[User] Please transcribe this text

[Assistant] Starting from here, the price of consum- if you want to buy


[User] Please transcribe this text

[Assistant] At this point, consumption at this point, you're gonna have to pay more than a half to get it.


[User] Please transcribe this text

[Assistant] You see app, the land here is producing an entire apple, and on average going forward, you know, it's gonna be producing one apple. The geometric average of all these numbers is one starting at this point. Once you've gone down to here,


[User] Please transcribe this text

[Assistant] Production has deteriorated. The land is only producing half an apple here, and in the future, instead of producing twelve, it's gonna produce three apples here and only three quarters of an apple. So it's crumbier land down here. So you're gonna pay less for the land. Okay, any other questions? But I'm gonna tell you exactly what you'll pay for the land. So we're gonna solve it. So need to solve.


[User] Please transcribe this text

[Assistant] First of all, you notice that the arrow securities are always the key. Once you solve for those, you can figure out everything else.


[User] Please transcribe this text

[Assistant] So you just have to solve for these. The price of the up and down arrow securities in every state.


[User] Please transcribe this text

[Assistant] So, what do you mean, what do I mean by that? I mean, here, how many apples would you pay to get an apple here? I'm guessing that that's the same amount you would pay here to get an apple here. Okay, and what you would pay an apple


[User] Please transcribe this text

[Assistant] At this point, to get an apple down here is going to be the same as this generation would pay to get an apple down here.


[User] Please transcribe this text

[Assistant] It seems like everything's sort of homogeneous, and so guess.


[User] Please transcribe this text

[Assistant] Land is proportional to dividends.


[User] Please transcribe this text

[Assistant] Okay, well the whole thing is sort of homogeneous and everything, so, you know, the land here is four times as good as the land here. The land here is four times as good as the land here.


[User] Please transcribe this text

[Assistant] Because the dividends are always four times higher if you go up one thing. So this is four times that, that's four times that. So, why not guess that the price of land is four times here than it is here, four times higher here than it is here? Sounds possible anyway, and of course these are going to turn out to be correct guesses. So once you make these guesses, it's going to be very simple to find the equilibrium, and then we can verify that the guesses are correct. Okay, so,


[User] Please transcribe this text

[Assistant] What does this guy want to do here? So assume you only work when you're young. So what's your income? What's anybody's income? Like this young guy here. Young at time, at time one. So this is time one here.


[User] Please transcribe this text

[Assistant] Time two, time three, time four. So yeaah, time one, their income


[User] Please transcribe this text

[Assistant] They want to consume when they're young, their utility, remember, so what's their budget set? Their budget set is they can consume y, that's consuming here, or they can consume here or here. Okay, but, so they can consume a p, uh, up times z up plus p down times z down. That's what they can do with zero security.


[User] Please transcribe this text

[Assistant] Plus one quarter log z down. That's what they're trying to do.


[User] Please transcribe this text

[Assistant] Okay, so now what do we have to do? We have to clear this market and this market, given what the young are going to do in the next period.


[User] Please transcribe this text

[Assistant] But let's see what this guy's gonna do when he's young. Everybody's gonna face a similar, basically a scaled up version of this problem. So this is what the guy's doing here.


[User] Please transcribe this text

[Assistant] He's got an income of three, and there's the price of up and down arrow securities. That's going to tell him what to consume here, and what to consume planned to consume up here and down here.


[User] Please transcribe this text

[Assistant] So the utility function is the same one as this, I just multiplied through by a half. Right? So I put a half, a quarter, a quarter.


[User] Please transcribe this text

[Assistant] To get that. I just multiplied it all by a constant.


[User] Please transcribe this text

[Assistant] Okay, so now the coefficients add up to one, a half, a quarter, and a quarter.


[User] Please transcribe this text

[Assistant] It's exactly the same. He's gonna maximize the same utility function.


[User] Please transcribe this text

[Assistant] And his budget constraint is the prices of the air security for the same. The only difference is going to be that he's only got three halves here.


[User] Please transcribe this text

[Assistant] Okay, so they're pretty similar problems. So let's see what the guy at time one is going to do. That's going to imply that y equals


[User] Please transcribe this text

[Assistant] Cobb-Douglas, a half times his income, three, divided by the price of y, which is just one, so half of three over one, which is three-halves.


[User] Please transcribe this text

[Assistant] Okay, and what's he going to do when he's old? Z up is going to be, well his cob dog is a quarter. Okay, and then his income is still three.


[User] Please transcribe this text

[Assistant] And then the price is P up of the arrow security. And in the down case, he's gonna have a quarter times three, that's his income, divided by P down. Okay?


[User] Please transcribe this text

[Assistant] So to clear this market, what do we have to do? We have to clear the market by doing what? Z up, that's this guy.


[User] Please transcribe this text

[Assistant] So one quarter, so this is the old guy, times three, divided by P up, that's what this guy wants to consume up here, plus


[User] Please transcribe this text

[Assistant] What the young are gonna do there, but we know what the young are gonna do here. This guy, this young person, is doing exactly what this guy did, except instead of having a starting wealth of three, is a starting wealth of six.


[User] Please transcribe this text

[Assistant] So instead of consuming young 3 halves, he's going to consume young 6 halves. So this is going to be plus 3.


[User] Please transcribe this text

[Assistant] He's gonna consume half of his wages, just like this guy consumed half of his wages when he was young.


[User] Please transcribe this text

[Assistant] So this old, this new young guy, the young guys are always gonna consume in the first period half of what their wages are. That's what this equation tells us. If this is your budget set, this is your income, this is what your budget set is, and this is what you're maximizing, this is a very special Cobb-Douglas case where you have endowments only in one good.


[User] Please transcribe this text

[Assistant] So it's clear that in that good, you'll always consume the Cobb-Douglas coefficient fraction of it. A half of that till three halves. And this guy's gonna consume a half of six, which is three.


[User] Please transcribe this text

[Assistant] So the old up here are gonna consume that number, the young are gonna consume three, and what's the total that's available for them to consume?


[User] Please transcribe this text

[Assistant] The young got six apples and the land paid two apples in dividends. So the total number of apples is eight.


[User] Please transcribe this text

[Assistant] Okay, because if you put 3 over 20 in here, you get the 3's cancel.


[User] Please transcribe this text

[Assistant] And you get 20 over 4, which is 5, and 5 plus 3 is 8. So it's 3 over 20. Now meanwhile, what are we gonna have in the up state? In the up state, down state, that was the up state. In the down state, uhm,


[User] Please transcribe this text

[Assistant] In the down state, the guy's gonna do 1, 4 to 3 over pd. Okay, what is this other guy gonna do? The young person here is gonna consume what?


[User] Please transcribe this text

[Assistant] So in the down state, what are we going to have? The equation of the down state.


[User] Please transcribe this text

[Assistant] Well his income, his wages are three halves, and he's gonna consume half of that, so it's three quarters.


[User] Please transcribe this text

[Assistant] Alright, because when you're young, you always end up consuming half of your, of your wages. So three quarters, and that has to equal what's the total.


[User] Please transcribe this text

[Assistant] So, this arrow security was a half, and the probabilities are a half half, but the prices, the arrow security prices are going to be less than a half and more than a half, forgot what they were already, three twentieths.


[User] Please transcribe this text

[Assistant] Those are the arrow prices. By the way, you notice they don't add up to one. So what's the interest rate?


[User] Please transcribe this text

[Assistant] If you want to get a riskless bond pays one in each of these two states. So how much is that worth?


[User] Please transcribe this text

[Assistant] So therefore 1 plus r equals 4 thirds, so r is 33 percent. Okay, so that's what we did. We've got the probabilities 3 20ths and 3 fifths.


[User] Please transcribe this text

[Assistant] Just as we thought, the price of the down arrow security is gonna be much bigger than the price of the up arrow security, even though objectively they have 50-50 probability. And why is that? Because everyone who's rational is gonna realize that they have to plan to consume much more up here than down here. But they're not gonna wanna do that unless the price is much cheaper.


[User] Please transcribe this text

[Assistant] Okay, so that's how we got the price. Now what's the price of land? That's the only last thing we have to figure out. So the price of land. So the price of land at time one equals what?


[User] Please transcribe this text

[Assistant] If you get the downstate, you get a dividend, so the downstate has, is worth three-fifths to you. Right? And then what do you get? What's your dividend? A half.


[User] Please transcribe this text

[Assistant] Okay, so that's what the price of land is at once. It depends, of course, on what you can sell the price of land for in the future. But we made a guess here that the price of land is just proportional to the price of the, uh, dividend.


[User] Please transcribe this text

[Assistant] So whatever the price of land is here when the dividend is one, it should be twice as high here and half as high here. So therefore this, instead of looking at it as the price of land went up,


[User] Please transcribe this text

[Assistant] It's gonna be the price, twice the price of land was at the very beginning.


[User] Please transcribe this text

[Assistant] So it's just 2 times the price of land at the very beginning, times that same price.


[User] Please transcribe this text

[Assistant] Price of land is down, the dividend's half, so the land is sort of half as good from here on out as it was in the beginning. So let's guess the price of land is half of what it was at the beginning.


[User] Please transcribe this text

[Assistant] Okay, but now you see we've just got P land in terms of other stuff and so we can solve for it. So the price of land is going to equal, well now it's complicated. So 3 tenths plus 3 tenths times the price of land plus, uhm, 3 tenths plus 3 tenths times the price of land.


[User] Please transcribe this text

[Assistant] Minus six-tenths of four-tenths the price of land.


[User] Please transcribe this text

[Assistant] Uhm, so let's just review what we did, okay? And so, I don't think it's that complicated a story. The story is that you've got land which has risky dividends. So, instead of having one


[User] Please transcribe this text

[Assistant] Straight line of what can happen in the future. We've got this infinitely expanding tree. So it's very hard to imagine solving it, but then we've built in so much special assumptions. We've got all this homogeneity, and that allows us to solve it because it's the same problem getting repeated over and over again, just like we did before. So the price of land turns out to be three halves. So what is this guy going to do?


