WEBVTT 00:01.690 --> 00:03.410 Prof: Okay, but now I want to move to the 00:03.408 --> 00:05.978 next topic, which is the topic called the 00:05.975 --> 00:10.145 Capital Asset Pricing Model, and it's in some points the 00:10.153 --> 00:12.123 high point of the class. 00:12.120 --> 00:15.090 It used to be the high point of finance. 00:15.090 --> 00:19.770 The theory hasn't worked out as well as people thought in recent 00:19.771 --> 00:22.001 times, but it's quite a great 00:21.997 --> 00:25.677 achievement and a lot of it was done here at Yale, 00:25.680 --> 00:27.810 so I want to explain it to you. 00:27.810 --> 00:29.970 So you see we have a problem so far. 00:29.970 --> 00:34.200 If everybody's trying to hedge that means everybody's trying to 00:34.201 --> 00:36.591 get a completely riskless payoff. 00:36.590 --> 00:39.580 It's impossible because, I mean, there's real risk in 00:39.583 --> 00:40.393 the economy. 00:40.390 --> 00:43.300 And what do we mean by real risk? 00:43.300 --> 00:46.470 Well, something in one state is just going to be bad for the 00:46.471 --> 00:48.731 whole economy compared to another state. 00:48.730 --> 00:50.860 Maybe we'll run out of oil or something like that. 00:50.860 --> 00:54.420 It's impossible that everybody can consume the same thing in 00:54.415 --> 00:56.705 every state, so it's impossible that 00:56.708 --> 00:59.678 everybody can perfectly hedge, but everybody wants to 00:59.683 --> 01:00.493 perfectly hedge. 01:00.490 --> 01:01.990 So what has to happen? 01:01.990 --> 01:02.860 What gives? 01:02.859 --> 01:04.759 How does the theory have to change? 01:04.760 --> 01:09.640 Well, the theory's going to change in a simple way, 01:09.640 --> 01:12.480 which Shakespeare himself already knew and already told us 01:12.483 --> 01:14.633 about in The Merchant of Venice. 01:14.629 --> 01:18.139 What's going to happen is everybody is going to try and 01:18.137 --> 01:21.057 hedge as much as they can by diversifying, 01:21.060 --> 01:24.290 but because there's some real risk in the economy, 01:24.290 --> 01:29.640 in some states things will be in the aggregate worse than they 01:29.638 --> 01:31.918 will be in other states. 01:31.920 --> 01:34.850 So what's the consequence of that? 01:34.849 --> 01:37.869 The consequence of that is if you're going to buy an asset 01:37.867 --> 01:40.937 that pays something that's riskless you're going to pay the 01:40.937 --> 01:43.317 discounted expected return of the asset, 01:43.319 --> 01:46.649 but if you're going to buy an asset that's risky you're going 01:46.646 --> 01:50.026 to need a higher rate of return so the price will be less than 01:50.029 --> 01:51.969 the expected discounted payoff. 01:51.970 --> 01:54.330 So Shakespeare, remember, said exactly that. 01:54.330 --> 01:58.780 When the play begins with Antonio looking melancholy his 01:58.778 --> 02:02.498 interlocutor says, Salerio or somebody asks him 02:02.498 --> 02:06.218 whether he's worried about his businesses. 02:06.218 --> 02:08.368 He says, "No, I've got a different 02:08.372 --> 02:10.922 ship--every ship's on a different ocean so I'm 02:10.923 --> 02:11.833 diversified. 02:11.830 --> 02:13.210 I'm not that worried." 02:13.210 --> 02:15.750 So Shakespeare knows about diversification and that's what 02:15.746 --> 02:18.556 everybody should do, but then when it comes time to 02:18.556 --> 02:21.886 pick the caskets to get to marry the beautiful Portia, 02:21.889 --> 02:24.449 who by the way is not just beautiful but she's rich so 02:24.450 --> 02:28.320 they're looking for a prize, but they sign a contract that 02:28.318 --> 02:33.298 whoever picks the wrong casket not only doesn't get Portia, 02:33.300 --> 02:35.940 he can never marry anyone in the future. 02:35.940 --> 02:37.960 So what's the purpose of that contract? 02:37.960 --> 02:40.310 It's to make it a very risky gamble. 02:40.310 --> 02:42.770 And so why did Shakespeare want to make it a risky gamble, 02:42.765 --> 02:45.045 so he could explain he understands risk and return. 02:45.050 --> 02:46.650 So you remember the conversation where everybody 02:46.646 --> 02:47.956 says, "Well, I'm not going to 02:47.955 --> 02:49.675 pick this unless, you know, it's only because 02:49.676 --> 02:51.936 she's so rich and so beautiful that I'm willing to do this. 02:51.940 --> 02:55.070 The return is so high that it's worth the risk to me." 02:55.068 --> 02:58.048 So Shakespeare already understood that things that are 02:58.054 --> 03:01.434 risky are going to have to be priced less than their expected 03:01.433 --> 03:03.463 return-- expected payoff--so the 03:03.455 --> 03:06.055 expected return, that is the payoff per dollar 03:06.056 --> 03:09.096 put into it looks higher to compensate you for the risk. 03:09.098 --> 03:11.308 So Shakespeare almost had the whole story. 03:11.310 --> 03:13.130 What's missing from Shakespeare? 03:13.128 --> 03:16.468 Well, what is the definition of risk is missing from 03:16.470 --> 03:20.470 Shakespeare, and it will turn out that it's going to be a very 03:20.466 --> 03:22.166 surprising definition. 03:22.169 --> 03:25.329 So the purpose of the model I'm going to explain is, 03:25.330 --> 03:28.070 how do you measure risk, and how should that affect the 03:28.066 --> 03:30.846 price of things, and how does that affect all 03:30.845 --> 03:32.915 the analysis we've done so far? 03:32.919 --> 03:35.449 So that's the topic of the next couple lectures. 03:35.449 --> 03:40.189 So the first person to confront this problem and propose a 03:40.186 --> 03:44.756 solution, a mathematical solution, was the mathematician 03:44.757 --> 03:47.247 Bernoulli and his brother. 03:47.250 --> 03:50.220 So the Bernoullis were a famous mathematical family, 03:50.221 --> 03:52.611 and one of the brothers went off to St. 03:52.610 --> 03:58.660 Petersburg where he ended up dying shortly afterwards, 03:58.662 --> 04:05.062 but he noticed the following puzzle, and some of you have 04:05.057 --> 04:08.597 heard this, it's called the St. 04:08.598 --> 04:11.338 Petersburg Paradox. 04:11.340 --> 04:13.410 So suppose I offer you a bet. 04:13.408 --> 04:18.898 I say I'm going to flip a coin, and I keep flipping the coin 04:18.898 --> 04:23.908 until it comes up tails, and I count how many coin flips 04:23.913 --> 04:26.743 I've counted until you get tails, 04:26.740 --> 04:30.680 and if that's N flips you get 2 to the N dollars. 04:30.680 --> 04:34.080 So if you flip it 1 time and it comes up tails right away, 04:34.083 --> 04:37.013 which is probability 1 half, you get 2 dollars. 04:37.009 --> 04:41.019 If I flip it 2 times and it gets heads and then tails, 04:41.016 --> 04:45.326 that's with probability 1 quarter, you've flipped it twice 04:45.327 --> 04:48.047 you get 2 to the 2 or 4 dollars. 04:48.050 --> 04:50.490 If I flip it three times and I get heads, 04:50.490 --> 04:53.510 heads, tails the odds of that are 1 half times 1 half times 1 04:53.507 --> 04:55.037 half, which is 1 eighth, 04:55.038 --> 04:57.548 but then you'll get 2 to the 3 dollars. 04:57.550 --> 05:07.560 So 4 8 1 over 2 to the N times 2 to the N .... 05:07.560 --> 05:10.100 So Bernoulli, the one who died, 05:10.101 --> 05:14.341 told his brother Daniel about this and he said, 05:14.338 --> 05:16.598 "Well, I've offered this bet to a bunch of people and I 05:16.603 --> 05:19.503 asked them, how much would they be willing 05:19.502 --> 05:22.262 to pay for this risky asset?" 05:22.259 --> 05:25.309 I mean, what would you pay? 05:25.310 --> 05:26.130 Let's hear some numbers? 05:26.129 --> 05:29.639 How many dollars would you pay if I offered you this bet? 05:29.639 --> 05:33.299 I'm just going to keep flipping a fair coin, count the number of 05:33.300 --> 05:36.380 flips until a tails and pay you 2 to the N dollars. 05:36.379 --> 05:41.219 So this is the expectation which obviously equals infinity. 05:41.220 --> 05:43.890 So according to what we've said so far you should pay infinite 05:43.886 --> 05:46.236 amount of dollars for it, but Bernoulli couldn't get 05:46.240 --> 05:47.850 anyone to offer him that much money. 05:47.850 --> 05:50.210 How much would you offer for this bet? 05:50.209 --> 05:53.499 I just want to hear some numbers. 05:53.500 --> 05:54.720 Student: 1 dollar fifty. 05:54.720 --> 05:56.630 Prof: 1 dollar fifty. 05:56.629 --> 06:01.059 Anybody else have any-- it's pretty conservative, 06:01.064 --> 06:05.134 I mean, that's almost ridiculous, in fact. 06:05.129 --> 06:08.089 You're guaranteed 2 dollars no matter what happens, 06:08.086 --> 06:08.556 right? 06:08.560 --> 06:10.850 So you're paying 1 dollar fifty and you're going to get 2 for 06:10.845 --> 06:13.115 sure, so that's a pretty conservative 06:13.115 --> 06:15.465 number to say, anybody a little more 06:15.466 --> 06:16.926 venturesome than that? 06:16.930 --> 06:19.810 You can't do worse than 2 dollars in this bet. 06:19.810 --> 06:20.540 Student: 4 dollars. 06:20.540 --> 06:21.170 Prof: What? 06:21.170 --> 06:21.780 Student: 4 dollars. 06:21.779 --> 06:23.169 Prof: 4 dollars. 06:23.170 --> 06:27.580 All right, so Bernoulli asked a bunch of people and the average 06:27.576 --> 06:31.126 of what they say happens to have been 4 dollars. 06:31.129 --> 06:33.989 That's what they said on average. 06:33.990 --> 06:35.720 And so Bernoulli said, "Well, this is amazing. 