] Nobody ever taught us how to light a cigarette in a storm of rain, nor how a fire could be made with wet wood, nor that it is best to stick a bayonet in the belly because there it doesn't get jammed as it does in the ribs.


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[Assistant] If a woman saves money on herself, it means that it makes men want to save money on her.


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[Assistant] Everyone saves someone at least once, just as he kills someone at least once, even though he may not know it.


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[Assistant] But every soldier believes in chance and trusts his luck.


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[Assistant] What comfort there is in the skin of someone you love.


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[Assistant] Good or ill, life is life. You only realize that when you have to risk it.


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[Assistant] Human beings are a much worse poison than schnapps or tobacco.


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[Assistant] Okay, so time to start. So let me begin by reviewing a little bit of the dynamic hedging, which I could have


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[Assistant] Knows that the probability the Yankees are going to win the World Series is 60%.


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[Assistant] And therefore, a game, each game of the World Series is 60% and knows that if he, the probability the Phillies win the game is 40%.


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[Assistant] So suppose that he finds somebody who's willing to bet with him on the Phillies.


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[Assistant] So he can win $100, if the Yankees win he has to pay $100 if the Yankees lose. Now his expected payoff is 20.


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[Assistant] He should take the bet, because if the person's willing to bet at even odds, the Phillies fan's willing to bet at even odds, and he knows the odds are 60-40, on expectation he's gonna make 20.


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[Assistant] Even though he's done the smart thing, he could still end up losing $100, which could be a disaster for him. So what he would like to do is to hedge his bet. Now what does it mean to hedge his bet?


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[Assistant] Well, suppose there was another bookie who was willing to bet


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[Assistant] He should bet with the other bookie $80, uh, he'd be willing to give up $80 if the Yankees won in order to win $120 if the Phillies win. And that's a fair bet according to the other bookie because it's 60-40 odds.


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[Assistant] 60% of this and 40% of that are, you know, this is 3 to 2 odds, so it's a, it's a fair bet.


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[Assistant] So in other words, he makes money by taking advantage of the Philly fan to bet on the Yankees.


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[Assistant] He hedges his bet, that's where the expression came from. He hedges that bet by betting in the opposite direction on, betting on the Phillies with his bookie in the opposite direction, but, you know, standing to lose less than he, you know,


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[Assistant] He's betting in this proportion, in this amount, so that he gets 20 no matter what. So he's locked in his $20 profit. Okay, so that was where we began.


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[Assistant] Now let's just think about it a little bit more carefully than I did before. Somebody in fact basically asked this question.


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[Assistant] Is he really making money because he understands better the odds of the Yankees winning the game? Is he making money because he knows the odds are 60-40 and the poor Philly fan thinks they're 50-50? The answer is no, that's not why he's making money.


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[Assistant] It doesn't matter what the odds are of the Yankees winning. He's winning because he's managed to arbitrage


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[Assistant] So what is an arbitrage? An arbitrage is you find two bets, uh, which are more or less the same thing,


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[Assistant] Uh, in fact they're exactly the same thing, but trade at a different price. Okay, so what are the, uh,


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[Assistant] What are the two bets? Well, there's the Philly bet, the Philly fan bet.


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[Assistant] Okay, so, if he's gonna make 20 for sure, so he could make the 20 at the beginning just by borrowing against his winnings, then he'd have 20 at the very beginning, and his payoff would, uh, you know, would be


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[Assistant] Uh, the payoff from the Philly fan bet would be exactly


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[Assistant] He'd owe the twenty, so he'd have minus a hundred, you know, so this bet is, uh, he'd get twenty at the beginning.


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[Assistant] He'd be able to get twenty at the beginning, but he'd have to pay minus a hundred and a hundred.


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[Assistant] He'd pay a hundred dollars if the Yankees won and get a hundred dollars if the Phillies won.


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[Assistant] Okay, and so that way he'd get, he'd just cancel his bet at the end and end up with $20 for sure. Okay, so just to say it again,


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[Assistant] And, and it's the presence of the other trader who's really giving him his profit opportunity, not his superior knowledge.