[User] Please transcribe this text

[Assistant] He's going to take his land, his income, which is three, eat half of it, which is three halves, that's what we said he always does when he's young, spend half the consumption, half his income on consumption, the rest of his income, three halves, he used to buy all the land.


[User] Please transcribe this text

[Assistant] Now with the land, he gets a payoff of two, okay, but he gets to sell the land now for double the price he bought it, namely for three. So two plus three is five. And that, by the way, the price of up was three-twentieth, so this is five. Right, because if I put in three-twentieth up here,


[User] Please transcribe this text

[Assistant] If that's the arrow price of up, then Z up is gonna be 5. And so that's exactly how we got it. With the land, he gets the dividend of 2, plus he sells the land for 3. That's 5, so he does consume 5. Z down, incidentally, the price of down was 3 fifths. So the consumption of down, what's the consumption of down gonna be?


[User] Please transcribe this text

[Assistant] Five-fourths. Thank you. That'll be five-fourths. So let's see if that's going to work out too. So the guy buys the land, he gets his dividend of a half, and then he sells the land.


[User] Please transcribe this text

[Assistant] For half of this price, which is three quarters, and a half plus three quarters is five fourths. Exactly what you're supposed to have. You see it works out that everybody's doing exactly what they're supposed to and all the markets are clearing. Anyhow.


[User] Please transcribe this text

[Assistant] The point is not so much the numbers, the point is the following, that sure enough, just as we said before,


[User] Please transcribe this text

[Assistant] The price of arrow securities are going to be expensive when things are going down and cheap when things are going up. The upshot of that is the return on, on


[User] Please transcribe this text

[Assistant] Stocks, like land, is going to be very high. You put in three halves.


[User] Please transcribe this text

[Assistant] Okay, what happens with the three halves? With probability a half,


[User] Please transcribe this text

[Assistant] That's what you put in. With probability a half, you get 2 plus 3 or 5, you get 2 plus you get to sell the land for 3, which is 5. And with the other probability a half,


[User] Please transcribe this text

[Assistant] You get to, the dividend is a half, plus you sell the land for three quarters, that's what you're eating, five and five fourths, so that's equal to, you know, five halves plus, uhm, plus five fourths, plus five, I don't know what this is, five fourths


[User] Please transcribe this text

[Assistant] What is this? 1 half plus 3 quarters is 5 fourths times a half, 5 eighths, okay, divided by 3 halves. So this is a very high number, so what is this? 4 times that, that's 25 over 8, 25 over 8 divided by 3 over 2, which is, uh, is what?


[User] Please transcribe this text

[Assistant] Right? So your return is 100%. Whatever you, whatever you put in, you more than doubled your money in expectation. Whereas the interest rate, we said, was 33 and a third percent.


[User] Please transcribe this text

[Assistant] Okay, so of course, you're getting a tremendous return by putting your money into capital, because that's the whole point of the theory. It's risky, the whole economy's gonna, you know, be rich or poor together, so they're gonna, the price of capital on the margin is a very risky thing to do, it just adds to this


[User] Please transcribe this text

[Assistant] That's all I wanted to do. I just solved it out concretely, so you can see very clearly that you get a high return in the good state, and uh, you get a high return on capital, much higher than the riskless rate of interest. So there we are, we're back at the beginning now, we're George Bush, we can see that somewhere in the middle here, things are really disastrous.


[User] Please transcribe this text

[Assistant] These people who are getting social security, which means they're getting a fixed amount here and here, are doing really badly, at least in expected return, compared to what they would do in the stock market. Okay? And so, what should we do about that? And is it true that, uhm, just because stocks make a higher return, it's, uh, we should privatize?


[User] Please transcribe this text

[Assistant] Okay, so George Bush is saying, and many Republicans say, that we should


[User] Please transcribe this text

[Assistant] We should let people, you know, get them invested in stocks as a higher expected return. What do we think about that?


[User] Please transcribe this text

[Assistant] This is a real policy issue here. We've got this problem in America right now. We've got this huge deficit.


[User] Please transcribe this text

[Assistant] In Social Security, you know, the rate of return has gotten really poor for your generation. You're looking forward to not making very much money. And now we know, theoretically, confirming what we've seen in the past, stocks are making higher returns than bonds.


[User] Please transcribe this text

[Assistant] And Social Security is paying people a fraction of the wages of the young and, uh, you know,


[User] Please transcribe this text

[Assistant] So what do we, you know, so, let me remind you the argument. So the issue has come up because social security is running out of money. So it came up because of this problem.


[User] Please transcribe this text

[Assistant] You know, there's gonna be a deficit. If we continue to pay people the same rules that we've used up until now, we're not gonna be able to afford to do it much longer. So we're gonna have to lower what we pay the, the people going forward.


[User] Please transcribe this text

[Assistant] And the reason, and because we're gonna have to make changes like that, that's why the whole issue of social security has come up. And we've now, you know, deeper thinkers are questioning the whole idea of social security. So let me just remind you that America was ahead of everyone in the world when we created social security under Roosevelt. Nobody had anything like it.


[User] Please transcribe this text

[Assistant] And now, everybody's facing the same problem we are, which we know why you have to face it. You're giving away stuff to the people at the beginning. And so, they're facing it by changing Social Security.


[User] Please transcribe this text

[Assistant] In some crazy way or another, and we haven't changed anything at all, but obviously we have to, and the question is how should we change it and what's the right thing to do.


[User] Please transcribe this text

[Assistant] And just to remind you again of the issues, the Democrats and Republicans seem to be totally, uh, opposed to each other. The Democrats say, oh, we want to, you know, continue to give people, uh, who have done poorly in their lives a better deal on Social Security because we want to, uh,


[User] Please transcribe this text

[Assistant] You know, we want to redistribute. We want to help the poor. The Republicans, actually some of them agree with that, but some of them don't agree with that. But anyway, even the ones who agree, that's not the most important thing they think of in Social Security.


[User] Please transcribe this text

[Assistant] Then the Democrats say Social Security is so important because, you know, we're sharing risks across generations. We don't want


[User] Please transcribe this text

[Assistant] Uh, you know, if the next generation has poor wages, so they end up down here with bad wages, you know, the economy takes a turn south with bad wages, then the old people should suffer along with them. And if the young do better with high wages because things went up,


[User] Please transcribe this text

[Assistant] Then the old should also get better social security dividends, and that's what the current situation promises. Okay? The uhm,


[User] Please transcribe this text

[Assistant] And then another important thing with Democrats, you don't have an opportunity to make a mistake. If you invest in stocks and stuff like that, you know, and the stock market suddenly collapses, you're not gonna lose money in Social Security because Social Security depends on the wage of the next generation and that moves much slower than the stock.


[User] Please transcribe this text

[Assistant] Okay, the Republicans on the other hand say, you know, this is just terrible. Nobody knows what their property rights are in Social Security. No one actually understands how much money they're getting when they're old. They just know now it's gonna be bad.


[User] Please transcribe this text

[Assistant] We have to, you know, delay benefits or something to put it back into balance, and we're gonna take away what you thought you were gonna get.


[User] Please transcribe this text

[Assistant] Okay, really what they want to know is what's the value of my social security account and I want it to be mine.


[User] Please transcribe this text

[Assistant] And then, uh, I can also then tell what the redistribution is. You know, if I know what I'm paying in taxes and I know the value of the benefits I'm getting, I can see that my taxes are much more than my benefits.


[User] Please transcribe this text

[Assistant] And if I happen to be a nice guy and think it's a good idea to have redistribution, fine. But if I'm a bad, you know, not a bad guy, if I just don't believe in that much redistribution, I should be aware of it and have the chance to vote against it.


[User] Please transcribe this text

[Assistant] Okay, and also a key thing is these Republicans say people should get equity-like returns. Not the wages of the next generation. Those grow slowly. Stocks might have a much higher return on average. People should get equity-like returns. So let's put them in the stock market. And then Republicans think choice is always great.


[User] Please transcribe this text

[Assistant] You know, even if people make mistakes, it's their own mistakes. So let them make mistakes.


[User] Please transcribe this text

[Assistant] Tell you my plan of RECSO. Everyone's got the problem straight. We're running out of money.


[User] Please transcribe this text

[Assistant] And the philosophies of the two groups seem to be totally opposed, so what should we do about it?


[User] Please transcribe this text

[Assistant] Alright, so my plan in five minutes, because I want to end with Black Should, my plan in five minutes is, uhm,


[User] Please transcribe this text

[Assistant] Okay, to make a few observations. First of all, yes it's true that you get a higher return on stocks than you get on bonds, but you face a higher risk.


[User] Please transcribe this text

[Assistant] So it's not like you get it for free, it's a, you know, it's a bigger risk. And so, uhm, okay, so that's number one.


[User] Please transcribe this text

[Assistant] So, you, you, actually, point zero is you still got the problem because you gave money to all those people in the 40s and the 50s and the 60s and the 70s. That money's gone, we have to, you know, we've essentially borrowed to give them money and somebody's got to pay back that debt. And so there's no way around that. There's this huge,


[User] Please transcribe this text

[Assistant] 17 trillion dollar debt hanging out there that we have to pay back and you can't get around that. So putting the money into equities or wherever you put the money, there's still that tremendous debt.


[User] Please transcribe this text

[Assistant] Okay, so that's the first thing. So the next thing is, uhm,


[User] Please transcribe this text

[Assistant] Maybe it's not such a big difference, you know, this equities and not equities, because notice in this model,


[User] Please transcribe this text

[Assistant] You know, when the stock is paying higher, when the stock dividends are higher too, instead of one, the price of stocks goes up. That's a great return from the stock market. But the wages were also higher here as well.


[User] Please transcribe this text

[Assistant] So in fact, I claim that in the long run, and it's amazing that Republicans haven't noticed this, in the long run, you know, over 30 or 40 years, it's obvious that the stock market and wages are correlated. I mean, you know, if the stock market collapses,


[User] Please transcribe this text

[Assistant] You know, that's disaster for America, and you know, you can be sure the wages are going to go down too. If our stock market's booming and America's incredibly successful, it's a sure thing that in 40 years, if the stock market is just booming along, wages are going to be higher as well.