06:35.720 --> 06:38.880 The expectation is infinite and they're only willing to pay me a 06:38.882 --> 06:40.742 miserable 4 dollars for this." 06:40.740 --> 06:43.720 Now, the real reason might have been that they didn't believe 06:43.723 --> 06:46.163 Bernoulli was actually going to pay the money, 06:46.160 --> 06:48.260 and they'd give up their money and they weren't going to get 06:48.262 --> 06:49.952 anything back, but let's ignore that 06:49.949 --> 06:51.749 temporarily and take it seriously. 06:51.750 --> 06:55.060 Bernoulli said, the solution must be that 06:55.064 --> 06:58.714 people don't care about the dollar payoff. 06:58.709 --> 07:01.119 They care about the utility of the dollar payoff. 07:01.120 --> 07:03.500 So let's put in a utility function. 07:03.500 --> 07:10.440 So the utility of the dollar payoff would be [one half] 07:10.437 --> 07:17.627 U of 2 1 quarter U of 4 1 eighth U of 8 1 over 2 to the N 07:17.634 --> 07:20.594 U of 2 to the N ... 07:20.589 --> 07:23.719 And so then he said--well, of course why is that going to 07:23.718 --> 07:24.108 help? 07:24.110 --> 07:26.790 Well, because if the utility function, 07:26.790 --> 07:31.090 say, looks like this--so here's X and here's U of X-- 07:31.089 --> 07:33.909 the more dollars you get, maybe it increases utility but 07:33.913 --> 07:36.803 by less and less, so you're not really gaining 07:36.798 --> 07:39.718 much by getting these numbers way out here. 07:39.720 --> 07:42.250 They're not adding really much to utility, so you only care 07:42.254 --> 07:43.484 about these small numbers. 07:43.480 --> 07:46.860 I mean, it's good to get more, but not much better to have 07:46.855 --> 07:47.265 more. 07:47.269 --> 07:52.179 So he said, lo and behold, if I put in log natural as my 07:52.175 --> 07:55.645 utility function, which this looks like, 07:55.654 --> 08:00.564 that's the graph of log natural, and I put this in. 08:00.560 --> 08:07.100 Now, you see this is easy to solve, to compute, 08:07.101 --> 08:13.931 because log of 2 to the N is N times log of 2. 08:13.930 --> 08:16.960 So the log of 2s come out and it's just the sum. 08:16.959 --> 08:27.059 It's log of 2 times the sum of 1 over 2 to the N times N. 08:27.060 --> 08:28.600 So it's N over 2 to the N. 08:28.600 --> 08:33.810 So this is equal to log of 2 times the sum, 08:33.812 --> 08:39.772 N equals 1 to the infinity, N over 2 to the N. 08:39.769 --> 08:45.939 That's what it turns out, that this thing is that. 08:45.940 --> 08:48.490 So it's not just 1 over 2 to the N which would have added up 08:48.491 --> 08:49.791 to 1, but N over 2 to the N. 08:49.788 --> 08:54.268 So the point is because you have the log function here this 08:54.273 --> 08:56.673 actually equals the log of 4. 08:56.668 --> 08:59.738 So anyway, I've worked out the arithmetic. 08:59.740 --> 09:00.540 It's very simple. 09:00.538 --> 09:03.028 You all know how to sum 1 over 2 to the N. 09:03.028 --> 09:05.538 You probably don't know how to sum N over 2 to the N. 09:05.538 --> 09:08.028 You never thought of doing it before, but the same trick gets 09:08.028 --> 09:09.768 you to be able to sum N over 2 to the N. 09:09.769 --> 09:12.249 It's obviously more than 1 over 2 to the N, in fact, 09:12.253 --> 09:13.133 it's equal to 2. 09:13.129 --> 09:17.409 So this sum is equal to 2 and 2 times log of 2 is log of 2 09:17.414 --> 09:19.524 squared which is log of 4. 09:19.519 --> 09:24.659 So by plugging in--instead of caring about expectation you 09:24.662 --> 09:27.732 care about the expected utility. 09:27.730 --> 09:30.860 You can explain why the average person was willing to pay 4 09:30.860 --> 09:34.210 dollars because the expected log of this is equal to the log of 09:34.206 --> 09:34.526 4. 09:34.529 --> 09:36.859 This is equal to the log of 4. 09:36.860 --> 09:40.750 So Bernoulli thought he'd brilliantly explained his 09:40.750 --> 09:41.530 paradox. 09:41.529 --> 09:42.919 So this is the other brother, Daniel. 09:42.918 --> 09:44.998 The dead brother posed the problem, 09:45.000 --> 09:47.340 maybe solved it too for all I know, 09:47.340 --> 09:53.550 but the other brother who still lived came up with the solution, 09:53.548 --> 09:56.988 maybe with his brother, that people don't look at 09:56.994 --> 10:00.124 expected payoffs, they look at expected utility 10:00.120 --> 10:03.000 of payoffs, and the utility should have 10:03.000 --> 10:07.480 this concave feature that more and more payoff adds on the 10:07.475 --> 10:09.985 margin less and less utility. 10:09.990 --> 10:16.770 So this function satisfies d squared U (X) / dX squared is 10:16.773 --> 10:18.443 less than 0. 10:18.440 --> 10:20.470 The second derivative is negative. 10:20.470 --> 10:23.210 So the marginal utility is declining as you get more and 10:23.211 --> 10:23.561 more. 10:23.558 --> 10:30.438 So that was the first advance on how to deal with risk. 10:30.440 --> 10:35.470 Now, actually Bernoulli didn't really solve the problem because 10:35.474 --> 10:40.594 just saying that you replace the payoff with expected utility of 10:40.591 --> 10:43.871 a concave function-- this log wouldn't have really 10:43.874 --> 10:47.164 solved the problem, because suppose that Bernoulli 10:47.158 --> 10:52.028 had offered instead a bet not of 2 to the N but of 2 to the 2 to 10:52.027 --> 10:54.617 the N, a much more generous bet? 10:54.620 --> 10:58.890 Then even with logs you would have gotten an infinite number. 10:58.889 --> 11:03.299 So basically he should have said that people care about a 11:03.297 --> 11:08.177 concave function of payoff where the function is bounded unlike 11:08.177 --> 11:10.457 log which is not bounded. 11:10.460 --> 11:11.820 But anyway, let's leave that aside. 11:11.820 --> 11:14.880 It should be a concave function. 11:14.879 --> 11:18.049 So to put it another way, a concave function has the 11:18.052 --> 11:20.022 property, if you look at it, 11:20.023 --> 11:24.103 let's say it looks like that, that if you have this payoff 11:24.099 --> 11:28.389 X_A and this payoff X_B and you've got-- 11:28.389 --> 11:33.349 so this is the utility now here, X_A, 11:33.350 --> 11:36.060 and this is the utility U of X_B, 11:36.058 --> 11:40.078 and this is the utility U of X_A. 11:40.080 --> 11:43.430 If you have a 50/50 bet of either getting X_A or 11:43.426 --> 11:46.536 getting X_B you're going to end up with this 11:46.537 --> 11:47.767 expected utility. 11:47.769 --> 11:50.249 Your utility is going to be 1 half of U X_A 1 half 11:50.248 --> 11:51.128 of U X_B. 11:51.129 --> 11:53.969 That's what we had down here, 1 half of this utility plus 11:53.970 --> 11:56.810 half of that utility assuming nothing else can happen. 11:56.808 --> 12:02.658 But if you give the person half the amounts of money for sure 12:02.658 --> 12:08.798 then he gets this utility which is much bigger than that utility 12:08.798 --> 12:12.598 because this is a concave function. 12:12.600 --> 12:15.230 The extra you gain by winning the bet, 12:15.230 --> 12:17.910 compared to getting the average for sure, 12:17.908 --> 12:20.738 the extra you gain doesn't drive the utility up very much 12:20.735 --> 12:22.245 because it's flattening out. 12:22.250 --> 12:24.430 Whereas losing the bet, even though you're losing the 12:24.432 --> 12:27.032 same number of dollars because from here to here is the same as 12:27.033 --> 12:29.443 from here to here, the loss of the same number of 12:29.443 --> 12:32.533 dollars is more important to you than the gain of an equal amount 12:32.527 --> 12:35.017 of dollars, and that's why you'd rather get 12:35.023 --> 12:38.513 the middle for sure than having a 50/50 chance of going on the 12:38.513 --> 12:39.203 extremes. 12:39.200 --> 12:43.750 So Bernoulli pointed the way to the modern theory of risk 12:43.754 --> 12:49.814 aversion, which is to just assume--risk 12:49.806 --> 13:00.306 aversion in modern economics means people care about expected 13:00.308 --> 13:05.548 utility, maybe discounted, 13:05.552 --> 13:16.282 expected discounted utility where the utility is concave. 13:16.278 --> 13:19.788 So whatever utility function we wrote in here, 13:19.791 --> 13:24.241 maybe it shouldn't be log, it should be something else. 13:24.240 --> 13:25.760 How would people evaluate this? 13:25.759 --> 13:27.669 They'd evaluate it log of 4. 13:27.668 --> 13:31.028 In other words, they'd say take whatever that 13:31.028 --> 13:34.078 constant utility was, which was log of 4, 13:34.082 --> 13:38.362 that produces the same utility as the random gamble. 13:38.360 --> 13:42.220 So this random gamble gives this expected utility which is 13:42.221 --> 13:44.731 equivalent to having that for sure. 13:44.730 --> 13:46.950 So here's the 4. 13:46.950 --> 13:51.060 So 4 for sure gives a utility, log of 4, that puts you here 13:51.062 --> 13:54.962 which is the same as the expected utility of getting the 13:54.962 --> 13:56.242 random gamble. 13:56.240 --> 14:01.730 That's the modern theory of risk aversion, 14:01.730 --> 14:05.580 and it explains why people would rather have things for 14:05.578 --> 14:08.458 sure, but it's now quantifiable 14:08.462 --> 14:13.742 because if you can't have something for sure then you know 14:13.735 --> 14:18.685 that it's more dangerous, but with this concrete utility 14:18.686 --> 14:22.916 function you can find out exactly how much you're willing 14:22.917 --> 14:27.447 to pay to transform this risky gamble into a safe gamble. 14:27.450 --> 14:31.390 You'd give up this much expectation in order to get the 14:31.394 --> 14:32.714 payoff for sure. 14:32.710 --> 14:37.130 So we're going to turn a vague theory into something 14:37.128 --> 14:41.198 quantifiable and get a surprising conclusion. 14:41.200 --> 14:42.490 So that's step one. 14:42.490 --> 14:45.620 We now think about people maximizing utility. 