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[Assistant] And the fact that he has two different traders. And so why does that give him an opportunity? Because he can arrange a bet with the Philly fan and a bet with the, uh, bookie, the other bookie, but combined with, uh, you know, borrowing the money so that in the end he's, uh,


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[Assistant] He has nothing at the end. So whatever losses he got from the Phillies, he made up from the bookie bet. Whatever losses he had from the bookie bet, he made up for it with the Phillies fan bet.


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[Assistant] So arbitrage relies on replication. Somehow, through some


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[Assistant] Other bat, he was able to hedge his other bat, he was able to hedge his filly's bat.


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[Assistant] Uhm, by replicating the bet with somebody else.


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[Assistant] Yet for a different price at the beginning. The Philly fan didn't demand any money up front to make this 100 minus 100 bet with him.


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[Assistant] And, and with the, uh, bookie, he's able to get twenty dollars up front and make this, uh, you know, the opposite bet, the minus hundred hundred bet.


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[Assistant] Okay, so by getting more money up front and doing the opposite bet with the bookie, he can end up completely cancelling the Philly bet. Philly fan bet. And that's how he makes his profit. Okay, so uhm,


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[Assistant] That was the first idea. So what have I clarified? What I've clarified is that it wasn't his superior knowledge, it was the presence of two different traders that enabled him to make his profit.


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[Assistant] And I've clarified the method that he made the profit. The profit wasn't just kind of mixing in another bet. It was really cleverly arranging to undo totally the original bet.


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[Assistant] Okay, so he got actually zero in the end and ended up with twenty dollars at the beginning. So, let's just see now what you would do in a dynamic situation.


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[Assistant] Was willing to bet him a hundred dollars on the Phillies. So he gets a hundred of the Yankees win, he pays the fan a hundred of the Phillies win.


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[Assistant] So what could he do, since he knows he's gonna make $20 for sure in the future, he could just borrow it at the beginning, if the interest rate is zero, have $20 at the beginning, end up with zero, no matter what the uncertainty is, and put the $20 in his pocket.


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[Assistant] So that's another way of saying the same thing, which is that he could, by combining the bets, put money in his pocket at the beginning, and no matter what happens in the future, he doesn't have to give anybody any more money.


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[Assistant] So that's what he's done. That's what an arbitrage is. You put money in your pocket and in the future, in the present and in the future, you never have to face a problem. You've always got


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[Assistant] Okay, now in a dynamic situation, you can do the same thing. Okay, so I'm just reinterpreting what we did before a little bit. So in the dynamic situation, what are you doing? You've got, the Philly fan now is betting on the whole world series. Even though you know the odds are 60-40.


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[Assistant] So by backward induction, we did this calculation to say your expected payoff is $42.


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[Assistant] So, let's suppose now, but now the key is that you're making money not just because you know better what the odds are, you're making money because there's another bookie standing there, always ready to trade with you at 60-40 odds in any game.


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[Assistant] So the presence of the second bookie, even if you knew nothing, would enable you to make money, in this case, lock in a profit for sure of $42.


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[Assistant] Now how do you know that you can lock in a profit per sure of $42? Well, it's now done by dynamic hedging. You have to...


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[Assistant] So it seems almost hopeless to figure out what to do, you know, but I'm betting on the whole series.


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[Assistant] You know, many different things can happen, you know, the Phillies could win the first game, the Yankees could win the first game, lots of things could happen.


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[Assistant] By the end, I want to make sure I've locked in my profit for sure.


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[Assistant] How can I do that? Well the key is to think about marking to market.


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[Assistant] If you were reporting to your investors, you're the...


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[Assistant] You are a hedge fund. You're reporting to your investors how you're doing. You should be telling them after the first game, you know, at the beginning you're telling them, look, I'm going to make $42. I expect to make $42. After the Yankees win the first game, looks like you're going to make 60, you know, now things are better. You expect to make $64 on average.


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[Assistant] But if the Yankees lost the first game you'd only expect to make nine dollars on average. So after the first game you go from


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[Assistant] So, if you were telling your investors properly how you stood after the very first game,


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[Assistant] You might have to tell them you've lost 33 of the 42 dollars that you told them before you were gonna make.


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[Assistant] So you'd already be taking, you know, be running a risk. You really only care, maybe, about


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[Assistant] Getting money for sure by the end, but if you're doing proper accounting, you should be worried all the way along whether you're making or losing money. And that's the key to how to


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[Assistant] Uh, have no risk by the end. You just run no risk at all the in-between stages.