[User] Please transcribe this text

[Assistant] So if you get the wages of the generation 30 years from now, you know, in the Democratic plan, you're actually getting equity-like returns.


[User] Please transcribe this text

[Assistant] But you're getting an advantage, which is that the wages move slowly. So although the stocks are bouncing up and down, so somebody who retires in the year 2007, if he held stock and sold it when he retired, he could get a huge pension, whereas someone who retired in 2008,


[User] Please transcribe this text

[Assistant] Okay, so those are some preliminary observations and I'll give you the plan.


[User] Please transcribe this text

[Assistant] Okay, and then as I said, this is obviously controversial. I think I've identified the problem in a, in a scientific way that is everyone who, you know, would agree with me if they


[User] Please transcribe this text

[Assistant] Okay, as I said, other plans just say, well, things are bad, let's just keep the same system, and so there are two basic ideas.


[User] Please transcribe this text

[Assistant] We'll do all kinds of things like that. Maybe we'll raise some taxes, reduce some benefits, balance the system.


[User] Please transcribe this text

[Assistant] You know, force them to save, but put it in the stock market in their own account.


[User] Please transcribe this text

[Assistant] Uhm, I don't have time to talk about Bush's plan. Uhm, okay. So here's my plan. Number one.


[User] Please transcribe this text

[Assistant] The fact that we gave all this money away to people in the 40s and 50s and 60s and 70s, why should that be the responsibility of the, uhm, the workers today? I mean,


[User] Please transcribe this text

[Assistant] Our government decided it was a good idea. Roosevelt and Francis Perkins decided we had to rescue the old of the 40s. And our congresses kept that up for the next 20, 30 years.


[User] Please transcribe this text

[Assistant] Even though the old weren't in such dire shape, but they kept up, maybe it was not a good idea. Probably a very good idea. But because we have that huge debt, why should workers be the only people responsible for paying back that debt?


[User] Please transcribe this text

[Assistant] I would start by imposing a legacy tax on everybody, you know? Uhm, not for social security, just for the debt we accumulated by giving it to those old generations.


[User] Please transcribe this text

[Assistant] So I believe that would be something like one percent, uh, I wrote two to three percent, but I think it's closer to one percent actually, uh, on all income. If I did it on all income, it would be like one percent. And that fund would pay off for the Social Security legacy.


[User] Please transcribe this text

[Assistant] So then if we were starting the system afresh, how should we do it?


[User] Please transcribe this text

[Assistant] Okay, so uhm, and that's some countries like Chile decided they're gonna start afresh. So how should we start afresh?


[User] Please transcribe this text

[Assistant] Having gotten rid of this old debt that was hanging over our heads, well, uhm,


[User] Please transcribe this text

[Assistant] What would I do? I would have what I call progressive personal accounts.


[User] Please transcribe this text

[Assistant] Personal annuitized average wage security. So what this does, this is the security that just pays proportional to the wage in the whole country at the year you retire.


[User] Please transcribe this text

[Assistant] So whenever you, uh, earn money on Social Security, so I'm, I'm running out of time, so I don't wanna, I'm gonna skip over this pretty quickly. So if you earn a dollar and you pay, let's say, 12% in Social Security's taxes, that income now, I'm going to, uhm,


[User] Please transcribe this text

[Assistant] That income, I'm going to say that's your income, okay, it's your 12%, you're going to be forced to save it, and what you're forced to do is to buy these securities.


[User] Please transcribe this text

[Assistant] So it's a new kind of security that pays proportional to the wage. So it might pay ten


[User] Please transcribe this text

[Assistant] You know, 1% of the average wage. So it would pay 12 cents here, 3 cents here, and 3 quarters of a cent there. That's the wage in the economy. And it would pay, you know, 1% of that. And how much of it would you get to buy? As much as you could afford to buy with your security, with your tax contribution. Now, that


[User] Please transcribe this text

[Assistant] It doesn't replicate the current system, there's nothing progressive about it yet. So what I would do is I would make it a progressive tact.


[User] Please transcribe this text

[Assistant] You know, if you're making a hugely high income, I wouldn't give you the whole 12%. I would take part of it. And if you're making a really low income,


[User] Please transcribe this text

[Assistant] I would have the government subsidize it a little bit. So instead of getting 12% of your low income, you'd get a little more than 12%. If you had a high income, your 12% tax, you know, some of the government would take one of those 12 away from you, and you'd only have 11% that you got to put in your personal account.


[User] Please transcribe this text

[Assistant] Okay, and then, uh, now, but it has a tremendous advantage, this system, which is that if you can price the pause, if you know what the market price of pause is, then you're balancing the system.


[User] Please transcribe this text

[Assistant] Okay, because every time you hand somebody money, you don't make some crazy promise that when you get to be old, I'm gonna pay you such and such dollars like the current system is, you're letting the guy


[User] Please transcribe this text

[Assistant] The uh, balancing the system going forward is occurring because you're forcing people to buy their social security benefits. So anyway, that's my, in a nutshell, my system. And the last thing is, how are you going to get paws priced by the market? Well, I'm going to get the market to trade them. And I think that would be a tremendous


[User] Please transcribe this text

[Assistant] Uh, boom to the economy if we had these securities paid. Why is that? Because they're tremendous. Okay, so how would I do that? I'd force everybody to invest only in pause, which I claim are like stocks in the long run, but in the short run they're less risky, so it's a better investment vehicle.


[User] Please transcribe this text

[Assistant] What they think future wages are gonna be. So anybody giving a pension plan that's indexed to future wages would know what the market prices of those things are, so it would dramatically help pension plans as well. Anyway, that's my idea of the, uh, for Social Security.


[User] Please transcribe this text

[Assistant] So I have four minutes left and I want to end with one.


[User] Please transcribe this text

[Assistant] Uh, last thought, okay, one last idea, which is black shoals. Now we've already done black shoals in a few problem sets.


[User] Please transcribe this text

[Assistant] So what is the idea of Black Should? It's just like the idea of the example we gave. By the way, you know, if you do come next week and you want to question me about my social security plan and criticize it, I'd be thrilled to be criticized because, uhm,


[User] Please transcribe this text

[Assistant] You know, you can't learn anything unless you're criticized, and I, I, I think I can defend it too, but, uh,


[User] Please transcribe this text

[Assistant] So let me just end with, uh, with the black shoal.


[User] Please transcribe this text

[Assistant] So in 1972, Black and Scholes, uh, wrote a famous paper. And what did they do? They started off by saying, this model we had, you see, of the stock market, which follows a geometric random walk. It can go up by a certain percentage or down by a certain percentage. We've used that with interest rates too.


[User] Please transcribe this text

[Assistant] It was not as mathematical as Fisher Black. Anyway, he started the hedge fund long-term capital, Fish, uh, Myron Scholes was one of the people. They looked in 1972 at the returns each day on the S&P 500.


[User] Please transcribe this text

[Assistant] And they binned them up, like this. So here, the number of times the return was between, you know, .47% and some other number. Okay, and then they compared that to a normally distributed random variable.


[User] Please transcribe this text

[Assistant] And look how close to normally distributed it is. It's practically like that. So this thing over here is exactly normally distributed, and this is the frequency, you know, graph of what actually happened that year. Incredibly close to normally distributed. If you take the cumulative occurrence of each thing and compare it to the normal, it's incredibly close to the normal.


[User] Please transcribe this text

[Assistant] So they said, gosh, isn't it great this model of things going up or down by a certain percentage of a year, maybe with a drift, that's exactly the model the stock market seems to follow. And maybe interest rates follow that sort of model too, and that's why we've worked with all those models.


[User] Please transcribe this text

[Assistant] It turns out if you make those assumptions and then you try to solve for, uh,


[User] Please transcribe this text

[Assistant] Option prices, you can do it very quickly through some formula, which I'm not going to have.


[User] Please transcribe this text

[Assistant] Time to present. But, so, it'll only take one minute, but I'm gonna skip that. So what happened then? So this, so this, uh, so this, uhm,


[User] Please transcribe this text

[Assistant] This was the second high point of, you know, there's the CAPM model and then the Black-Scholes model. You've done problems with Black-Scholes now and the problem set to figure out the value of a call option.


[User] Please transcribe this text

[Assistant] So I'm just pointing out that you could put it in a spreadsheet, like I've done, which will be on the, on the web, and you can do, you know, for ten, you know, daily returns and fix the standard deviation and stuff, and do backward induction incredibly quickly, and figure out the value of the call option, even get a closed form formula for it.


[User] Please transcribe this text

[Assistant] So that was, and then explain why call option prices have the form that they do. So it's a great triumph and it relied on their things being normally distributed, uhm, down to 30 seconds. So, what happened? If you do the same thing,


[User] Please transcribe this text

[Assistant] Sorry, not in 1972, but, you know, the last five years, say, they only did one year, but you do the last five, I actually did this a couple years ago. So, from seven years ago to two years ago, the pink thing is normally distributed, and we do the same binning that Black & Scholes did, look every day at what the return is, and stick it in the bin, and, you know, just do the frequency thing.


[User] Please transcribe this text

[Assistant] What do you see? You see that there are a lot more times where in reality the move was very small, but there are a lot more times in reality where the move was very big. So this is precisely what you call a fat


[User] Please transcribe this text

[Assistant] Okay, so I'm ending now with this thought, that finance produced remarkable theories, remarkably precise predictions, and for decades at a time, it seemed those were born out in practice.


[User] Please transcribe this text

[Assistant] But then looking back 50 years later or 30 years later on these discoveries, we see that they don't do quite as well as they seem to be doing at first glance.


[User] Please transcribe this text

[Assistant] So something's missing in the theory. And I think the subject is so exciting because it's so connected to the world, you know. Everybody talks about finance nowadays. You can't, I mean, any dinner part, anyone in the world is spending half their time thinking about finance.


[User] Please transcribe this text

[Assistant] And the theory is still up in the air because the old, you know, classics of the theory no longer hold and so it's an exciting time to be developing a new theory and maybe you'll think about it in the future. So I'll see you, some of you next week.