14:45.620 --> 14:47.240 Well, of course we thought about that from the beginning. 14:47.240 --> 14:50.570 The very first class you had utility and diminishing marginal 14:50.573 --> 14:51.133 utility. 14:51.129 --> 14:54.719 So actually this risk aversion with diminishing marginal 14:54.721 --> 14:56.381 utility, fortunately for us, 14:56.383 --> 14:59.363 is exactly the same thing we've been thinking about all along 14:59.363 --> 15:01.333 anyway, diminishing marginal utility 15:01.327 --> 15:02.187 for consumption. 15:02.190 --> 15:04.980 So the very assumption of diminishing marginal utility 15:04.975 --> 15:07.815 that we made from the beginning is also explaining risk 15:07.816 --> 15:08.496 aversion. 15:08.500 --> 15:12.280 So it's incredibly fortunate that we don't actually have to 15:12.275 --> 15:15.785 change any of our mathematics and we've explained a new 15:15.793 --> 15:16.773 phenomenon. 15:16.769 --> 15:22.339 Now, the most simple utility function is either the log one 15:22.344 --> 15:24.174 or the quadratic. 15:24.168 --> 15:32.668 So remember, U (X) = a b X - c X squared. 15:32.668 --> 15:37.708 Adding a constant is never going to change anything, 15:37.714 --> 15:43.254 so I'm always going to write this as a X - 1 half alpha X 15:43.253 --> 15:44.443 squared. 15:44.440 --> 15:49.650 That's going to be my utility function, my quadratic utility. 15:49.649 --> 15:51.469 It could be like this or it could be like that. 15:51.470 --> 15:54.320 If you add a constant which doesn't depend on X that's not 15:54.320 --> 15:56.870 changing what anybody does so that's irrelevant, 15:56.870 --> 16:00.720 and if I divide it by a constant like B that's not going 16:00.722 --> 16:04.792 to change what everybody does so I might as well assume the 16:04.785 --> 16:08.495 quadratic utility is X - 1 half alpha X squared, 16:08.500 --> 16:09.530 quadratic utility. 16:09.528 --> 16:12.298 So that's about as simple as we can get and we're used to 16:12.297 --> 16:14.667 working with those kinds of utility functions. 16:14.668 --> 16:19.268 Now, why is that such a good convenient thing for us to use? 16:19.269 --> 16:22.539 It's because let's suppose now that you've got this random 16:22.544 --> 16:25.934 payoff where with probability gamma_1 you're going 16:25.934 --> 16:28.684 to get X_1, probability gamma_2 16:28.682 --> 16:30.242 you're going to get X_2, 16:30.240 --> 16:32.690 probability gamma_S you're going to get 16:32.690 --> 16:35.140 gamma_S [correction: X_S]. 16:35.139 --> 16:38.119 So what's the expected utility, the analog of Bernoulli? 16:38.120 --> 16:45.300 That means U is going to equal summation, s = 1 to S of 16:45.302 --> 16:51.292 (gamma_s X_s- 1 half alpha 16:51.288 --> 16:54.878 X_s squared). 16:54.879 --> 16:56.649 So that's all we're doing. 16:56.649 --> 16:59.049 We're just saying that people don't care about the payoffs. 16:59.048 --> 17:02.218 They have to evaluate getting X_1, 17:02.217 --> 17:04.647 X_2 or X_S. 17:04.650 --> 17:09.710 They're going to multiply the payoff by the expectation but 17:09.707 --> 17:14.587 not look at the payoff itself, look at the utility of the 17:14.589 --> 17:15.549 payoff. 17:15.548 --> 17:18.658 Now, quadratic is very simple and the reason why we're going 17:18.657 --> 17:21.927 to get such a beautiful theory out of it is because this number 17:21.925 --> 17:24.975 you don't have to keep track of all the X's to express this 17:24.981 --> 17:25.721 utility. 17:25.720 --> 17:29.270 We're going to be able to summarize it incredibly simply. 17:29.269 --> 17:35.699 This is going to equal some function F of the expectation of 17:35.703 --> 17:38.543 X and the variance of X. 17:38.538 --> 17:41.188 So all we're going to have to worry about is the expectation 17:41.191 --> 17:43.311 of X and the variance of X, and so many, 17:43.314 --> 17:47.254 many very complicated things we can think about very simply. 17:47.250 --> 17:50.530 So more generally if you put the log instead of the quadratic 17:50.530 --> 17:53.810 utility we couldn't get this simplification and so the theory 17:53.811 --> 17:55.891 would have to be more complicated. 17:55.890 --> 17:59.180 So the beautiful theory, the Capital Asset Pricing 17:59.178 --> 18:03.068 Model, comes out of using this simple quadratic utility. 18:03.068 --> 18:06.088 So why does it get so simplified? 18:06.088 --> 18:11.408 Well, if I just write it out, U is going to equal the 18:11.409 --> 18:16.319 summation of gamma_s X_s, 18:16.318 --> 18:21.368 so this is s = 1 to S (I've let my probabilities be gamma, 18:21.368 --> 18:25.168 I don't know why I chose that) - 1 half alpha, 18:25.170 --> 18:32.640 summation s = 1 to S, of gamma_s 18:32.644 --> 18:37.024 X_s squared. 18:37.019 --> 18:41.359 Well, here we have the expectation of X already. 18:41.358 --> 18:44.398 Now, what is this 1 F alpha gamma_s X_s 18:44.404 --> 18:44.934 squared? 18:44.930 --> 18:51.150 Well, if I wrote X_s - the expectation of X and I 18:51.150 --> 18:57.160 summed this squared that's equal the variance of X--oops, 18:57.155 --> 19:00.155 times gamma_s. 19:00.160 --> 19:04.190 That by definition is the variance, but if I wrote this 19:04.192 --> 19:05.912 out what would I get? 19:05.910 --> 19:10.660 I'd get summation s = 1 to S, gamma_s X_s 19:10.660 --> 19:15.490 squared which is what I have over there and then I'd have-- 19:15.490 --> 19:23.600 well, what would I have-- minus 2 times summation 19:23.599 --> 19:34.249 gamma_s X_s expectation of X summation s = 1 19:34.246 --> 19:43.536 to S of gamma_s expectation of X squared. 19:43.539 --> 19:47.209 But what's this? 19:47.210 --> 19:50.010 The second term minus 2 times that, 19:50.009 --> 19:53.489 the expectation of X is a constant so I can take that out, 19:53.490 --> 19:57.450 minus 2 expectation of X, can take this out and notice 19:57.452 --> 20:01.192 that summation gamma_s X_s, 20:01.190 --> 20:03.940 that's the expectation of X as well. 20:03.940 --> 20:08.480 So that's just minus 2 times the expectation of X squared. 20:08.480 --> 20:13.520 And so this I can take the expectation of X squared out and 20:13.519 --> 20:17.169 the summation of the probabilities is 1. 20:17.170 --> 20:22.620 So therefore I just get equal to summation s = 1 to S, 20:22.623 --> 20:28.693 gamma_s X_s squared - (expectation of X) 20:28.694 --> 20:29.934 squared. 20:29.930 --> 20:35.750 So therefore, this up here is equal to the 20:35.750 --> 20:38.450 expectation of X. 20:38.450 --> 20:39.460 So what have I got here? 20:39.460 --> 20:42.650 I've got this term. 20:42.650 --> 20:45.760 So I've got the variance of X equals this summation 20:45.760 --> 20:49.180 X_s X_s squared - (expectation of X) 20:49.182 --> 20:49.932 squared. 20:49.930 --> 20:54.080 So this term equals the variance of X (expectation of X) 20:54.084 --> 20:54.844 squared. 20:54.838 --> 21:02.858 So therefore I've got this minus 1 half alpha (expectation 21:02.858 --> 21:09.188 of X) squared - 1 half alpha variance of X. 21:09.190 --> 21:13.280 That's what the algebra gives me, so why is that again? 21:13.278 --> 21:18.298 Because given quadratic utility up here, that thing--getting 21:18.297 --> 21:23.647 old--given the quadratic utility up there I can write it as this 21:23.653 --> 21:25.103 in this term. 21:25.098 --> 21:28.288 This term is obviously the expectation of X, 21:28.294 --> 21:31.794 but this term is just the variance of X plus the 21:31.785 --> 21:33.935 expectation of X squared. 21:33.940 --> 21:37.920 So when I subtract it I keep the expectation of X minus the 21:37.920 --> 21:39.910 (expectation of X) squared. 21:39.910 --> 21:43.030 That's the first term, and then I've got minus the 21:43.026 --> 21:46.456 variance of X from that term times the 1 half alpha. 21:46.460 --> 21:51.130 So you see that depends on the expectation of X in a positive 21:51.134 --> 21:55.894 way, assuming alpha's a small number, and in a negative way on 21:55.885 --> 21:57.595 the variance of X. 21:57.598 --> 22:01.668 So, just as I said, somewhere, I said it was going 22:01.667 --> 22:05.567 to turn out like that, and it did right here. 22:05.568 --> 22:08.878 The utility is equal to the expectation of X and the 22:08.884 --> 22:12.794 variance of X in a positive way on the expectation of X and a 22:12.785 --> 22:14.925 negative way on the variance. 22:14.930 --> 22:20.060 So now we're ready to start the analysis. 22:20.058 --> 22:24.118 So far we've assumed people only care about the expectation 22:24.124 --> 22:27.354 and then we said, well, we know they don't only 22:27.346 --> 22:29.586 care about the expectations. 22:29.588 --> 22:32.108 Hedge funds and everybody else, if they know what they're doing 22:32.105 --> 22:34.695 and they're trying to keep their investors happy they're going to 22:34.703 --> 22:35.113 hedge. 22:35.108 --> 22:36.888 We didn't say why they're going to hedge. 22:36.890 --> 22:40.500 We just asserted they like to hedge so their investors don't 22:40.498 --> 22:43.198 get mad at them, but really what we had in mind 22:43.202 --> 22:45.522 is the investors have some utility function. 22:45.519 --> 22:49.339 They don't like risk so the hedge fund is going to try and 22:49.337 --> 22:51.077 keep the payoffs steady. 22:51.079 --> 22:51.979 But there's a tradeoff. 22:51.980 --> 22:54.380 You can't eliminate all risk. 22:54.380 --> 22:57.570 So, how much is the hedge fund and the investor going to suffer 22:57.573 --> 22:59.173 if all risk isn't eliminated? 