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[Assistant] So the night before the first game, you, you bet with the Philly fan on the entire series, which is, you know, giving you a huge advantage, but you're willing to bet with the other bookie


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[Assistant] The other bookie we're assuming is only standing by game by game, you're betting 33 to 22 on the Phillies. You're betting on the team you expect to lose. You're betting on that team with the other bookie, but in a smaller quantity. So this is gonna mean that your profit at the end of the first day


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[Assistant] So you can guarantee, you can lock in 42 after every game. And so as I said last time, the crucial thing is


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[Assistant] So then, so then one thing I didn't say, so one thing to, uh, which I forgot to observe last time, which I think is interesting, is


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[Assistant] So is that what you should do? Should you do that after every game? Always bet 33 to 22.


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[Assistant] Uh, you know, against the Yankees with the other bookie? Well the answer is no. Okay, what should you do if the Yankees win the first game?


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[Assistant] By winning the first game, your expectation goes up to 64. Now, it could be either 82


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[Assistant] Or 30, but, but after, but so, and the night after the first game, you're much better off than you were before. Of course, you laid off part of the bet with the other bookies. You had to pay off 22, so you're still only 42 ahead.


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[Assistant] But the Yankees have now won the first game, so it's now the night after the first game, before the second game. You're subject to more risk. What should you do now?


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[Assistant] You should go back to the book, you make another bet on the second game. But should the other bet be 22 to 33? Absolutely not, because 64...


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[Assistant] What should you do now? So I've done some rounding here, which is going to screw everything up a little bit. But now what should you do here? After the first game, you're here. Here you are after the first game.


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[Assistant] With the bookie the first game, you bet 33 against 22. Now what should you do the second game? What should your bet be with that bookie?


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[Assistant] He's still willing to give you three to two odds on the Yankees. Either direction. What should you do?


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[Assistant] You're not gonna bet 33-22 again, you're gonna make a different bet. And what should it be? Yes.


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[Assistant] Rounding up, she conveniently rounded a little bit because I had rounded a little bit.


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[Assistant] So the book is still willing to do that for you, and you're locking in the 64, because it's gonna be 64 again.


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[Assistant] In the first step, you made a bet to lock this number in. In the second step, you made a bet to lock this number in. You're always locking in where you were before.


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[Assistant] So that bet is changing each time, but you're always locking in what you were before, because after the Yankees have won a game now, you know, the next game is, uh, not quite as important. You know, the time is running out on the Phillies, so it's not as, uh, you know, revealing the next game about what's gonna happen. So, uhm,


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[Assistant] You make a smaller bet up here than you did before.


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[Assistant] Okay, so you just keep proceeding down the tree, locking in where you were before, and so by induction you've locked in 42.


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[Assistant] So that's what you're doing, and the bet is changing every time. So what do you think the bet is here, by the way? The bet here, you could go from nine, you could go from nine to, uhm,


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[Assistant] 36 or minus 32. So what should you do here?


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[Assistant] What's the bet that you're gonna make with the bookie after the Phillies win the first game?


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[Assistant] Okay, and that's, you know, three to two odds with some rounding, right? Thirteen and a half times three is forty and a half. So still, fair odds.


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[Assistant] And therefore you've locked in 42 all the way along. You know, it sounds like we're locking in 9, but we're not locking in 9, because remember the first day you won 33 dollars from the bookie when the Phillies won the first game. So you've locked in the 9, but don't forget the 33 from before.


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[Assistant] Okay, so that's dynamic hedging. You, you, you, uh, have a complicated bet along with, that depends on a lot of things along the way, but you bet each time along the way, maybe changing your bet each time, depending on what's happened so far, and that way you can lock in whatever profit you had. So that's the thing that sports, uh,


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[Assistant] Bookies have known for a long time, you know, the expression hedging your bet comes from that.


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[Assistant] Okay, but remember the crucial, okay, so the, the step,


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[Assistant] That I didn't emphasize very much is that although you're, you know, you seem to be locking in where you were before, there's a slightly more abstract way of saying that.