[User] Please transcribe this text

[Assistant] Never argue with stupid people. They will drag you down to their level.


[User] Please transcribe this text

[Assistant] Let us live, so that when we come to die, even the undertaker will be sorry.


[User] Please transcribe this text

[Assistant] Only two things will regret on deathbed.


[User] Please transcribe this text

[Assistant] If you pick up a starving dog and make him prosperous, he will not bite you.


[User] Please transcribe this text

[Assistant] Forgiveness is the fragrance that the violet sheds on the heel that has crushed it.


[User] Please transcribe this text

[Assistant] The fear of death follows from the fear of life.


[User] Please transcribe this text

[Assistant] The human race has only one really effective weapon, and that is laughter.


[User] Please transcribe this text

[Assistant] A person with a new idea is a crank until the idea succeeds.


[User] Please transcribe this text

[Assistant] The question is what did our investors, uh, say about, uh, about us, uhm, taking the big risk of adding programmable shaders and reinventing


[User] Please transcribe this text

[Assistant] First of all, it's not a, it's not a conversation you really have with your, with your shareholders, but you do have the conversation with your, uh, your management team first, uh, your employee second, your board of directors third, typically is the way, the process that I, I take. Uhm, and uhm, when you're in, when you're in high tech,


[User] Please transcribe this text

[Assistant] In this video we'll go over the neural network, the architecture of it, and how we can incorporate additional information into it.


[User] Please transcribe this text

[Assistant] So the neural network architecture that we use for diffusion models is a unit. And the most important thing that you need to know about a unit is that it is, uh, taking as input this image and it's producing as output, uh, something of the same size as that image, but here it is that predicted noise.


[User] Please transcribe this text

[Assistant] Units have been around for a very long time, uh, since 2015, and it was first used for image segmentation. It was first used to take an image and actually segment it into, uh, either a pedestrian or a car, so it's used a lot in self-driving car research.


[User] Please transcribe this text

[Assistant] But what's special about Unets is just that its input and outputs are the same size.


[User] Please transcribe this text

[Assistant] And what it does is it first embeds information about this input. So it down samples with a lot of convolutional layers into an embedding that compresses all that information into a small amount of space.


[User] Please transcribe this text

[Assistant] And then it upsamples with the same number of upsampling blocks into the output back out for its task. And in this case, that task is to predict the noise that was applied to this image.


[User] Please transcribe this text

[Assistant] And if you want to look a little bit deeper, which we'll do together, is that each of these named blocks here are also shown in the code with the same names.


[User] Please transcribe this text

[Assistant] What's also great about this unit is that it can take in additional information. So it's compressed that image to understand what's going on, but it can also take in more information. And so what information do we want to include? Well, one thing that's really important for these models is the time embedding. And so this is an embedding that


[User] Please transcribe this text

[Assistant] It kind of tells the model what the time step is and therefore what kind of noise level we need.


[User] Please transcribe this text

[Assistant] And all you have to do for this time embedding is you embed it into some kind of vector and you can add it into these up sampling blocks.


[User] Please transcribe this text

[Assistant] Another piece of information that could be useful is context embedding. We'll do more of this later, but all that context embedding does is it helps you control what the model generates. For example, a text description like you really want it to be Bob or a some kind of factor like it needs to be certain color. We'll discuss this a bit more later. And for that context embedding, you can just multiply it in.


[User] Please transcribe this text

[Assistant] Great, so what does that look like in code? Here you can see a context embedding, this is just one of them, and then here you see the time embedding, and in the upsampling block, all you have to do again, just like in this diagram, you multiply the context embedding with the upsampling block, and you add the time embedding.


[User] Please transcribe this text

[Assistant] Cool, so now in the notebook, in the forward pass of the model, so this is running the model, you can see, uh, some of these down, down, down blocks and then also these up, up, up blocks here.


[User] Please transcribe this text

[Assistant] And again, here are your context and time embeddings. We have two of them here for each of those up blocks. And how these, uh, down and up blocks are defined is up here.


[User] Please transcribe this text

[Assistant] And so for the down, this is what a unit down looks like. And we actually do have that in our helper functions if you want to look at them in greater detail. Uh, but they are just convolutional blocks.


[User] Please transcribe this text

[Assistant] And in the next video, you'll learn how to train this neural network.


[User] Please transcribe this text

[Assistant] Tire Town School Episode 6 The Car Wash


[User] Please transcribe this text

[Assistant] Then he looked at himself. He saw lots of dirt.


[User] Please transcribe this text

[Assistant] So we're, uh, at the point in the course now where we're talking about the capital asset pricing model and the economist


[User] Please transcribe this text

[Assistant] Understanding of risk aversion and its consequence for the financial sector.


[User] Please transcribe this text

[Assistant] So much of what economists, uh, know was already known, you know, to businessmen because it's common sense and even to literary writers.


[User] Please transcribe this text

[Assistant] Uh, already had the idea of diversification.


[User] Please transcribe this text

[Assistant] In The Merchant of Venice, Antonio says, uh, each boat on a different ocean


[User] Please transcribe this text

[Assistant] Each sail on a different Argosy or something, I forgot what it was. But anyway, each boat on a different ocean, so he wasn't very worried because he was diversified.


[User] Please transcribe this text

[Assistant] And he also had the idea of, uh, risk and return.


[User] Please transcribe this text

[Assistant] You know, nothing ventured, nothing gained. Nothing ventured. So ventured means risk, nothing gained.


[User] Please transcribe this text

[Assistant] So what have economists done that wasn't already known to every business person and every clever literary writer? Well, economists have quantified this and turned this into a usable, practical piece of advice.


[User] Please transcribe this text

[Assistant] So the diversification theorem in cap M, so I should have moved cap M, I should have called this Shakespeare.


[User] Please transcribe this text

[Assistant] And this cap M, cap M becomes, uh, the mutual fund theorem.


[User] Please transcribe this text

[Assistant] The mutual fund theorem has very practical advice. It says if you're investing in the stock market, don't try to pick out individual stocks. Hold every stock in proportion to its value in the whole economy.


[User] Please transcribe this text

[Assistant] If you want to be more venturesome, don't pick riskier stocks, just simply leave less money in the bank. So divide all your money between the index and the bank.


[User] Please transcribe this text

[Assistant] It says hold all your money in stocks in the same proportion everyone else is holding them. Hold the market, in other words. Hold an index like the S&P 500 of all the stocks in proportion to how big they are. And if you want to get more venturesome,


[User] Please transcribe this text

[Assistant] You know, if you're, if you're cautious, put some of your money in the, in the index and some of it in the bank. If you want to be more venturesome, take some money out of the bank and put it into the stocks.


[User] Please transcribe this text

[Assistant] So if you're interested in risk, you shouldn't pick high technology startup companies. You should pick the same mixture of, uh,


[User] Please transcribe this text

[Assistant] You know, blue chip companies like General Electric and these startup companies that everybody else is choosing. So that's what it means to diversify. Hold a little bit of everything in the same proportion as everyone else. So it's much more precise, much more surprising, uh, than Shakespeare's advice. Okay, then the, uh,


[User] Please transcribe this text

[Assistant] Then the, you know, then the second thing, risk and reward, it says that the price of a security, so we'll call it, uh, Pi J, say,


[User] Please transcribe this text

[Assistant] Which we used to think of as the expectation, of course you need the probabilities of the payoff of the security J.


[User] Please transcribe this text

[Assistant] Okay, discounted. That's what we would have thought the price was. So this is called the Covariance Pricing Theorem.


[User] Please transcribe this text

[Assistant] Okay, the shocking thing is it's not the expectation discounted, which is what we thought it was before. Okay, we know that you have to adjust for risk somehow, which we haven't taken into account before. But what's so shocking is you might have thought you'd subtract maybe something times the variance of


[User] Please transcribe this text

[Assistant] The return of acid J, but that's not going to be the case. It's going to be


[User] Please transcribe this text

[Assistant] Surprisingly, subtracting something times the covariance with the market of asset J. Okay, so you penalize


[User] Please transcribe this text

[Assistant] The price of a stock should go lower if it gets riskier. Its risk is not defined by what its variance is, but by what its covariance is with the market. So I want to develop these two ideas.


[User] Please transcribe this text

[Assistant] Uh, in this lecture. Spend the whole lecture on just developing these two ideas, which is a modern mathematical version of what Shakespeare already knew 400 years ago, but made much more concrete, much more precise with, uh, uhm, mathematics wholly behind them.


[User] Please transcribe this text

[Assistant] So what have we got here? We imagine a world, the situation I'm going to describe is the one we had last time.


[User] Please transcribe this text

[Assistant] Where we know that there are different things that can happen. These are the states with different probabilities. So there's states.


[User] Please transcribe this text

[Assistant] Of course the sum of the gamma s equals 1. So everybody knows the probabilities of the states.


[User] Please transcribe this text

[Assistant] And then they know what the payoff is of the different assets. So here's A1, here's A2, here's Aj, say.


[User] Please transcribe this text

[Assistant] And they know what the payoffs are going to be. A1 in state 1, A1 in state 2, A1 in state S, A2 in state 1, A2 in state S, AJ in state 1, AJ in state S.


[User] Please transcribe this text

[Assistant] So given these probabilities, and given the payoff, you can compute the expectation of each thing. You can compute the expectation of aj, say, just the summation from s equals 1 to s of gamma s aj s. Okay, you can also compute the variance


[User] Please transcribe this text

[Assistant] So I'll write that sometimes as A bar J. You can compute the variance of A J.


[User] Please transcribe this text

[Assistant] Which is the summation s equals one to s, remember of gamma s.


[User] Please transcribe this text

[Assistant] And the covariance between any pairs, those are numbers that anybody could compute knowing the payoffs of the assets and knowing the probabilities.


[User] Please transcribe this text

[Assistant] Okay, so we wrote that last time, we said if you had quadratic utility for consumption in every state,


[User] Please transcribe this text

[Assistant] You could summarize these, you know, there could be millions of states and thousands of securities, you add them all together, what you hold, it only depends on the end about what the expectation and the variance of your final portfolio is. You like expectation, you hate variance.