22:59.170 --> 23:00.720 Now we have a way of quantifying it. 23:00.720 --> 23:04.630 People care about the utility and not about just the expected 23:04.627 --> 23:07.427 payoff and so you add more risk to them-- 23:07.430 --> 23:10.200 you replace a sure thing with a risky thing with the same 23:10.198 --> 23:12.468 expectation-- they think it is worse, 23:12.469 --> 23:13.599 that much worse. 23:13.598 --> 23:17.568 And so we've said that of all the myriad of utility 23:17.567 --> 23:20.517 functions-- we could use log, 23:20.517 --> 23:24.687 some exponential e to the minus aX, 23:24.690 --> 23:27.770 X to the r, there are lots of different utilities we could 23:27.770 --> 23:29.690 use-- we're going to deal with the 23:29.692 --> 23:32.512 quadratic because it has the simple property that in 23:32.505 --> 23:35.535 evaluating an entire risky proposition people care about 23:35.537 --> 23:38.847 the expectation, which is what they cared about 23:38.848 --> 23:41.638 before, but they're punishing 23:41.638 --> 23:45.648 themselves for getting a bad variance. 23:45.650 --> 23:49.450 So because that's such a simple thing to say we're going to get 23:49.449 --> 23:52.819 a simple conclusion and a very surprising conclusion. 23:52.818 --> 24:01.288 So let's now analyze a problem and see what would happen. 24:01.288 --> 24:06.758 So the problem I'm going to choose to analyze is this one. 24:06.759 --> 24:18.249 I'm going to say that three things can happen in the 24:18.247 --> 24:20.497 economy. 24:20.500 --> 24:22.920 Anyway, those are the probabilities. 24:22.920 --> 24:25.090 Now, there are many firms in the economy A, 24:25.088 --> 24:28.698 B and C, and let's say the first firm, 24:28.700 --> 24:32.460 I don't want to invent the numbers here so I might as well 24:32.462 --> 24:34.842 just write down the ones I picked. 24:34.838 --> 24:41.928 The first one's going to be 50,100 and 75. 24:41.930 --> 24:51.430 B, the other firm's going to be 150,180 and 365, 24:51.434 --> 24:59.124 and C is going to be 300,220 and 60. 24:59.118 --> 25:03.668 So those are the three things that can happen in the payoff of 25:03.666 --> 25:05.006 the three firms. 25:05.009 --> 25:16.719 Let's say there are two agents, agent alpha owns A and also 25:16.720 --> 25:22.980 133.5 units of X_0. 25:22.980 --> 25:32.770 And beta owns B & C and--I may have reversed 25:32.773 --> 25:42.363 these two guys--66.5 units of X_0. 25:42.359 --> 25:43.279 So here we go. 25:43.279 --> 25:46.199 So by alpha and beta, I mean, there are a million 25:46.199 --> 25:49.849 alpha agents and a million beta agents so everything could be 25:49.848 --> 25:53.188 scaled by a million because I want a big economy like we 25:53.194 --> 25:54.294 always have. 25:54.289 --> 25:55.359 So there we have it. 25:55.359 --> 25:58.079 We've got now a risky world. 25:58.079 --> 25:59.569 Things can happen. 25:59.569 --> 26:02.189 So what are the utilities? 26:02.190 --> 26:07.250 Let's say utility now of alpha is going to equal--sorry, 26:07.246 --> 26:09.816 I left out the main point. 26:09.818 --> 26:15.868 Utility of alpha is going to equal 1 half X_0-- 26:15.868 --> 26:17.428 I just made up these numbers, by the way, 26:17.430 --> 26:22.330 they're not--so summation s = 1 to 3, 26:22.328 --> 26:25.108 gamma_s, that's the gamma_s up 26:25.108 --> 26:29.438 there, times X_s - 1 over 26:29.440 --> 26:34.740 400 X_s squared, the states. 26:34.740 --> 26:37.180 So there's s_1. 26:37.180 --> 26:38.130 So here are the states. 26:38.130 --> 26:43.440 This is state 1, s = 1, s = 2 and s = 3. 26:43.440 --> 26:49.960 So those are the three possible states just like we had before 26:49.961 --> 26:56.591 with this payoff and those are the payoff of all the assets. 26:56.588 --> 27:00.988 Alpha owns firm A which is producing that output in the 27:00.990 --> 27:04.330 three states, and also owns 133.5 units of 27:04.332 --> 27:05.802 X_0. 27:05.798 --> 27:11.968 So over here alpha's owning 133.5 and beta's owning 65.5 of 27:11.967 --> 27:14.517 consumption at time 0. 27:14.519 --> 27:18.079 So this is time 0. 27:18.078 --> 27:22.458 This is time 1, the end of the year. 27:22.460 --> 27:26.420 So by the end of the year something is going to happen. 27:26.420 --> 27:28.630 There's a lot of uncertainty between now and then. 27:28.630 --> 27:33.250 Some of the firms are going to be paying off in some of the 27:33.247 --> 27:36.827 states and badly in other states and so on. 27:36.828 --> 27:39.718 And so the utility function for alpha, 27:39.720 --> 27:42.810 so he cares about consumption at time 0 and also in each of 27:42.813 --> 27:45.533 the three states, but now he's going to have 27:45.528 --> 27:47.238 these quadratic utilities. 27:47.240 --> 27:55.560 He's going to say to himself, if I just hung onto my A in 27:55.563 --> 28:00.773 state 1 this would be-- if I never traded, 28:00.772 --> 28:05.922 I just hung onto A the utility function would be this quadratic 28:05.915 --> 28:07.405 thing of 133.5. 28:07.410 --> 28:14.530 So it would be 133.5 - 1 over 400, 28:14.528 --> 28:23.438 133.5 squared, plus he would end up with (50 - 28:23.442 --> 28:34.732 1 over 400 times 50 squared) times 1 quarter (100 - 1 over 28:34.731 --> 28:46.621 400 100 squared) times 1 quarter (75 - 1 over 400 75 squared) 28:46.615 --> 28:50.375 times 1 half. 28:50.380 --> 28:52.510 That would be his utility if he never traded. 28:52.509 --> 28:58.679 If he just stuck to A he'd eat his own endowment 133.5 at time 28:58.684 --> 28:59.094 0. 28:59.089 --> 29:00.339 Oh, that isn't true. 29:00.338 --> 29:02.668 I wrote down the wrong utility at time 0. 29:02.670 --> 29:05.050 I said his time 0 utility is 1 half X_0. 29:05.049 --> 29:08.369 So it's 1 half 133.5. 29:08.368 --> 29:13.868 So we get 1 half 133.5, but in the future he'd get 50 29:13.869 --> 29:18.529 in state 1,100 in state 2, and 75 in state 3, 29:18.525 --> 29:23.175 and that's the utility he'd end up with. 29:23.180 --> 29:26.050 But that's not very good for him because he's running this 29:26.053 --> 29:26.863 gigantic risk. 29:26.858 --> 29:30.318 He's got this risk at time, you know--state 1 is a disaster 29:30.317 --> 29:33.597 for him if he just sticks to that, so he doesn't want to 29:33.596 --> 29:34.666 stick to that. 29:34.670 --> 29:37.460 So how should he evaluate the shares of firm A? 29:37.460 --> 29:40.750 How should he evaluate the shares of firm B which he could 29:40.749 --> 29:44.039 get if he gave up some of A, or how should he evaluate the 29:44.038 --> 29:45.248 shares of firm C? 29:45.250 --> 29:46.460 What should he do? 29:46.460 --> 29:52.980 So beta has a similar utility. 29:52.980 --> 29:57.920 Beta's utility, U_beta is going to 29:57.916 --> 30:03.426 equal 3 quarters X_0 the summation, 30:03.430 --> 30:18.310 s = 1 to 3, (gamma_s times X_s - 1 over 800 30:18.308 --> 30:24.308 X_s squared). 30:24.308 --> 30:28.428 So I've made these guys--far from being impatient they seem 30:28.431 --> 30:31.701 to prefer consuming in the future until now. 30:31.700 --> 30:34.320 That was a poor choice of numbers. 30:34.318 --> 30:39.158 This number should be bigger than 1 and this should be bigger 30:39.157 --> 30:43.107 than 1, but anyway I put 1 half and 3 quarters. 30:43.108 --> 30:46.108 So there's impatience built in except it goes the wrong way. 30:46.108 --> 30:49.928 That was just poor choice of numbers, but the rest of it 30:49.928 --> 30:52.148 expresses their risk aversion. 30:52.150 --> 30:55.920 So alpha is looking at the expected payoff of what he gets 30:55.916 --> 30:59.486 to consume in the future, but he's punishing himself by 30:59.486 --> 31:00.606 the variance. 31:00.608 --> 31:03.148 So you look at this formula you see it's not just the 31:03.154 --> 31:05.554 expectation, but he loses something because of the 31:05.553 --> 31:06.193 variance. 31:06.190 --> 31:08.210 And similarly, beta, he's looking at 31:08.213 --> 31:10.553 consumption today, he's adding to that the 31:10.554 --> 31:13.394 expectation of his consumption tomorrow for his utility, 31:13.390 --> 31:19.010 but he's punishing himself for having variance in the future. 31:19.009 --> 31:22.769 So it's exactly what we formalized, 31:22.769 --> 31:28.009 Shakespeare's idea of people not liking to be exposed to 31:28.007 --> 31:30.527 variance, to uncertainty, 31:30.532 --> 31:34.592 which we've quantified by calling variance. 31:34.589 --> 31:35.779 Is everyone with me now? 31:35.779 --> 31:36.199 Yes? 31:36.200 --> 31:37.550 Good, I'm glad you have a question. 31:37.548 --> 31:39.168 Student: I don't understand how you got 1 over 31:39.165 --> 31:39.815 400 and 1 over 800. 31:39.818 --> 31:41.718 Prof: I just made up those numbers. 31:41.720 --> 31:44.150 That's the utility of alpha and that's the utility of beta. 31:44.150 --> 31:45.620 I could pick any people I wanted. 31:45.619 --> 31:47.979 I just picked those two people. 31:47.980 --> 31:49.630 Now, how do they differ? 31:49.630 --> 31:56.100 Which person is more afraid of risk than the other? 31:56.098 --> 31:58.548 Is alpha or beta more afraid of risk? 31:58.549 --> 32:00.609 Student: Alpha. 32:00.608 --> 32:01.768 Prof: Alpha is more afraid of risk, 32:01.772 --> 32:02.002 right? 32:02.000 --> 32:05.910 This 1 over 800 is smaller than 1 over 400, 32:05.910 --> 32:09.230 so beta doesn't really care that much about risk, 32:09.230 --> 32:12.910 well cares, but is not going to punish himself too much by being 32:12.907 --> 32:13.957 exposed to risk. 32:13.960 --> 32:16.670 Alpha is not going to punish himself too much, 32:16.673 --> 32:19.513 but is going to punish himself somewhat more. 32:19.509 --> 32:21.879 So alpha is more risk averse. 