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[Assistant] That at the end, you're gonna get 42 dollars, uh, for sure. So of course you could borrow all the way back from the beginning and get 42 dollars immediately and then


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[Assistant] Never owe anything in the future, because whatever bet you were winning or losing with the Philly fan, you were making up for with your myriad bets with the bookie.


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[Assistant] Okay, so it's a replication argument.


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[Assistant] That by betting sequentially with the bookie a different amount, you're undoing the single long bet you had with the Philly fan.


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[Assistant] Okay, so now let's move from sports, as we did last time, to, uhm, the market. So here we had the prepayment spreadsheet. And so we had, remember, this was a, this was a bond.


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[Assistant] A mortgage, a 30-year mortgage, 8% mortgage.


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[Assistant] Uh, so the interest rates could follow this process. There you see it.


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[Assistant] They start at 6%, but they can go up or down by 20% basically every period.


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[Assistant] So you assume that everybody knows this and knows that the probabilities are 50-50. That's like saying, you know in advance, there's a bookie out there, that is the market, that's willing to give you, no matter what day it is, 50-50 odds of the interest rate going up or down from then on. That's just like the bookie standing by.


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[Assistant] Now this time though, you're gonna have to, okay, so that's how, so you now are going to take advantage of the fact that you can anticipate that the market will stand there ready, uh, to bet with you.


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[Assistant] Now if the whole market freezes or something, you can be totally screwed. You're doing your complicated dynamic, you know, betting with the market.


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[Assistant] Okay, so you're relying on the market, so it's not surprising that when the market freezes up, lots of people suddenly are exposed to more risk than they were before. Alright, now, let's, uh, let's suppose that


[User] Please transcribe this text

[Assistant] So now in the market, you're not going to be able to find these pure arbitrages. In the sports example, it was a pure arbitrage. There was a Philly fan who was betting at even odds, and a bookie who was betting at three to two odds.


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[Assistant] Now, you shouldn't expect that to happen. I mean, by what miracle did this fan come to you instead of to the other bookie?


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[Assistant] You know, why do you deserve to meet this fan and take advantage of him?


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[Assistant] What have you got that no other bookie has? It's a miracle he came to you and not to one of the other hundred bookies. Well, it's probably because you actually know something that the other bookies don't that you can take advantage of. So in the background here, there's another thing that you knew. You knew the fan was going to pay off if he lost.


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[Assistant] Maybe the other bookies weren't willing to bet with him because they didn't think he was going to pay off if he lost. Okay, so that's


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[Assistant] The extra thing that you knew, you knew he was gonna pay off. Now it makes sense that he came to you and not to the other bookies because you actually knew something that they don't. Okay, and so that's how it really works.


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[Assistant] I'm assuming that there's a bookie, that the rest of the market is standing by, ready, as I said, to trade, make interest rate bets with you. Okay, with everybody. At even odds, you know, whether the market, whether the


[User] Please transcribe this text

[Assistant] Interest rates gonna go up or down. So no matter where you find yourself in the future, like here,


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[Assistant] Someone's willing to bet at 50-50 odds that it'll go up or down.


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[Assistant] Now this is a year-long bet. You're going to have to take into account the interest rate. It's not just going to be 50-50, you know.


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[Assistant] You have to discount all the cash flows by the interest rate, but it's basically a 50-50 bet discounted by the interest rate.


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[Assistant] What's new here? Well you don't know what the cash flows are.


[User] Please transcribe this text

[Assistant] Okay, so here's the extra piece of information you're adding. You know that the mortgage homeowner is rational and is going to prepay rationally. So you know what these cash flows are. You know, some of the time the mortgage homeowner is going to pay the $8 COs, some of the time he's just going to pay $8


[User] Please transcribe this text

[Assistant] In fact, here's what you know is gonna happen. Down here, where the ones are, he's gonna pay his coupon of 8, pay off the remaining principal, and that's the end of it. So in fact, the very first day, if interest rates went down, the rational thing to do is to pay off the entire mortgage. Okay, so you know that's gonna happen. Now, that's the extra information you have. You know what the cash flows are. If other people knew those cash flows,


[User] Please transcribe this text

[Assistant] They would figure out the price of the mortgage too. It's just they don't know those. You know the cash flows and you can now figure out that the price better be 98 because if it's


[User] Please transcribe this text

[Assistant] Why? If there's someone else willing to trade with you to buy that mortgage for more than 98.