[User] Please transcribe this text

[Assistant] So under that hypothesis, we're going to have a very concrete interpretation of what Shakespeare was talking about from the beginning, which is going to allow us to quantify what you should do, how people in the market are doing. We'll be able to answer a series of questions.


[User] Please transcribe this text

[Assistant] In 1950, it began in 1950, when Koopmans, who's a professor,


[User] Please transcribe this text

[Assistant] At the Coles Foundation. The director of the Coles Foundation was in Chicago. I told you the Coles Foundation.


[User] Please transcribe this text

[Assistant] was started by Alfred Coles who'd been a Yale undergraduate and his family owned the Chicago Tribune and a bunch of other newspapers.


[User] Please transcribe this text

[Assistant] You know, newspapers and realized they've been predicting everything was going to be fine too, except for one guy, you know, Jones.


[User] Please transcribe this text

[Assistant] The Dow Jones guy. Except for that, everybody had been making the same, you know, rosy predictions. And so he decided the whole subject of economic forecasting, in fact economics altogether, was a fraud. And he wanted to put money in encouraging the use of mathematics in economics instead of the touchy-feely, uh,


[User] Please transcribe this text

[Assistant] You know, more qualitative analysis that was the norm at the time. So he took, he went to see Irving Fisher, who was the great Yale economist, and said, what should I do with all the money? Where should I put this institute? And who should I invite to the institute? So Fisher and he set up the whole thing.


[User] Please transcribe this text

[Assistant] And it started in Colorado Springs because Fisher believed in fresh air and he wrote a lot of books about fresh air and how it staved off tuberculosis and things like that.


[User] Please transcribe this text

[Assistant] The director then in the 50s was selling coupons, and coupons had this idea of trying to examine risk by making people with quadratic utilities, and he said, okay, student Markowitz,


[User] Please transcribe this text

[Assistant] Why don't you try to develop a theory? And Markowitz wrote, began the theory of the capital asset pricing model. Now the theory, just to get a little ahead of the story,


[User] Please transcribe this text

[Assistant] Markowitz went quite a ways, won a Nobel Prize many years later for what he did, but didn't quite get to the end.


[User] Please transcribe this text

[Assistant] And while he was working on it, Koopmans didn't want to be director anymore of Kohl's, and he invited Tobin, the great Yale economist, to come and be


[User] Please transcribe this text

[Assistant] Uh, director of COALS and Tobin got there and the COAL, the people at the University of Chicago said


[User] Please transcribe this text

[Assistant] To Tobin, well we're glad you're coming, you'll be right next to us, but you know, don't try and get us mixed up with all those mathematical people at the Kohl's Foundation, they're all crazy. We would like to keep the mathematics, you know, that high tech mathematics out of our department. That's just Kohl's crazy idea.


[User] Please transcribe this text

[Assistant] So, Tobin refused the job, came back to Yale, and, uh, Markowitz was so angry, not Markowitz, uh, Kutman, his advisor, was so angry that he went to Coles and said, they're not treating us right at the University of Chicago, let's move the entire Coles Foundation to Yale. So, in 1955, they moved it to Yale.


[User] Please transcribe this text

[Assistant] But in these conversations Tobin had with Koopmans and the Coles Foundation, he got to know what Markowitz was doing, and so he added the final, uh, crowning achievement to the mutual fund theorem to the Markowitz, uh, uh, setup.


[User] Please transcribe this text

[Assistant] And he subsequently won the Nobel Prize for that. In fact, the New York Times said in his, uh, right after, in his description of what he won the Nobel Prize for, the New York Times said, James Tobin won the Nobel Prize for showing that you shouldn't put all your eggs in one basket.


[User] Please transcribe this text

[Assistant] So anyway, I will, so that's the theory that I have to explain now, that it's a little bit less trivial than just saying don't put all your eggs in one basket. So.


[User] Please transcribe this text

[Assistant] The idea is that you like expectation, you hate variance. So the first thing we're going to do is notice that


[User] Please transcribe this text

[Assistant] Covariance, I mean sorry, standard deviation is defined as the square root of variance.


[User] Please transcribe this text

[Assistant] So if we had a portfolio that you could choose among some combination of putting your money in the A's, that's okay, putting your money in the A's.


[User] Please transcribe this text

[Assistant] You get some expectation and some standard deviation for your total portfolio. And people like more expectation and


[User] Please transcribe this text

[Assistant] So the indifference curves are going in that direction unlike the usual picture. That was the first step. Now the first thing we noticed last time


[User] Please transcribe this text

[Assistant] We made a very, and this is, uh, three weeks ago or so, we made an extremely important observation. We said, what is the covariance of part of your money in X


[User] Please transcribe this text

[Assistant] And the variance of part of your money in X


[User] Please transcribe this text

[Assistant] And one minus t and y. For example, half an x and half a y.


[User] Please transcribe this text

[Assistant] Well that is by definition the same as the covariance of this portfolio with itself.


[User] Please transcribe this text

[Assistant] And now we notice from this theorem, from this definition of covariance,


[User] Please transcribe this text

[Assistant] That if you look at the covariance of J with K, so fixing K, this is just linear, this is the expected, you know, this is a constant multiplying the differences from J and its expectation. So this is linear in each variable. So therefore this is going to equal the covariance of Tx with Tx plus twice the covariance of


[User] Please transcribe this text

[Assistant] Then this with that, then that with that, and that with the thing on the end. Okay, but it's linear again.


[User] Please transcribe this text

[Assistant] So I can pull out all these things. So, okay, so that's the formula. Now what we noticed is that


[User] Please transcribe this text

[Assistant] If two stocks, x and y, are independent, their covariance is zero.


[User] Please transcribe this text

[Assistant] Okay, so I'm assuming you remember that. So those terms just go away. And then it's also linear, so this is just going to equal t squared


[User] Please transcribe this text

[Assistant] Then you can, then that is what allows me to cancel these two terms like that. So I get t squared times the covariance of x plus one minus t squared times the variance, variance of x and variance of y. If I take t equals a half, say, so if I take t equals a half,


[User] Please transcribe this text

[Assistant] Okay, the variance of a half x plus a half y equals a quarter times the variance of x


[User] Please transcribe this text

[Assistant] Plus a quarter times the variance of y. We derived this three weeks ago, I'm just repeating it.


[User] Please transcribe this text

[Assistant] So the amazing thing is, if X and Y have the same expectation and the same variance, but are independent of each other, by splitting your money between two independent things, you're obviously going to have the same expectation, but only a half the variance.


[User] Please transcribe this text

[Assistant] So amazingly, I mean it's not so amazing I guess if you've thought about it, but if you haven't thought about it, you've got two equal risks. Neither one is better than the other. So why bother with one rather than the other? They're both the same. But if they're independent, you should put half your money in each of them. You get the same expectation, but you're gonna get only half the variance.


[User] Please transcribe this text

[Assistant] So if you have two dice, let's say the first die will give you...


[User] Please transcribe this text

[Assistant] You pay $100 for a dice, you can get $100, you can get $100 plus the number of dots showing.


[User] Please transcribe this text

[Assistant] So if it gets a 1, it's 101 dollars, you get a 6, it's 106 dollars. So if you held one dice,


[User] Please transcribe this text

[Assistant] One die, you'd get on average a hundred three and a half dollars.


[User] Please transcribe this text

[Assistant] If you pay a hundred, get a die, which on average pays a hundred three and a half.


[User] Please transcribe this text

[Assistant] Instead, you could put $50 on the first die, get half of that, $50 on the second die, get half of that, so on average you're still getting $103.50, but the variance you get is going to be half of what it was


[User] Please transcribe this text

[Assistant] Okay, so in this picture, you start off with X and Y with the same standard deviation and the same expectation, but they're independent and now you put half your money in each and you cut the variance in half, which means you divide


[User] Please transcribe this text

[Assistant] You know, the standard deviation is the square root if you divide it by the square root of two. Okay, so you've improved, you've moved


[User] Please transcribe this text

[Assistant] To the left, you've gotten less standard deviation without constricting your expectation at all. Okay, so that's the most important thing to notice.


[User] Please transcribe this text

[Assistant] And I'm gonna say it, uh, one more thing, one more way. I'm gonna say, I'm gonna go back to this formula. Okay, which is where the variance was here.


[User] Please transcribe this text

[Assistant] Uh, which I'm gonna switch now. X and Y. That was most unfortunate. But suppose I take two stocks.


[User] Please transcribe this text

[Assistant] And I'm at the point where all my money is in y, so I'm going to take the case t equals zero.


[User] Please transcribe this text

[Assistant] Okay, but before I take t, so if I took t equals zero, I would just have the variance of y. Okay, so this would be


[User] Please transcribe this text

[Assistant] Oh, that's actually quite good, the variance of y. Okay, so I'd have all my money in y, I'd have this standard deviation and this expectation. Now what happens if the two have the same


[User] Please transcribe this text

[Assistant] Uh, what if the two are independent, x and y? Let's drop the fact that they have the same variance now. Maybe x has way more variance than y does. Okay, so look it, x is much riskier than y.


[User] Please transcribe this text

[Assistant] Maybe X has a higher variation, a higher expectation than Y, but it's so risky. You know, people who are risk averse wouldn't ever want to subject themselves to all this risk, say. Okay, so maybe they should just stick with Y, you would think. But no, the answer is you shouldn't. Because what happens if you start to mix some of your money


[User] Please transcribe this text

[Assistant] Your expectation, if you take a little money out of Y and put a little into X, your expectation is going to go up.


[User] Please transcribe this text

[Assistant] Because it's going to be the average of x and y. The expectation's going up, but then you say, oh, but my variance is going up, so I wouldn't want to do it. But no, that's wrong, the variance is not going up, because they're independent.


[User] Please transcribe this text

[Assistant] Even though x has a way higher variance than y, mixing a little between the two lowers your variance. So how could that be? Seems quite shocking. Well, let's take the derivative of this with respect to having all your money originally in y.