32:21.880 --> 32:32.900 Alpha is more afraid of risk, it seems. 32:32.900 --> 32:36.430 So I've taken two agents who are afraid of risk, 32:36.430 --> 32:39.540 one's more afraid than the other, and I've put them in an 32:39.537 --> 32:42.697 economy where there are risky things that could happen. 32:42.700 --> 32:48.780 And so we now want to work out a more sophisticated version of 32:48.777 --> 32:53.557 pricing and of equilibrium than we had before. 32:53.558 --> 32:58.278 So let me remind you that what we sort of have been supposing 32:58.277 --> 33:02.867 up until now is that the price-- what would the price of A be if 33:02.873 --> 33:05.173 we didn't think about risk aversion? 33:05.170 --> 33:13.450 So far what we would say--what would you say the price of A is, 33:13.453 --> 33:15.863 price of firm A? 33:15.858 --> 33:19.938 If we were naive you might say it is 1 quarter-- 33:19.940 --> 33:23.600 by the way, I hope I have those probabilities right-- 33:23.598 --> 33:34.098 you'd say it is 1 quarter times 50 1 quarter times 100 1 half 33:34.095 --> 33:36.015 times 75. 33:36.019 --> 33:40.709 Is that what we would have said up until now? 33:40.710 --> 33:43.090 Even up until now we would have been more sophisticated than 33:43.086 --> 33:43.366 that. 33:43.369 --> 33:44.279 Student: Discounted. 33:44.279 --> 33:50.319 Prof: Discounted, times discounted. 33:50.318 --> 33:53.018 That's what we sort of figured up until now. 33:53.019 --> 33:54.889 That's the logical thing to do. 33:54.890 --> 33:59.790 Well, but we ignored risk aversion, and we ignored it at 33:59.785 --> 34:03.785 our peril because it's obviously important. 34:03.788 --> 34:05.848 I mean, Shakespeare, a literary person, 34:05.848 --> 34:09.058 he understood already 400 years ago that risk aversion was 34:09.057 --> 34:11.497 important, and there are facts that 34:11.498 --> 34:14.418 confirm what Shakespeare's intuition is. 34:14.420 --> 34:17.590 The stock market historically has had a lot higher return than 34:17.592 --> 34:18.532 the bond market. 34:18.530 --> 34:21.040 Even with the last stock market crash, 34:21.039 --> 34:24.419 of course it came back a lot, averaged since 1926 the stock 34:24.416 --> 34:28.026 market's made something like 9 percent a year compared to 2 and 34:28.027 --> 34:30.237 1 half percent in the bond market. 34:30.239 --> 34:33.489 So there's a huge disparity and after over such a long period of 34:33.487 --> 34:36.427 time it can't just be it was luckier every year after year 34:36.425 --> 34:37.195 after year. 34:37.199 --> 34:40.019 Somehow people must have realized the stock market is 34:40.021 --> 34:42.211 riskier, and so as Shakespeare said they 34:42.211 --> 34:45.501 wanted a higher return meaning they were paying a lower price, 34:45.500 --> 34:46.890 but how much lower? 34:46.889 --> 34:48.659 How can you figure out how much lower? 34:48.659 --> 34:50.349 So in this example, in other words, 34:50.347 --> 34:51.587 what is the price of A? 34:51.590 --> 34:56.440 So this is the wrong price of A, apparently, 34:56.442 --> 35:01.522 because it doesn't recognize risk aversion. 35:01.519 --> 35:04.509 So that's where we are. 35:04.510 --> 35:06.720 So any questions about what the question is? 35:06.719 --> 35:07.939 We're about to give an answer. 35:07.940 --> 35:11.490 So you see what the question is, that our old methodology for 35:11.489 --> 35:15.049 figuring out prices-- that's taking expectation and 35:15.054 --> 35:17.284 discounting-- obviously can't be right 35:17.284 --> 35:19.424 because it doesn't recognize risk aversion. 35:19.420 --> 35:21.460 On the other hand, we always had a utility 35:21.458 --> 35:23.448 function in there from the beginning, 35:23.449 --> 35:26.229 even a quadratic one, so all we have to do is do what 35:26.228 --> 35:29.058 we did before and put in a quadratic utility and we'll 35:29.061 --> 35:30.881 probably get the right answer. 35:30.880 --> 35:33.360 So that's exactly what we're going to do. 35:33.360 --> 35:37.720 So Arrow, in 1951, this is the same guy who proved 35:37.721 --> 35:42.261 with Debreu the Pareto efficiency of equilibrium. 35:42.260 --> 35:44.570 He was my thesis advisor. 35:44.570 --> 35:48.820 He said we can do the same trick that Fisher did, 35:48.820 --> 35:53.070 only for some reason he never credited Fisher. 35:53.070 --> 35:54.760 I could never quite figure that out. 35:54.760 --> 35:57.580 He had some obscure Danish guy he credited. 35:57.579 --> 36:17.159 But anyway, apply Fisher trick and assume firm dividends are 36:17.159 --> 36:24.129 part of endowments. 36:24.130 --> 36:32.250 Look for GE, the general equilibrium, 36:32.246 --> 36:39.006 trading outputs, trading goods, 36:39.010 --> 36:48.480 then go back to deduce value of firms. 36:48.480 --> 36:50.530 Now, what goods are we trading here? 36:50.530 --> 36:53.400 That was a conceptual advance. 36:53.400 --> 36:59.380 We call them Arrow and then Debreu got involved too. 36:59.380 --> 37:03.200 Arrow-Debreu, so Debreu was the Yale 37:03.195 --> 37:09.185 Assistant Professor while Arrow was a Stanford Assistant 37:09.190 --> 37:15.730 Professor, so Arrow-Debreu State Contingent Commodities. 37:15.730 --> 37:18.730 So, just as Fisher said, an apple today and an apple 37:18.726 --> 37:20.696 next year, even though they're identical 37:20.695 --> 37:22.395 apples, are different commodities with 37:22.403 --> 37:24.893 different prices because they come at different places in 37:24.893 --> 37:25.253 time. 37:25.250 --> 37:28.370 In fact, most people would prefer the apple today to the 37:28.369 --> 37:29.389 apple next year. 37:29.389 --> 37:35.029 So Arrow said an apple in the top state is a different 37:35.025 --> 37:40.235 commodity from the same apple in the second state, 37:40.235 --> 37:44.485 so it should have a different price. 37:44.489 --> 37:46.759 So we've got just our conventional equilibrium 37:46.762 --> 37:49.842 according to Arrow where as long as you have these Arrow state 37:49.844 --> 37:52.224 contingent commodities that you can trade-- 37:52.219 --> 37:55.189 trading today, you can imagine today buying an 37:55.186 --> 37:59.336 apple if state 1 occurs but not having to get the apple if state 37:59.340 --> 38:02.240 2 or state 3 occurs, and that'll have a price 38:02.239 --> 38:03.019 P_1. 38:03.018 --> 38:05.588 And today you could imagine buying the apple if state 2 38:05.585 --> 38:07.615 occurs, a different price from the 38:07.619 --> 38:10.899 apple if state 1 occurs, and also an apple if state 3 38:10.896 --> 38:12.826 occurs, which obviously is going to be 38:12.827 --> 38:15.357 more expensive, or it looks like it'll be more 38:15.360 --> 38:18.920 expensive than the other apples because it's 50 percent likely 38:18.922 --> 38:21.252 to happen, and those are the prices we 38:21.246 --> 38:22.246 have to look for. 38:22.250 --> 38:26.000 And that's going to solve our problem because the prices of 38:25.998 --> 38:29.228 the Arrow securities are going to be different, 38:29.230 --> 38:32.770 maybe, from the probabilities and that's what will reflect the 38:32.766 --> 38:35.606 fact that when everybody's trying to hedge and not 38:35.608 --> 38:39.608 everybody can do it you're going to have to change the tradeoffs. 38:39.610 --> 38:41.930 So we've already seen this in our gambling thing, 38:41.927 --> 38:42.987 at least the prices. 38:42.989 --> 38:45.989 Remember with our bookies the bookies were effectively 38:45.994 --> 38:49.634 willing--remember there were two outcomes, the Yankees win or the 38:49.625 --> 38:50.585 Phillies win. 38:50.590 --> 38:55.550 You could get the bookie who thought the odds were 60/40, 38:55.550 --> 38:58.740 by paying 60 cents today the bookie was going to give you 1 38:58.742 --> 39:02.262 dollar if the Yankees won, or paying 40 cents today the 39:02.257 --> 39:05.747 bookie will give you 1 dollar if the Phillies won. 39:05.750 --> 39:08.320 So we've already had these Arrow contracts, 39:08.317 --> 39:11.677 these Arrow securities implicitly in our equilibrium. 39:11.679 --> 39:15.589 And those 60/40 odds those were the opinions of the bookie, 39:15.592 --> 39:18.092 maybe not the actual probabilities. 39:18.090 --> 39:21.820 We said the final betting odds depended on what the other 39:21.822 --> 39:23.892 bookies were willing to give. 39:23.889 --> 39:25.889 It didn't have to correspond to reality. 39:25.889 --> 39:27.609 There might not be a reality even. 39:27.610 --> 39:29.810 So here there's a reality, 25,25, 39:29.809 --> 39:32.209 50, but that doesn't mean that the odds, 39:32.210 --> 39:34.630 the prices they're going to quote in the market are going to 39:34.628 --> 39:35.528 turn out to be that. 39:35.530 --> 39:41.170 We have to solve for equilibrium and see what they 39:41.168 --> 39:41.858 are. 39:41.860 --> 39:43.300 So what's going to happen? 39:43.300 --> 39:48.910 Well, we can solve for equilibrium very easily because 39:48.909 --> 39:53.249 we've done this a million times before. 39:53.250 --> 39:56.280 And I've chosen linear quadratic utilities, 39:56.277 --> 39:59.587 the kind we did on the very first day of class, 39:59.594 --> 40:03.564 because those are easiest to solve for equilibrium. 40:03.559 --> 40:06.439 You don't have to get involved in the budget set or anything 40:06.436 --> 40:07.116 complicated. 40:07.119 --> 40:10.049 You just set marginal utility equal to price. 