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[Assistant] Okay, like, why might that happen? Because someone else might have the erroneous opinion that the whole world will never


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[Assistant] Call the mortgage. We'll never prepay because he's just going to be asleep all the time. So let's say


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[Assistant] Let's say it's a, the homeowner's a she. Let's say the homeowner's are all she's. This other trader thinks


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[Assistant] Sorry, we're here. You know the cash flow. You know how to value the bond. You know it's worth 98.


[User] Please transcribe this text

[Assistant] Okay, but you can, so, and let's say the other guy's willing to pay you 120 for it. So you can sell it short for, uh, you can sell it to the guy short for 120. That means you owe whatever cash flows the mortgage pays. Okay, and you know what it's gonna pay down there. So,


[User] Please transcribe this text

[Assistant] How can you lock in your profit? Well because with $98 at the very beginning, you can replicate those cash flows. You can go out into the market and buy those cash flows, the very cash flows you know the mortgage is going to make. How can you do that? Well because it's the first step


[User] Please transcribe this text

[Assistant] The interest rate can go up or down. If the interest rate goes up, the payment is 8, that's the coupon, plus 92 something, 92.6, and if it goes down, the payment is 8.


[User] Please transcribe this text

[Assistant] Okay, so what is this? The payment is actually 8. That's the cash flow that has to be made. The mortgage wasn't called here at the beginning, so no matter what,


[User] Please transcribe this text

[Assistant] She does, the homeowner, she's gotta pay the 8. So you know you're gonna owe the guy 8 the first period. Now, down here, she's actually gonna...


[User] Please transcribe this text

[Assistant] Up here, she's not going to prepay, she's going to hang on to the mortgage. So what is this 92.6? This is the present value, assuming those cash flows, of her future payments. So if you get $98,


[User] Please transcribe this text

[Assistant] So 1 half times this divided by 1.06 plus this thing, remember, is 107.11 divided by 1.06 times a half. That's how we got the number 98. 98.8, whatever it was.


[User] Please transcribe this text

[Assistant] That's how you calculated 98.8 by backward induction. So this value,


[User] Please transcribe this text

[Assistant] Times a half discounted plus this value times a half discounted. That's how you got 98.8. So therefore, if people are willing to trade at these prices, in other words, what's the price here? It's a half times


[User] Please transcribe this text

[Assistant] You can buy a dollar here at a cost of one half times 1.06. You can buy a dollar here at a cost of a half.


[User] Please transcribe this text

[Assistant] Paying this amount at the beginning will buy you a dollar here. Paying this amount at the beginning will buy you a dollar here. So by paying this amount, you can get this payoff, 102.6.


[User] Please transcribe this text

[Assistant] 100.6, sorry, 100.6. You can get this 100.6 and you can get 107.11. Because, precisely, this is the cost that you have to pay for each dollar up here, and this is the cost


[User] Please transcribe this text

[Assistant] Uh, you know, we'll be willing to promise you a dollar here in exchange for this amount of dollars here. It'll be willing to promise you a dollar here in exchange for this amount of dollars here. So with this amount of dollars, you can buy $8 and have 92.6 left over and you can also buy


[User] Please transcribe this text

[Assistant] Okay, she's prepaying, just as we said, at the very first step.


[User] Please transcribe this text

[Assistant] If things went up, you'd have to pay 8, okay, which you've also bought, but you've got 92.6 left over, and that's gonna be enough to buy everything that comes after that, because the next day, you're gonna have to buy more stuff, and so you just keep buying going forward.


[User] Please transcribe this text

[Assistant] So with 98.8 dollars you can buy all the future cash flows and you sold the things so you can make all your payments that you're obliged to by selling it short and you received 120 to begin with. So you locked in 21.2 dollars. Okay, so that's the gist of the whole thing.


[User] Please transcribe this text

[Assistant] A little bit better this time, still not brilliantly, but you get, now get it.


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[Assistant] Okay, so I'll say the punchline again. The punchline is that


[User] Please transcribe this text

[Assistant] If you can find a favorite, so here, what was different about the World Series and the mortgage example? In the World Series, you took advantage of the fact that there was a bookie and a Philly fan


[User] Please transcribe this text

[Assistant] Willing to bet at different odds on the same thing.