[User] Please transcribe this text

[Assistant] So this is the variance of putting t dollars in x and 1 minus t in y. So we simply take the derivative of that.


[User] Please transcribe this text

[Assistant] Which, which I've done here, I suppose, I could just, why don't I just write it out, take the derivative of that.


[User] Please transcribe this text

[Assistant] Okay, but this is a minus, I have to, you know, I take the derivative, it's 2 times this, times minus 1, so minus 2t, 2 times 1 minus t, variance of y.


[User] Please transcribe this text

[Assistant] The first term is zero, the t is zero, so this doesn't matter. The second term, t is zero, so this is minus two times the variance of y. It's going down. Okay, the variance is actually going down here.


[User] Please transcribe this text

[Assistant] Okay, so just as this says, the variance starts down. Eventually it has to end up here. So if you move your money, all of it originally in Y, and start moving some of it into X, your variance is gonna go down.


[User] Please transcribe this text

[Assistant] And they're, they're identical, then you should put half of your money in each. But even if they're not identical, you shouldn't stick to Y, you should mix Y with some X. Okay, so there's why diversification is such a good idea. So it's not all diversification, it's diversification of independent risk. So there's a little bit of a mathematical nuance.


[User] Please transcribe this text

[Assistant] Diversification of independent risk is bound to help you. Everybody should diversify a little bit. Because you can reduce your risk. Okay? So that's the first important thought.


[User] Please transcribe this text

[Assistant] Do too much in the mathematics, but, uh, so I prove in the notes, but I won't do it here. Suppose we didn't assume x and y were independent. Suppose x and y were arbitrarily chosen.


[User] Please transcribe this text

[Assistant] You know, they have big covariance, whatever you want.


[User] Please transcribe this text

[Assistant] Okay, then what is going to happen when you combine your money between x and y? So there's a proof


[User] Please transcribe this text

[Assistant] Which I'm not going to give, because I'm a little bit behind. So the proof is this.


[User] Please transcribe this text

[Assistant] Suppose we have exactly the same situation where you started with Y and you've got X, and you could think of moving your money in between X and Y. Okay, well,


[User] Please transcribe this text

[Assistant] If they, it turns out if they're perfectly correlated, so really they're the same thing, y is just, you know, 80% of x, it's the same stock, so moving back between them is not really changing, you know, x is just a bet, you know, a scalar improvement on y, so it has,


[User] Please transcribe this text

[Assistant] If they were perfectly correlated, you would just be moving along this line.


[User] Please transcribe this text

[Assistant] Okay, but no matter how correlated they are, as long as they're not perfectly correlated, it's gonna turn out the curve you get by moving between them is gonna look like that. If they're negatively correlated, then you can really improve yourself and you'll get a curve that looks like that. Independent will be like this.


[User] Please transcribe this text

[Assistant] Uh, positive correlated but not perfectly will be like that. And, uh, perfectly positively correlated, no diversification gains, you get the straight line. But you're always above the straight line. Okay, so that's a kind of, anyway, that's, uh, okay, it takes a little bit of algebra.


[User] Please transcribe this text

[Assistant] Uh, to prove that, you know, the Cauchy-Schwartz inequality, but we're not, so I'm, I'm gonna skip that. But anyway, that's a mathematical fact, okay? And you see an independent case, we basically, uh,


[User] Please transcribe this text

[Assistant] You know, we proved, we proved that the line has to start going that way, and eventually it has to end up here, so it shouldn't be no surprise that it looked like that in the independent case.


[User] Please transcribe this text

[Assistant] And then it turns out that every other case is sort of in between that or at the other extreme. It always has this bowed out shape.


[User] Please transcribe this text

[Assistant] Okay, so there's always gains to diversification, uhm,


[User] Please transcribe this text

[Assistant] Alright, so that's it. That's the mathematics and that's the general principle. Now, what can you get out of this? I want to derive


[User] Please transcribe this text

[Assistant] These two famous theorems, which we already saw in action in the example that we calculated, but it seemed like a miracle. So what did we find in the example we calculated last time in class?


[User] Please transcribe this text

[Assistant] Assuming there were arrow securities, it would end up that every consumer held the same mix of all these stocks plus positive or negative amounts of the bond. So the mutual fund theorem turned out to be true.


[User] Please transcribe this text

[Assistant] That the best thing everybody could do was hold the same amount, the same proportion of stocks and bonds, and maybe


[User] Please transcribe this text

[Assistant] More or less of the riskless asset, maybe even negative of the riskless asset. So I wanna show you why that's true now.


[User] Please transcribe this text

[Assistant] It seems, and then we're going to have to talk about whether you really believe it or not. So, let's see what an implication is. Well, suppose now what I do is I write down, I have all these stocks.


[User] Please transcribe this text

[Assistant] The prices are given, let's say prices, uh, of stocks are in equilibrium.


[User] Please transcribe this text

[Assistant] Okay, so I've solved for the equilibrium like we did last time, and I write down x, q, y, and z. These are the expectation and standard deviation per dollar of, uh, the stocks.


[User] Please transcribe this text

[Assistant] So let's say it's I dollars of each stock what their expectation and return and standard deviation would be.


[User] Please transcribe this text

[Assistant] If, if, uh, one dollar, you get some expectation, standard deviation, ten dollars of the same stock, you'd be able to buy ten shares, which means you would, uh, ten times as many shares, which means you'd have ten times the expectation and ten times the standard deviation. Okay, so, I can, you know, I, whatever the guy's income is, I'll fix the dollars, any, I'll fix these returns as,


[User] Please transcribe this text

[Assistant] Put all his money into X, he'd end up here, or all his money into Y, he'd end up here, or all his money into Z, and he'd end up here. But maybe once he divides part of his money in Y, and part of it, part in Z, and part in Y, he'll be somewhere like here.


[User] Please transcribe this text

[Assistant] Or maybe he wants to instead put part of his money in Z and part of it in X. Well he could end up here. But why should he stop there? Maybe he wants to take some combination of the money that was in Z and X


[User] Please transcribe this text

[Assistant] Okay, so there are lots of different combinations, uh, he could hold. So graphically it looks like that. Mathematically,


[User] Please transcribe this text

[Assistant] In fact, let's just say it's pi, it's a j divided by pi j. So r j is the return per dollar you put in the asset. r j is the payoff of the asset per dollar. Let's stick with that. Let's say the guy has a dollar to spend. So it's the return per dollar. Okay, and so those are all his possibilities. Now what does he want to do? He wants to choose


[User] Please transcribe this text

[Assistant] His money, how much money he's gonna put in each asset, okay, which is omega 1.


[User] Please transcribe this text

[Assistant] And if he does that, he's going to have a portfolio, which is, you know, I'll write as r omega, so it's going to have a certain expectation, a certain variance.


[User] Please transcribe this text

[Assistant] So the question is, how should he divide his money between the assets?


[User] Please transcribe this text

[Assistant] Okay, and here's the kind of stuff he could imagine doing, which is very complicated.


[User] Please transcribe this text

[Assistant] But it's gonna turn out to be very simple. So are there any questions about what we're doing so far?


[User] Please transcribe this text

[Assistant] So I want to get a simple rule to tell you what, what, practical advice about what to do.


[User] Please transcribe this text

[Assistant] So Markowitz posed this problem. This is the Markowitz problem. Okay?


[User] Please transcribe this text

[Assistant] Now, it turns out by this argument, you see that if I keep combining things over and over again, I'm getting this sort of blob, but the blob is always pushing itself further and further out. It's always convex, because given any two points, I can always do the thing to the left of it.


[User] Please transcribe this text

[Assistant] So if I could do this and I could do this and I look at the line connecting them, I can always be above that line by spreading my money between those two assets. We're given this combination of assets.


[User] Please transcribe this text

[Assistant] So I produce this expectation of standard deviation, and given that one up there, I can combine the two and do better than the straight line connecting them due to the left of it. So therefore, all the combinations put together have to geometrically look like this in the end.


[User] Please transcribe this text

[Assistant] Okay, so I'm gonna do this algebraically in a second. They have to look like that. So I did every possible variation.


[User] Please transcribe this text

[Assistant] Okay, it couldn't have, it couldn't end up with a picture where after I do every possible variation,


[User] Please transcribe this text

[Assistant] Whatever portfolio produced that, I'd put half my money into each of the stocks that this told me to do, half the money into each of the stocks this told me to do, I'd still end up investing one dollar, and instead of getting the line, I'd get something that looked like that.


[User] Please transcribe this text

[Assistant] Okay, so I couldn't have the thing going in like that, it has to go out. But then of course I can combine this and this and get that, and so I'm gonna end up with this kind of shape. Now,


[User] Please transcribe this text

[Assistant] I'm gonna prove that algebraically, but if I had that kind of shape, then what would the guy do? He'd choose something like that, where he's tangent. Okay, that's what Markowitz basically did. Except he did it algebraically. So let's... Alright, so that's what Markowitz did.


[User] Please transcribe this text

[Assistant] So that doesn't seem to have gotten us very far. So you notice that different people are going to make different choices. If somebody cares, is incredibly worried about


[User] Please transcribe this text

[Assistant] Uhm, someone's not very worried about the standard deviation and cares a lot about expectation. They're going to have a flatter indifference curve. So instead of looking like that, it'll be a flatter thing. And so if it's flatter, the guy who cares


[User] Please transcribe this text

[Assistant] This is the guy, sorry, if it's flatter it means he cares about expectation and not standard deviation because


[User] Please transcribe this text

[Assistant] You know, a little bit of expectation can compensate him for a lot more standard deviation. So his indifference curve is flatter, he's going to choose further up.


[User] Please transcribe this text

[Assistant] And so he's gonna choose a point like way up there. And so he's gonna get a higher expectation and higher standard deviation. Someone who's more risk averse is gonna choose like where that point is. So, that's qualitatively what's going on.