40:10.050 --> 40:13.710 So we know for alpha, sorry, we know that the 40:13.713 --> 40:18.713 marginal utility of alpha at time 0 divided by the price 0 is 40:18.710 --> 40:23.210 going to have to equal the marginal utility of alpha at 40:23.208 --> 40:27.288 each state s times the price P_s. 40:27.289 --> 40:35.579 So the equilibrium, the Arrow-Debreu equilibrium, 40:35.579 --> 40:38.019 is going to involve P_0, 40:38.021 --> 40:40.191 P_1, P_2 and 40:40.186 --> 40:42.156 P_3, the prices of the Arrow 40:42.159 --> 40:44.689 securities, the present value prices. 40:44.690 --> 40:48.000 P_0 is what you pay today to get the apple today. 40:48.000 --> 40:51.620 P_1 is what you pay today to get the apple a year 40:51.617 --> 40:52.987 from now in state 1. 40:52.989 --> 41:00.909 So these are the present value (that's what Fisher would say) 41:00.914 --> 41:04.354 state contingent prices. 41:04.349 --> 41:07.679 The state contingent is what Arrow added. 41:07.679 --> 41:10.089 Now, you may ask whether there really are these Arrow 41:10.088 --> 41:11.848 securities floating in the economy, 41:11.849 --> 41:13.179 and we're going to come back to that question, 41:13.179 --> 41:17.509 but you could imagine all these Arrow-Debreu state contingent 41:17.512 --> 41:20.792 prices and commodities, and those would be the prices 41:20.789 --> 41:22.139 we'd solve for equilibrium. 41:22.139 --> 41:26.909 So we get this over this, marginal utility of that. 41:26.909 --> 41:29.409 So what is this? 41:29.409 --> 41:33.989 And similarly for beta, marginal utility^(beta) at 0 41:33.987 --> 41:37.037 over the price of 0 equals marginal 41:37.041 --> 41:41.621 utility^(beta)_s over the price of s. 41:41.619 --> 41:42.359 So what is this? 41:42.360 --> 41:49.360 For alpha, his marginal utility of consumption is 1 half. 41:49.360 --> 41:53.080 We might as well assume one of the prices is 1. 41:53.079 --> 41:55.299 Let's take this price to be 1. 41:55.300 --> 42:00.830 So beta, her marginal utility is 3 quarters and the price is 42:00.833 --> 42:01.213 1. 42:01.210 --> 42:06.830 What's his marginal utility in any state s? 42:06.829 --> 42:17.679 It is gamma_s times (1 - 1 over 200 times 42:17.679 --> 42:21.369 X_s). 42:21.369 --> 42:22.549 So I just differentiated this. 42:22.550 --> 42:26.310 I got 1 - 2 over 400 times X_s. 42:26.309 --> 42:28.909 And what's her marginal utility? 42:28.909 --> 42:36.449 It is gamma_s in state s times (1 - 1 over 400 42:36.447 --> 42:39.597 times X_s). 42:39.599 --> 42:52.699 So I know in equilibrium that's going to imply that 1 half-- 42:52.699 --> 42:54.549 well, now I have to screw around here, 42:54.550 --> 43:00.400 so how am I going to--so I've got this thing over here 1 half 43:00.396 --> 43:03.316 equals this thing over here. 43:03.320 --> 43:04.020 What? 43:04.018 --> 43:05.018 Student: Over P_s. 43:05.019 --> 43:06.649 Prof: Over P_s. 43:06.650 --> 43:08.610 Ah, glad that appeared. 43:08.610 --> 43:10.170 I was getting worried there. 43:10.170 --> 43:10.840 Thank you. 43:10.840 --> 43:17.360 Over P_s, that helps a lot. 43:17.360 --> 43:21.730 So that implies that something like X--so this is what alpha is 43:21.731 --> 43:25.331 going to do and this is what beta is going to do. 43:25.329 --> 43:29.349 So this implies X^(alpha)_s equals 43:29.346 --> 43:29.996 what? 43:30.000 --> 43:33.400 So if I multiply through by 200, and I bring P_s 43:33.402 --> 43:36.032 over gamma_s to the other side, 43:36.030 --> 43:41.150 and I do a bunch of stuff, I'm going to guess this is 200 43:41.146 --> 43:45.256 - 100 P_s over gamma_s. 43:45.260 --> 43:48.230 How do you think that's going to play in Peoria? 43:48.230 --> 43:48.760 Let's see. 43:48.760 --> 43:51.530 If I multiply through by P_s over 43:51.532 --> 43:54.242 gamma_s I get P_s over 43:54.239 --> 43:56.549 gamma_s times 1 half. 43:56.550 --> 43:59.190 Then I multiply everything through by 200. 43:59.190 --> 44:01.960 So I get 100 P_s over gamma_s, 44:01.960 --> 44:05.770 and then I get the 200 here, and the X_s goes to 44:05.766 --> 44:07.646 the other side, and the P_s over 44:07.646 --> 44:08.736 gamma_s goes to the other side. 44:08.739 --> 44:11.259 So it's 200 - 100 P_s over gamma_s. 44:11.260 --> 44:16.460 And this one is going to be--X^(beta)_s is 44:16.456 --> 44:19.766 going to equal-- well, I have to do the same 44:19.768 --> 44:22.998 trick here except I'm going to be multiplying through by 400 44:22.998 --> 44:25.078 and taking 3 quarters which is 300. 44:25.079 --> 44:36.579 So it'll be 300 minus--no, that was wrong, 44:36.577 --> 44:50.317 400 - 300 P_s over gamma_s. 44:50.320 --> 44:52.890 Because if I multiply through by 400, 44:52.889 --> 44:54.939 put P_s over gamma_s on the other 44:54.936 --> 44:56.506 side I have 3 quarters P_s over 44:56.510 --> 44:59.010 gamma_s times 400, which is 300 P_s over 44:59.005 --> 44:59.735 gamma_s. 44:59.739 --> 45:02.469 This becomes a 400 and the P_s gamma_s 45:02.467 --> 45:03.757 went away so I have that. 45:03.760 --> 45:08.290 So I know now if I could figure out what the prices are I know 45:08.288 --> 45:11.628 what everybody would demand in every state. 45:11.630 --> 45:16.140 So let me pause here. 45:16.139 --> 45:17.809 That was the first critical step. 45:17.809 --> 45:18.919 So what did I do? 45:18.920 --> 45:21.220 I said it's a long story. 45:21.219 --> 45:23.129 A lot of years went into this. 45:23.130 --> 45:25.010 I said people are risk averse. 45:25.010 --> 45:26.400 Shakespeare knew that. 45:26.400 --> 45:29.910 We want to quantify it so we say people have concave utility 45:29.905 --> 45:30.615 functions. 45:30.619 --> 45:32.069 That quantifies risk aversion. 45:32.070 --> 45:34.570 We want to make a simple concave utility function. 45:34.570 --> 45:36.740 We pick quadratic, but of course we don't know 45:36.739 --> 45:37.559 what quadratic. 45:37.559 --> 45:39.839 Different people could have different quadratic utility 45:39.842 --> 45:40.352 functions. 45:40.349 --> 45:43.469 Then we do the Fisher trick and say that any equilibrium, 45:43.469 --> 45:46.849 as long as you can buy and sell every contingent commodity in 45:46.851 --> 45:48.551 the future, because all the Arrow 45:48.550 --> 45:51.170 securities are there, it can always be reduced to 45:51.166 --> 45:53.856 general equilibrium just like we did before. 45:53.860 --> 45:59.380 And so now you have to feed the endowments into the agents'--I 45:59.378 --> 46:03.898 mean the payoffs and the dividends into the agents' 46:03.902 --> 46:05.262 endowments. 46:05.260 --> 46:06.650 So we haven't done that yet. 46:06.650 --> 46:09.070 And then we solve for supply equals demand. 46:09.070 --> 46:14.910 So all we have to do is we have to have X^(alpha)_s 46:14.911 --> 46:20.561 X^(beta)_s has to equal the endowment of alpha in 46:20.559 --> 46:24.259 s plus the endowment of beta in s. 46:24.260 --> 46:27.500 All right, so we have to do that for every s. 46:27.500 --> 46:33.110 So this is 200 - 100 P_sS over 46:33.112 --> 46:41.752 gamma_s equals--now we have to do it in state 1. 46:41.750 --> 46:43.670 So it equals whatever they are. 46:43.670 --> 46:48.510 So what is endowment of alpha of s plus endowment beta of s? 46:48.510 --> 46:51.280 We have to look at each state separately. 46:51.280 --> 46:55.390 And lo and behold I picked the numbers so that if you add these 46:55.385 --> 47:00.155 all together you get 500, and here you get 280 and 220 is 47:00.155 --> 47:05.235 also 500, and here you get 500 again. 47:05.239 --> 47:09.359 So lo and behold there is no aggregate risk in the economy, 47:09.355 --> 47:13.325 although the individual stocks are risky the aggregate is 47:13.329 --> 47:14.819 totally un-risky. 47:14.820 --> 47:19.810 So no matter what s is, I could put in 500 here. 47:19.809 --> 47:22.519 It's going to turn out that the total endowment of both people, 47:22.518 --> 47:24.608 because I've plugged the dividends into their personal 47:24.608 --> 47:27.088 endowments, added up the two people, 47:27.086 --> 47:27.886 it's 500. 47:27.889 --> 47:33.769 So it means that P_s over gamma_s equals... 47:33.768 --> 47:37.048 Student: > 47:37.045 --> 47:38.105 you forgot. 47:38.110 --> 47:38.790 Prof: What? 47:38.789 --> 47:41.139 Student: You forgot the second term. 47:41.139 --> 47:42.979 Prof: I've forgotten something for sure, 47:42.980 --> 47:43.260 what? 47:43.260 --> 47:44.510 Student: X^(beta). 47:44.510 --> 47:46.650 Prof: Oh, X^(beta). 47:46.650 --> 47:51.630 So that's alpha. 47:51.630 --> 47:53.790 Thank you. 47:53.789 --> 48:02.759 Plus 400 - 300 P_s over gamma_s = 500. 48:02.760 --> 48:08.710 So if when I add this up I get 600 - 400 P_s over 48:08.710 --> 48:13.100 gamma_s = 500, so then I flip them to the 48:13.103 --> 48:15.483 other side and I get P_s over 48:15.476 --> 48:19.136 gamma_s = 1 quarter, because 500 from here is 100 48:19.141 --> 48:22.171 and put the 400 on the other side and divide by it I get 48:22.170 --> 48:24.590 P_s over gamma_s equals 1 48:24.592 --> 48:25.862 quarter for all s. 48:25.860 --> 48:31.020 So what did I find? 48:31.019 --> 48:32.629 So it's the same. 48:32.630 --> 48:39.540 P_s over gamma_s is the same, 48:39.543 --> 48:42.573 same in all states. 48:42.570 --> 48:46.300 So what would the price of A be here? 48:46.300 --> 48:55.950 What's the price of A in equilibrium? 48:55.949 --> 48:57.529 What's the price of A? 48:57.530 --> 49:05.830 I'm going to take the price of A should be P_1 times 49:05.827 --> 49:12.487 50 P_2 times 100 P_3 times 75, 49:12.494 --> 49:16.444 and what does that equal? 