[User] Please transcribe this text

[Assistant] And so, even though the Philly fan had a very complicated, you know, seven day bet, and the Yankee, the bookies, the other bookies were only willing to make day by day bets, you could take advantage of that disparity to lock in a profit for sure, and that's what you should try to do. In the market,


[User] Please transcribe this text

[Assistant] You're probably relying on something else. You're relying on the fact that you know more about the cash flows than the rest of the traders do.


[User] Please transcribe this text

[Assistant] The rest of the market does, maybe because you understand pre-payments better than they do.


[User] Please transcribe this text

[Assistant] Okay, and so that's why I said it was so important to, you know, model prepayments. So let's say you understand prepayments better than they do, so you know what the cash flows are going to be much better than the market does. You can take advantage


[User] Please transcribe this text

[Assistant] Okay, so you're taking advantage of, you need now this advantage.


[User] Please transcribe this text

[Assistant] Just like in the World Series, you might have needed the advantage of knowing the silly fan would actually pay off if he lost. So there's that advantage that you have. And now you have to rely on the fact that the market or the bookie from then on, day by day, year by year, is willing to make bets.


[User] Please transcribe this text

[Assistant] So therefore you've locked in your profit, because you can anticipate, assuming the market stays there willing to trade at even odds, you'll always be able to produce the cash flow with the $98 you started with.


[User] Please transcribe this text

[Assistant] To exactly meet all your payment obligations you've incurred by selling the mortgage short. And since you sold it at the beginning for $120, you've made $21 at the beginning and you'll never need those dollars, you just put them in your pocket. Okay, so that's the, uh, Okay, so that's how to dynamically hedge a winning position to eliminate the chance of loss.


[User] Please transcribe this text

[Assistant] Of course you still bear a chance of loss if you're wrong about the prepayment.


[User] Please transcribe this text

[Assistant] You're going to lose money, and if the rest of the market suddenly stops being there to make these year by year bets on which way interest rates are going to go, you'll also be subject to a lot of risk. So you're relying on those two things. The whole market not collapsing, and you're being right about prepayments.


[User] Please transcribe this text

[Assistant] The blind luck of 100 minus 100 from the roll of the dice of the fillies, you got rid of that luck, but you still had to count on the bookie being there day after day after day, and on the fact, yeaah, and in the mortgage case on the fact that you knew how to predict the prepayment. Yes?


[User] Please transcribe this text

[Assistant] Okay, I never made 42 on the first bet, but here at the beginning, nothing's happened yet. The series hasn't been played yet.


[User] Please transcribe this text

[Assistant] So I say to myself, I've made this $100 bet with the Philly fan for the whole series on who's gonna win the series. That's gonna happen seven days later. So right now, I can anticipate on average I'll make $42. Some of the time I'll lose $100, some of the time I'll win $100, but on average I'll come out $42 ahead.


[User] Please transcribe this text

[Assistant] But, so I've done a smart thing betting with the naive Philly fan.


[User] Please transcribe this text

[Assistant] But I'm still potentially screwed. I could still lose a hundred dollars. I'll still have to face my investors and tell them I've lost all your money. They'll, you know...


[User] Please transcribe this text

[Assistant] My inclination. I bet not on the Phillies. I bet, I mean, not on the Yankees, who I think are gonna win. I'm betting on the Phillies. Cause I'm undoing part of the gamble that I did with this guy. I'm hedging that gamble.


[User] Please transcribe this text

[Assistant] But by doing that in a clever way, changing how much money I bet each game. So the night before the first game, I bet $33 to $22 on the Phillies winning.


[User] Please transcribe this text

[Assistant] After the Yankees win the first game, I bet $27 to $18 again on the Phillies winning.


[User] Please transcribe this text

[Assistant] Okay, so I decreased my bet going up, I increased it going down, so by doing these clever game by game bets, I locked in $42 for sure. Yes.


[User] Please transcribe this text

[Assistant] You have locked it in, so how could that be? So the good question, but you've overlooked something. So her question is,


[User] Please transcribe this text

[Assistant] You know, I'm, the game's gonna end after seven days. What if I ended with a minus a hundred? You know, I ended, the Phillies won all four games. Doesn't it look like I've lost? So why didn't I lose?