[User] Please transcribe this text

[Assistant] Alright, so Mark was played around with this, played around with this, then Tobin appears on the scene. Tobin says, all this gets so much easier if you have a riskless asset. Let's suppose


[User] Please transcribe this text

[Assistant] That there's a bond. You haven't mentioned the bond yet. Let's suppose we have a bond. Okay, that pays something for sure.


[User] Please transcribe this text

[Assistant] And we're going to ignore inflation, Tobin said, which is a big problem, because we have, we should come back to that if we have time. We're going to ignore inflation. So U.S. Treasury's going to pay a certain amount of money for sure. So it has an expectation like this, but no standard deviation. It's zero.


[User] Please transcribe this text

[Assistant] Now, why is this so important? Well, let's imagine combining this riskless bond with some other stock X, a dollar's worth of X.


[User] Please transcribe this text

[Assistant] Dollars worth of the riskless bond gets you 1.06, say.


[User] Please transcribe this text

[Assistant] What's your expectation going to be? Well, your expectation, let's say, you know, X is 12% and Y is 6%.


[User] Please transcribe this text

[Assistant] 1.06 and 1.12. Okay, the average is gonna be 1.09 if you put half your money in both. But what is the standard deviation gonna be?


[User] Please transcribe this text

[Assistant] Well, I claim it's just gonna be on that straight line. Why is that? Because the riskless thing has covariance zero. If x is riskless, it's covariance


[User] Please transcribe this text

[Assistant] Okay, look at the covariance. If, if x, if j is the riskless bond, it pays always its expectations. So these numbers are always zero. So therefore the covariance of that thing with anything, the covariance of y with any x is going to be zero.


[User] Please transcribe this text

[Assistant] So therefore, and not only that, but the variance of y itself is also zero. So all you get is, for the, for the variance of the


[User] Please transcribe this text

[Assistant] T dollars in X and 1 minus T dollars in Y, you just get T squared times the variance of X. But then when you take the standard deviation, it's just T times


[User] Please transcribe this text

[Assistant] So in other words, the standard deviation of the mixture of y and x lies right on the line. So unlike, to put it another way, something that has no


[User] Please transcribe this text

[Assistant] Variance is perfectly correlated with everything else. It makes no difference when the other thing goes up or down. It doesn't tell you, you still know what y is going to be.


[User] Please transcribe this text

[Assistant] Okay, so unlike everything else where you have this bowl-shaped thing,


[User] Please transcribe this text

[Assistant] The riskless asset just connects everything to a straight line. So if there were a riskless asset, and you could do this with some x, and here was a z, you could also do that, or you could do that.


[User] Please transcribe this text

[Assistant] Okay, so the punchline, which we're now going to see algebraically, the punchline is, and by the way, if I combine, what does it mean to extend that line? I should have extended that line, that was a huge mistake. Suppose instead of


[User] Please transcribe this text

[Assistant] Here's half the money in the riskless asset and half in X. Here's all my money in X.


[User] Please transcribe this text

[Assistant] What would happen if I extended the line here? What would that correspond to?


[User] Please transcribe this text

[Assistant] I'm now at a point on the extension of the line. What is that? How do I get that?


[User] Please transcribe this text

[Assistant] Right, that's negative y and, and bigger than one in x. So negative a half of x plus one and a half of y, that's still a dollar, I sold x short, and so I'm just gonna continue the line that way. Okay, so I shouldn't have stopped here, I could have kept going.


[User] Please transcribe this text

[Assistant] Okay, so it's putting money in the bank and sharing it between a single stock puts me on this line. Borrowing money from the bank to put in the stock, called leveraging my return, puts me on the extension of the line. We're gonna come back to that in a second. Okay, so knowing that, now what should I do? Is there something really simple?


[User] Please transcribe this text

[Assistant] What would you do here faced with this choice? I mean, which combination of stocks and bonds would you hold?


[User] Please transcribe this text

[Assistant] I say it's really simple. You just look for a line through the riskless point that's tangent to this blurb, which happens to be here.


[User] Please transcribe this text

[Assistant] And now you should choose your point anywhere along this line. Independent


[User] Please transcribe this text

[Assistant] of what your utilities, your mean variance utilities are. No matter how risk-averse you are or how risk-loving you are, you should choose somewhere along this line.


[User] Please transcribe this text

[Assistant] I've drawn, with this blob, I've drawn all the possible combinations I could get by mixing different risky assets. What I left out of the picture is all the combinations I could get by adding the riskless asset. I've got all the risky asset combinations, that's the blob.


[User] Please transcribe this text

[Assistant] So in particular, I can put part of my money here and part of my money there, or I can go short. So whatever this combination was, that's the best thing I should, I can do, because now I can get everything along this line.


[User] Please transcribe this text

[Assistant] Nothing else can do better because what other combination could there be? It has to be the riskless asset.


[User] Please transcribe this text

[Assistant] Plus a bunch of risky acids. But combining the risky acids just puts me in the blob. Okay? And then combining whatever, so it might put me here in the blob. And then once I combine that with the riskless acid, I'll be on that line.


[User] Please transcribe this text

[Assistant] This line is always below that line. So this is the best possible thing I can do. No combination of the riskless asset and any other combination of risky assets that always put me on some line from here to some point in the blob, which is going to be below this tangent point.


[User] Please transcribe this text

[Assistant] So this is the best thing I can do. And it's independent of whether I was risk averse or a risk lover. If I'm risk averse and want to play it safe, you know, my utility function, I forgot what those look like, my indifferent curve,


[User] Please transcribe this text

[Assistant] I'll put most of my money in the bank and not very many in the risky stock. If I'm more of a risk lover, I'll maybe be here and I'll choose to put a good fraction of my money in the stock and only a little in the bank. If I'm really reckless, I'll borrow money and put it in the stock market and be way up there.


[User] Please transcribe this text

[Assistant] Okay, so, but everybody will do the same thing. No matter what their preferences are, they'll all be along the same line. Okay, so that's the mutual fund theorem. Why is that the mutual fund theorem?


[User] Please transcribe this text

[Assistant] Because it says everybody should invest in the same index of stocks


[User] Please transcribe this text

[Assistant] And put more or less money in the bank. So every single person ought to be doing the same thing. This is the mutual fund that everybody should hold.


[User] Please transcribe this text

[Assistant] Whatever combination of stocks that got me to this point is the mutual fund that everybody should hold and combine that with putting money in the bank. Okay, so let's just prove that algebraically.


[User] Please transcribe this text

[Assistant] In a special case, which is going to illustrate the point, the idea of it anyway.


[User] Please transcribe this text

[Assistant] Okay, so I'm doing a very special case here, but just to illustrate it algebraically. So suppose that the r's are, as we said, the returns appear per dollar of stock.


[User] Please transcribe this text

[Assistant] And that's gonna be the riskless asset that pays like a dollar six all the time if the interest rate is six percent.


[User] Please transcribe this text

[Assistant] So now you have your money, you have i dollars, you're the investor with i dollars, you can divide that money into the riskless asset or any one of the j.


[User] Please transcribe this text

[Assistant] Uh, assets. So here you've got r0, you can get that return, r1, up to rj, and you can put money omega 0, omega 1, omega j, but of course this all has to add up to i. So how should you do it? Well, you care about your


[User] Please transcribe this text

[Assistant] expectation and you hate variance. So I'm doing just a special case to make it, uh, illustrate the idea.


[User] Please transcribe this text

[Assistant] Suppose I write down your utility as if it were the expectation minus alpha times the variance. It could be a more complicated function, you know, we said already,


[User] Please transcribe this text

[Assistant] When people have quadratic utilities over consumption, their induced utility over portfolio is this thing.


[User] Please transcribe this text

[Assistant] Okay, so it's not exactly the one I've written down, but the argument would be just the same, and this is shorter and simpler. So just to illustrate the flavor of it, let's say all you care about is your expectation minus some constant, how risk-averse you are, times the variance.


[User] Please transcribe this text

[Assistant] To keep it even simpler, that the assets are all independent.


[User] Please transcribe this text

[Assistant] Okay, the theory doesn't rely on that. I'm just saying suppose it were, so we can algebraically get it without having to work very hard.


[User] Please transcribe this text

[Assistant] So you're maximizing your expectations, so the bar above means the expectation. So what's your expectation of omega dollars in here, omega one in here, and omega j in here? Just omega zero times the expectation of this. Your expectation then is going to be omega zero.


[User] Please transcribe this text

[Assistant] Times r zero bar, that's the expectation of that, plus omega one r bar one, plus omega j r bar j. That's your


[User] Please transcribe this text

[Assistant] Is just going to be the variance of these, they're independent, so the variance is just going to be, uhm, what I've written.


[User] Please transcribe this text

[Assistant] So it's going to be sigma zero squared, that's the variance of sigma zero.


[User] Please transcribe this text

[Assistant] Times omega zero squared, remember, because for variance you have to square if you multiply by something, okay, plus omega one squared, sigma one squared, plus omega j squared, sigma j squared. Now of course remember,


[User] Please transcribe this text

[Assistant] That the variance of the riskless thing is zero, so that's actually zero.


[User] Please transcribe this text

[Assistant] Okay, so this is a variance you have to watch out for. This is the expectation you want. How should you pick omega 0, omega 1, omega j, uhm, when your penalty for having too much variance is alpha?


[User] Please transcribe this text

[Assistant] Okay, so different investors have different risk aversions, so they have different alpha. Should they be led, as Markowitz thought,


[User] Please transcribe this text

[Assistant] Without the riskless asset, remember, with a blob like this, one guy is gonna pick here, another guy is gonna pick there. They'd all do very different things.


[User] Please transcribe this text

[Assistant] Tobin added this riskless asset and lo and behold now everyone should do the same thing.


[User] Please transcribe this text

[Assistant] So why is that? We just reduce this algebraically, okay? Solve it algebraically. So the constraint is a nuisance, to maximize something is a constraint is a nuisance.


[User] Please transcribe this text

[Assistant] So you just notice that if you satisfy the constraint, omega zero has to equal i minus


[User] Please transcribe this text

[Assistant] So if I substitute for omega zero right up there, I substitute i minus the sum of the j's, then I'm going to get


[User] Please transcribe this text

[Assistant] r bar zero times i minus r bar zero times omega one minus r bar zero times omega two.