49:16.440 --> 49:21.660 Well, P_1 is just 1 quarter times gamma_1, 49:21.655 --> 49:22.335 right? 49:22.340 --> 49:24.970 P_1 over gamma_1 is 1 quarter. 49:24.969 --> 49:28.069 P_2 is 1 quarter times gamma_2, 49:28.070 --> 49:30.490 and P_3 is 1 quarter times gamma_3, 49:30.489 --> 49:34.179 so I just got this multiplied by 1 quarter. 49:34.179 --> 49:36.579 So in fact all I did is I did what I had always done. 49:36.579 --> 49:39.369 I took the expected payoff and discounted it. 49:39.369 --> 49:41.269 The discount rate is 1 quarter. 49:41.268 --> 49:44.368 What's the price of the riskless asset, 49:44.369 --> 49:49.329 pi of (1,1, 1), is just going to be 1 quarter, 49:49.329 --> 49:51.899 because it's 1 quarter times gamma 1 1 quarter times gamma 2 49:51.902 --> 49:55.202 1 quarter times gamma 3, gamma 1 gamma 2 gamma 3 is 1 so 49:55.197 --> 49:56.287 it's 1 quarter. 49:56.289 --> 50:02.659 So it implies the riskless interest rate is what? 50:02.659 --> 50:04.489 What's the riskless interest rate? 50:04.489 --> 50:05.339 Student: 300 percent. 50:05.340 --> 50:07.120 Prof: 300 percent. 50:07.119 --> 50:10.509 So we're discounting by 1 over 1 quarter because the interest 50:10.507 --> 50:11.747 rate is 300 percent. 50:11.750 --> 50:13.520 So basically nothing happened. 50:13.518 --> 50:16.618 We got all the prices exactly as we would compute them 50:16.621 --> 50:19.431 without, you know, just doing expectations we got 50:19.431 --> 50:21.071 the right discount rate. 50:21.070 --> 50:23.020 All we had to do was figure out the discount rate. 50:23.019 --> 50:25.159 So risk hasn't played any role. 50:25.159 --> 50:26.589 And why didn't it play any role? 50:26.590 --> 50:30.730 Because although alpha started off owning A alone which exposed 50:30.733 --> 50:34.883 her--forgot who was her and who was him, let's say her--exposed 50:34.878 --> 50:36.548 her to a lot of risk. 50:36.550 --> 50:39.420 She's not going to sit there stupidly just holding A. 50:39.420 --> 50:42.100 She's going to trade it for B and C for different shares. 50:42.099 --> 50:45.699 In fact she's going to end up holding her consumption, 50:45.699 --> 50:48.199 this is her consumption, 200 - 100 P_s over 50:48.199 --> 50:51.039 gamma_s, this number doesn't depend on s. 50:51.039 --> 50:54.589 She's going to consume the same thing in every state, 50:54.592 --> 50:56.372 and how can she do that? 50:56.369 --> 50:59.619 She can own equal shares of A, B and C. 50:59.619 --> 51:01.669 She'll own a share of the whole economy. 51:01.670 --> 51:12.200 So in other words, by diversifying alpha and beta 51:12.197 --> 51:18.117 each get rid of all risk. 51:18.119 --> 51:21.489 So instead of calling it diversifying I could call it 51:21.485 --> 51:23.165 hedging, the same thing. 51:23.170 --> 51:24.570 She doesn't just hold her A. 51:24.570 --> 51:27.390 She mixes B and C with it so that she gets a payoff of 51:27.385 --> 51:30.405 consumption that's exactly the same in every state because 51:30.414 --> 51:33.074 P_s over gamma_s is independent 51:33.070 --> 51:34.080 of the state. 51:34.079 --> 51:35.869 She'll always consume the same thing. 51:35.869 --> 51:38.689 Everybody can hedge perfectly and there's no problem because 51:38.693 --> 51:41.663 there's no aggregate risk that anyone has to be stuck with, 51:41.659 --> 51:45.639 and therefore the price is just going to be the same as the 51:45.637 --> 51:47.487 probabilities discounted. 51:47.489 --> 51:49.949 And that's the theory we've worked with so far. 51:49.949 --> 51:53.059 So, so far you could say that everything we did was kosher 51:53.056 --> 51:56.426 it's just that when we had these two different probabilities of 51:56.434 --> 51:59.494 things happening up or down we thought that the aggregate 51:59.487 --> 52:03.137 economy would have the same endowments here as it did there, 52:03.139 --> 52:06.629 and therefore the probabilities we used were the objective 52:06.634 --> 52:08.294 probabilities discounted. 52:08.289 --> 52:10.809 No reason to change them because nobody's going to be 52:10.811 --> 52:11.881 forced not to hedge. 52:11.880 --> 52:13.250 Everybody'll hedge. 52:13.250 --> 52:16.830 So are there any questions about what I've said? 52:16.829 --> 52:19.639 I'm sure there should be a question because I can't have 52:19.643 --> 52:21.693 said it as clearly as I ought to have. 52:21.690 --> 52:27.690 So would somebody like to say something? 52:27.690 --> 52:28.580 Yes? 52:28.579 --> 52:30.729 Student: > 52:30.731 --> 52:33.531 the old price that we found when we hadn't done this, 52:33.529 --> 52:35.519 but that also change the new one? 52:35.518 --> 52:37.508 Prof: This is the new price with the 1 quarter. 52:37.510 --> 52:39.500 This is the correct new price. 52:39.500 --> 52:42.790 So the theory so far hasn't changed in any interesting way. 52:42.789 --> 52:44.829 We just found the discount rate. 52:44.829 --> 52:46.849 It just looks like expected utility, 52:46.849 --> 52:49.729 but you shouldn't have expected it to change because the 52:49.733 --> 52:52.943 aggregate endowment was 500, the constant in every state. 52:52.940 --> 52:57.000 There's no reason why we can't have everybody perfectly hedged 52:57.003 --> 52:59.873 and consuming a constant in every state, 52:59.869 --> 53:01.349 and in fact that's what we did have, 53:01.349 --> 53:03.809 everybody--she consumed the same thing in every state. 53:03.809 --> 53:06.329 He consumed the same thing in every state. 53:06.329 --> 53:09.689 No reason why they both couldn't hedge themselves 53:09.690 --> 53:13.890 perfectly and in equilibrium that's exactly what they did. 53:13.889 --> 53:14.309 Any other... 53:14.309 --> 53:14.859 Yes? 53:14.860 --> 53:18.610 Student: If the total endowments in every state hadn't 53:18.608 --> 53:22.358 all added up to 500 would you create an expected endowment or 53:22.356 --> 53:24.726 would you just not do the problem? 53:24.730 --> 53:27.220 Prof: So the next step is going to be-- 53:27.219 --> 53:29.509 what I'm going to do now is I'm going to assume that the 53:29.514 --> 53:31.814 endowments don't add up to a constant in every state. 53:31.809 --> 53:33.309 Then what's going to happen? 53:33.309 --> 53:35.719 So this is not at all obvious how to solve this and what to 53:35.719 --> 53:37.219 do, but it's going to turn out to 53:37.222 --> 53:40.962 have a beautiful simple answer, shocking, not only be simple 53:40.958 --> 53:42.688 but also surprising. 53:42.690 --> 53:46.130 So before I do that I'm going to change the endowments so 53:46.128 --> 53:47.908 they're not all a constant. 53:47.909 --> 53:50.089 Any questions about where we're going? 53:50.090 --> 53:50.970 Yeah? 53:50.969 --> 53:53.209 Student: Could you just repeat what you said about 53:53.210 --> 53:54.650 hedging >? 53:54.650 --> 53:57.050 Prof: Yes. 53:57.050 --> 53:58.610 Thank you for the question. 53:58.610 --> 54:00.970 So I went a little quickly. 54:00.969 --> 54:04.679 I said that what we proved by solving for the general 54:04.681 --> 54:09.251 equilibrium is that the price in every state was just going to be 54:09.250 --> 54:11.820 1 quarter times the probability. 54:11.820 --> 54:14.420 That's what we showed had to happen in equilibrium. 54:14.420 --> 54:16.390 Now, what's the consequence of that? 54:16.389 --> 54:18.309 The consequences are twofold. 54:18.309 --> 54:21.539 Number one, the price of all the assets is the same 54:21.539 --> 54:25.349 expectation we naively would have taken before where we used 54:25.351 --> 54:27.421 the discount rate 1 quarter. 54:27.420 --> 54:31.410 That's the first implication. 54:31.409 --> 54:34.689 The second implication is that from the formula for consumption 54:34.688 --> 54:38.018 we noticed that she consumes the same amount in all three states 54:38.021 --> 54:40.981 because P_s over gamma_s is 1 quarter 54:40.983 --> 54:42.363 in all three states. 54:42.360 --> 54:44.550 Her consumption is going to be the same in all three states, 54:44.550 --> 54:46.950 and his consumption, which will be different from 54:46.949 --> 54:49.009 hers, but his will be the same in all 54:49.010 --> 54:50.190 three states as well. 54:50.190 --> 54:52.650 The two will add up to 500. 54:52.650 --> 54:56.180 So then I took a little bit of a leap and I interpreted that 54:56.175 --> 54:59.815 conclusion that her consumption doesn't depend on the sate. 54:59.820 --> 55:01.320 What's the interpretation of that? 55:01.320 --> 55:04.510 She has obviously, somewhere behind the scenes, 55:04.510 --> 55:08.710 given up some of her A to get B and C and held them in a mixture 55:08.710 --> 55:12.110 so as to get the same consumption in every state. 55:12.110 --> 55:13.810 What must the mixture be? 55:13.809 --> 55:17.389 Obviously she holds the same proportion of A, 55:17.387 --> 55:21.207 B and C because those add up to 500,500, 500. 55:21.210 --> 55:23.570 So she must have held the same proportion of A, 55:23.568 --> 55:26.178 B and C, a fraction of the market and got a riskless 55:26.184 --> 55:26.754 payoff. 55:26.750 --> 55:27.980 So she diversified. 55:27.980 --> 55:29.660 She didn't just stick with her A. 55:29.659 --> 55:32.609 She substituted a little bit of A, a little bit of B and a 55:32.614 --> 55:34.744 little bit of C, a different boat on every 55:34.739 --> 55:36.969 ocean, and now she runs no risk at all. 55:36.969 --> 55:39.439 So she diversified, but in the language we used 55:39.436 --> 55:42.116 last time I could call diversification hedging if I 55:42.119 --> 55:42.869 wanted to. 55:42.869 --> 55:45.389 She just, sort of, sold Arrow securities in the 55:45.385 --> 55:48.335 right proportions to turn her A into something that was 55:48.336 --> 55:49.646 completely riskless. 