[User] Please transcribe this text

[Assistant] So, but I can feed that into the other thing, so I get


[User] Please transcribe this text

[Assistant] Uhm, r0 times the i, and then the r1 term is going to be r bar 1 times omega 1 minus the thing that came from subtracting the omega 1 there in i minus omega, you know, by replacing omega 0 with i minus


[User] Please transcribe this text

[Assistant] So I'm going to notice all these other terms now. I get rid of the constraint and all the other terms get replaced. R bar 2 gets replaced by R bar 2 minus R bar 0. Okay?


[User] Please transcribe this text

[Assistant] Sigma zero because it's zero. So I'm maximizing something without a constraint by plugging in the constraint into omega zero.


[User] Please transcribe this text

[Assistant] Okay, so I have to choose my omega 1 through omega j.


[User] Please transcribe this text

[Assistant] Uhm, and then omega zero is determined from that. But I'm maximizing one thing. So I just take the derivative with respect to each omega i and set it equal to zero. So with respect to omega one, say,


[User] Please transcribe this text

[Assistant] It's going to be r bar 1 minus r bar 0.


[User] Please transcribe this text

[Assistant] Minus two alpha sigma two, well, differentiating with respect to omega one. So it's two alpha omega one times sigma one squared. Two alpha omega one times sigma one squared equals zero. Okay, so therefore omega one equals r one minus r zero over two alpha.


[User] Please transcribe this text

[Assistant] Sigma 1 squared. But if I did it now with respect to 2 or to 3 or to 4, nothing, I would get the same formula just with a j instead of a 1. Okay, so that's the formula everybody picks. And notice it depends on what alpha is.


[User] Please transcribe this text

[Assistant] So, people with different alphas are going to put different amounts of money in omega j.


[User] Please transcribe this text

[Assistant] The fraction, the relative amounts of money I put in i and j and divided this by omega i by omega j.


[User] Please transcribe this text

[Assistant] I would get that the alphas and the twos cancelled out. So everybody would put


[User] Please transcribe this text

[Assistant] Ri minus r bar zero divided by sigma i squared, r bar j minus r zero over sigma squared j. That number doesn't depend on who the person is. So the relative amounts of money you put in i and j are independent of what your alpha is, independent of the person.


[User] Please transcribe this text

[Assistant] So unlike this old Markowitz case, the Tobin case looks like that, and everybody chooses the same combination of risky acids.


[User] Please transcribe this text

[Assistant] And then, risk safe asset. So it may be that different people put different combinations of safe and risky, but everybody's proportion of risky assets is proportional to everybody else's. That's what we just proved algebraically. So graphically and algebraically, graphically a sort of general proof, algebraically a special case,


[User] Please transcribe this text

[Assistant] And then if you were in a graduate class, you'd get an algebraic proof in general, which is actually much faster than these, but you have to know a little linear algebra. And so, that's the proof. It's a very simple thing. Okay, it's a remarkable thing. And here's the famous Tobin diagram.


[User] Please transcribe this text

[Assistant] Everybody should do that. Put their money in the same combination of risky assets and the riskless assets. Okay, are there any questions about this?


[User] Please transcribe this text

[Assistant] You know, I could swear that, uh, most people think, maybe even rightly, that, you know, if you really want to go for it,


[User] Please transcribe this text

[Assistant] You know, put your money in the young Microsoft or something. Uhm, you know, it could be a tremendous success or it could go out of business. That's what you should do. But no, this says not at all. If you really want to go for it, take the same index


[User] Please transcribe this text

[Assistant] We're gonna talk, you know, tease out all the implications of it after this theory was created.


[User] Please transcribe this text

[Assistant] So that's what everybody started to do. So all these mutual funds and money managers, they all grew up.


[User] Please transcribe this text

[Assistant] Uh, originally, by guaranteeing to produce an index fund, the very same thing economic science recommended at almost no cost. If you read Swenson's book, and you, and he said, you know, his first most important piece of advice.


[User] Please transcribe this text

[Assistant] If you're a standard investor who doesn't know how to, you know, manipulate all the complicated instruments on Wall Street and has transactions costs and things like that,


[User] Please transcribe this text

[Assistant] Put all your money into an index. That's what he recommends doing. And he says, everyone else who tells you otherwise is just trying to steal your money. Rip you off. So that's the main financial advice.


[User] Please transcribe this text

[Assistant] That, uhm, economists give now. It's Shakespeare's advice.


[User] Please transcribe this text

[Assistant] Don't put all your eggs in one basket, but it's very specific. Put it in an index where you buy everything, uh, in the same proportion. Now of course if you're buying everything in the same proportion, and everybody's doing that, what does this have to be?


[User] Please transcribe this text

[Assistant] So I forgot to say that, if everybody's buying every stock in the same proportion, that proportion must be the market, because that's what everybody's holding. So this doesn't turn out to be any old portfolio, it's the market portfolio. So the index, that's a huge conclusion here,


[User] Please transcribe this text

[Assistant] The index that they should offer is not just some magical index that's better than everything else. Because if everyone's doing the right thing, everyone's choosing the right index, the same index. That's what we proved. If everyone's choosing the same index,


[User] Please transcribe this text

[Assistant] It must be the market index because that's all there is to choose. Supply is equal to demand. So therefore the, the advice is to put all your money into the market index. Not just an index, the market index. Plus maybe put money in a bank.


[User] Please transcribe this text

[Assistant] Or even borrow from the bank to put it into the market index.


[User] Please transcribe this text

[Assistant] That's the first, uh, surprising theorem that Tobin, uh, proved and it had dramatic, uh, dramatic, so it created the


[User] Please transcribe this text

[Assistant] I forgot the guy's name, who runs Vanguard, he's always giving these speeches.


[User] Please transcribe this text

[Assistant] Uhm, so that's how he describes how he got his start in his business. All he says is, I take all the recommendations of economic science, he says, and I allow people to carry them out. That's why I'm doing great good for society. While everybody else, all these hedge funds and stuff, they're ruining society. I'm just doing what Tobin told us to do. That's his basic speech. Uhm, okay. So Markowitz was a little bit more precise even than that. The Markowitz formula in Tobin's special case, he not only told you


[User] Please transcribe this text

[Assistant] What to do? He told you, you know, do what everyone else is doing. He told you what that should be. Okay, so let's suppose that I've got two stocks, i and j, with the same, uhm,


[User] Please transcribe this text

[Assistant] Same return, same expected return, but one has


[User] Please transcribe this text

[Assistant] Uh, standard deviation three times higher than the other.


[User] Please transcribe this text

[Assistant] That's standard picture. We've got the standard deviation. We've got the expectation.


[User] Please transcribe this text

[Assistant] We've got one stock here, and we've got another stock here, with three times the standard deviation.


[User] Please transcribe this text

[Assistant] Okay, so you might have thought, well these two stocks, same expected return.


[User] Please transcribe this text

[Assistant] What's the difference? Put all my money in one or the other, doesn't matter. A little bit further thought would say, well this one is much better than that one. I, you know, I've got smaller standard deviation. So I should put all my money into this one.


[User] Please transcribe this text

[Assistant] But then on further thought, given that these two are...


[User] Please transcribe this text

[Assistant] Independent, so they're independent, what should you do? How should you allocate your money between this stock and that stock?


[User] Please transcribe this text

[Assistant] Well, this is the, the ratio of money in one stock to the other stock looks like this ratio.


[User] Please transcribe this text

[Assistant] But I've told you that the expectations of the two stocks are the same.


[User] Please transcribe this text

[Assistant] So those things are the same. I've told you that one has three times the standard deviation than the other. So therefore, what should you do?


[User] Please transcribe this text

[Assistant] One sigma is three times the other, and you square it, it's gonna be nine times the other, and so obviously you're putting it, you know, and then you have to figure out what's in the numerator and what's in the denominator, but clearly you're gonna put more of your money in the lower standard deviation, so it's 90, 10. Okay, so very concrete advice. Okay, so the first concrete advice is,


[User] Please transcribe this text

[Assistant] First concrete thing we've gotten out of this is by the index.


[User] Please transcribe this text

[Assistant] Okay, now, the index typically in America, they're easy to buy, are indexes of stocks. But you know, the second piece of advice is,


[User] Please transcribe this text

[Assistant] And, and this is also Swenson's second most important point. I mean,


[User] Please transcribe this text

[Assistant] I'm quoting Swenson because we're here at Yale, not because he's the only one to recognize these things, but he was a student of Tobin. So the second important point is that, uhm, you should not just buy the index,


[User] Please transcribe this text

[Assistant] Chinese stocks, French stocks, Greek stocks, Israeli stocks. You should have all stocks in the index. You can't really very easily buy all those other things. You can only very easily buy index of American things. It's getting easier and easier now to buy index, you know, the global index. But anyway, you should buy the global index. That's what this says.


[User] Please transcribe this text

[Assistant] Okay, so every time you find a new independent risk, you should look for it. So you should, by the global index, seek out independent risk.


[User] Please transcribe this text

[Assistant] One of his most important things was Yale was always looking globally to find independent risk. So let's go back to the, uh, Okay, so you should always find independent risk.


[User] Please transcribe this text

[Assistant] Okay, and you suffer if you don't. So you see, by putting 90% here and 10% here, Markowitz is telling you, you're going to do better. So, alright. What else do I have to say?


[User] Please transcribe this text

[Assistant] Okay, so let's try, go back to our dice example. Suppose that you had a one dice. Now, by the way, I want to introduce one more term. Okay, so, which I'll call the sharp ratio. So the slope of this line


[User] Please transcribe this text

[Assistant] So it's the Tobin diagram, it should be called the Tobin ratio, but this other guy at Stanford snuck in and


[User] Please transcribe this text

[Assistant] Managed to give his name to it, and he ended up getting a Nobel Prize too. Uhm, so he, so here this line, the slope of this line, this is the riskless asset.


[User] Please transcribe this text

[Assistant] Okay, the slope of this line, what is the slope of this line?