55:49.650 --> 55:52.480 So whether you call it diversifying or call it hedging 55:52.476 --> 55:55.196 she's achieved the same end of totally balancing her 55:55.195 --> 55:56.045 consumption. 55:56.050 --> 55:59.850 He did the same thing and they both could do it because the 55:59.846 --> 56:02.396 aggregate consumption was a constant. 56:02.400 --> 56:03.310 Yes? 56:03.309 --> 56:05.999 Student: What would an Arrow security actually look 56:05.998 --> 56:06.328 like? 56:06.329 --> 56:07.089 Prof: In real life? 56:07.090 --> 56:07.790 Student: Yeah. 56:07.789 --> 56:12.489 Prof: The closest we've come to an Arrow security in 56:12.494 --> 56:15.754 real life is a CDS, and this is part of the reason 56:15.746 --> 56:18.096 why these economists, Larry Summers, 56:18.099 --> 56:21.959 my classmate, and Rubin who was the Secretary 56:21.963 --> 56:24.683 of the Treasury, and ran Citi Corp, 56:24.684 --> 56:27.364 and who was a Yale law school student and a Harvard 56:27.364 --> 56:29.564 undergraduate, and who I've sat on many 56:29.563 --> 56:32.983 committees with, they were seduced by the--so 56:32.976 --> 56:34.216 what's a CDS? 56:34.219 --> 56:40.929 A CDS pays 1 dollar if some bond defaults by 1 dollar. 56:40.929 --> 56:44.309 So that isn't an Arrow security because an Arrow security is a 56:44.311 --> 56:45.811 much more detailed thing. 56:45.809 --> 56:49.399 An Arrow security says I'll pay 1 dollar in state one. 56:49.400 --> 56:52.640 An Arrow security says you get an apple in state 1, 56:52.639 --> 56:56.089 but state 1, remember, is not described by a 56:56.086 --> 56:59.026 single firm, state 1, the states of nature 56:59.034 --> 57:02.414 are total descriptions of everything that could happen in 57:02.407 --> 57:03.307 the economy. 57:03.309 --> 57:07.059 So an Arrow security really says if it stops snowing in 57:07.056 --> 57:10.516 Siberia, if Khomeini loses power in 57:10.516 --> 57:17.046 Iran, if there's a favorable election outcome in Afghanistan, 57:17.050 --> 57:22.520 and if Obama wins reelection, and if the U.S. 57:22.518 --> 57:26.108 solves the energy problem then I'll give you 1 dollar. 57:26.110 --> 57:28.860 So the Arrow security lists an incredible number of contingent 57:28.856 --> 57:31.326 things, every contingency possible and says in that case 57:31.333 --> 57:32.553 I'll give you 1 dollar. 57:32.550 --> 57:35.910 A CDS says if this thing happens I'll give you 1 dollar 57:35.907 --> 57:38.267 whether or not Obama wins election, 57:38.268 --> 57:40.968 whether or not America discovers a new source of 57:40.974 --> 57:43.184 energy, whether or not Afghanistan 57:43.175 --> 57:44.045 turns around. 57:44.050 --> 57:46.770 Just so long as the bond defaults I'll give you 1 dollar. 57:46.768 --> 57:55.588 So the CDS is an event contingent security. 57:55.590 --> 57:58.390 That's the CDS, and an Arrow security is a much 57:58.391 --> 58:00.221 more finely specified thing. 58:00.219 --> 58:02.859 It's a state contingent--you say everything that happens in 58:02.856 --> 58:04.886 the economy, so we'll never get to Arrow 58:04.889 --> 58:07.679 securities, but CDS looks like we're on the 58:07.681 --> 58:08.581 way to them. 58:08.579 --> 58:11.739 And these guys blundered by thinking since CDSs are on the 58:11.740 --> 58:14.960 way to Arrow securities we should have as many CDSs and let 58:14.958 --> 58:17.398 people trade as much of them as we can, 58:17.400 --> 58:20.940 but we're going to get to that in the last lecture about how 58:20.940 --> 58:23.350 all this theory, what's wrong with all the 58:23.351 --> 58:23.761 theory. 58:23.760 --> 58:34.040 So, any other questions before we--so let's now make the change 58:34.041 --> 58:38.851 that he suggested up there. 58:38.849 --> 58:41.339 Let's now change the economy just a little bit. 58:41.340 --> 58:44.230 Let's eliminate C. 58:44.230 --> 58:48.010 So this just disappears. 58:48.010 --> 58:51.340 So obviously now the total endowment is very contingent. 58:51.340 --> 59:03.520 It's 200, it's 280 and it's 440. 59:03.519 --> 59:04.679 Now what do we do? 59:04.679 --> 59:10.849 So beta owns B. 59:10.849 --> 59:12.379 So now what's equilibrium going to be? 59:12.380 --> 59:14.060 What do you think is going to happen? 59:14.059 --> 59:16.159 We want to quantify this. 59:16.159 --> 59:19.189 We want to give a beautiful simple theory that's 59:19.193 --> 59:21.713 quantifiable, but what do you anticipate 59:21.708 --> 59:24.548 happening to P_1, P_2 and 59:24.548 --> 59:25.838 P_3? 59:25.840 --> 59:29.070 So everybody's going to say, alpha, she's going to say, 59:29.065 --> 59:30.375 look, my A is risky. 59:30.380 --> 59:32.250 I don't want to hold my risky thing. 59:32.250 --> 59:35.120 I want to start hedging and trading these Arrow securities 59:35.121 --> 59:37.341 so I get the same constant in every state. 59:37.340 --> 59:41.050 Of course beta who owns B, he's going to do the same 59:41.047 --> 59:41.627 thing. 59:41.630 --> 59:43.490 So they're both going to be trying to trade Arrow 59:43.487 --> 59:43.987 securities. 59:43.989 --> 59:49.489 What's going to happen, do you think? 59:49.489 --> 59:50.219 Yes? 59:50.219 --> 59:52.459 Student: Aren't they both just going to be exposed to 59:52.463 --> 59:54.103 whatever the total risk of the economy is in 59:54.099 --> 59:55.239 >? 59:55.239 --> 59:57.519 Prof: Yeah, there's no way that they can, 59:57.519 --> 1:00:00.289 exactly, there's no way that each of them can be perfectly 1:00:00.286 --> 1:00:00.816 hedged. 1:00:00.820 --> 1:00:04.050 So no matter what they do, they're going to be exposed to 1:00:04.045 --> 1:00:06.805 more risk in state, you know, state 3 is going to 1:00:06.809 --> 1:00:08.019 be a great state. 1:00:08.018 --> 1:00:10.608 State 1 is going to be a terrible state, 1:00:10.614 --> 1:00:14.014 so what do you think that means about the prices? 1:00:14.010 --> 1:00:17.640 Everybody can't be hedged, and so in fact what'll happen 1:00:17.637 --> 1:00:19.417 is nobody will be hedged. 1:00:19.420 --> 1:00:23.510 Although, alpha will be, who hates risk more than beta, 1:00:23.510 --> 1:00:26.920 will be closer to hedged than beta will be. 1:00:26.920 --> 1:00:30.320 So beta will end up bearing more of the risk than alpha. 1:00:30.320 --> 1:00:34.410 And what do you think will happen to the prices of the 1:00:34.414 --> 1:00:38.204 Arrow securities relative to the probabilities? 1:00:38.199 --> 1:00:39.199 Yes? 1:00:39.199 --> 1:00:40.069 Student: It won't be constant. 1:00:40.070 --> 1:00:43.260 Prof: There won't be a constant ratio of 1 quarter as 1:00:43.255 --> 1:00:45.735 we had before, but can you be more specific? 1:00:45.739 --> 1:00:51.349 Student: The price of the securities for state 1 will 1:00:51.353 --> 1:00:56.593 be greater relative to the probability than the price of 1:00:56.585 --> 1:00:59.245 the security in state 3. 1:00:59.250 --> 1:01:00.100 Prof: Exactly. 1:01:00.099 --> 1:01:02.929 So that's what's going to turn out. 1:01:02.929 --> 1:01:05.269 The world is short of commodities in state 1, 1:01:05.273 --> 1:01:06.983 there just aren't many apples. 1:01:06.980 --> 1:01:08.010 That's the disaster. 1:01:08.010 --> 1:01:10.630 That's when we can't solve the energy crisis. 1:01:10.630 --> 1:01:14.010 We're totally screwed. 1:01:14.010 --> 1:01:17.310 Everybody wants to consume more in that state. 1:01:17.309 --> 1:01:19.389 Everyone's going to try and hedge against that state. 1:01:19.389 --> 1:01:22.539 They're all going to be trying to buy Arrow securities in that 1:01:22.543 --> 1:01:24.163 state, which means that because there 1:01:24.164 --> 1:01:26.094 aren't as many to buy, there's just not enough apples 1:01:26.090 --> 1:01:29.340 to go around, the price of Arrow securities 1:01:29.335 --> 1:01:34.255 in state 1 is going to be high relative to state 3. 1:01:34.260 --> 1:01:35.990 There's plenty to go around there. 1:01:35.989 --> 1:01:40.119 So she is going to sell some of her A and get some B to 1:01:40.123 --> 1:01:42.863 diversify, but B's got so good in state 3 1:01:42.858 --> 1:01:46.218 that all of a sudden she's not going to be so worried about 1:01:46.219 --> 1:01:48.679 state 3 anymore, but state 1 she's still going 1:01:48.681 --> 1:01:50.961 to be worried about, and there's nothing to be done 1:01:50.961 --> 1:01:51.481 about that. 1:01:51.480 --> 1:01:54.640 So the price is going to have to be very expensive in state 1. 1:01:54.639 --> 1:01:56.159 So all right, that's all blah, 1:01:56.159 --> 1:01:56.839 blah, blah. 1:01:56.840 --> 1:01:59.570 Let's solve for equilibrium and see what happens. 1:01:59.570 --> 1:02:00.970 We can solve immediately. 1:02:00.969 --> 1:02:02.409 Nothing's changed. 1:02:02.409 --> 1:02:04.489 The utility functions are the same. 1:02:04.489 --> 1:02:07.849 None of this changed, so this board doesn't change at 1:02:07.851 --> 1:02:08.241 all. 1:02:08.239 --> 1:02:09.529 That's demand. 1:02:09.530 --> 1:02:11.700 Still depends on P_0, P_1, 1:02:11.695 --> 1:02:13.185 P_2 and P_3, 1:02:13.186 --> 1:02:16.216 but now we have to be a little bit more careful in state 1. 1:02:16.219 --> 1:02:19.689 So demand in every state is 600 - 400 P_1 over 1:02:19.690 --> 1:02:23.160 gamma_1 equals endowment of alpha endowment of 1:02:23.161 --> 1:02:23.731 beta. 1:02:23.730 --> 1:02:27.330 So in state 1, I'm going to now change this to 1:02:27.333 --> 1:02:31.983 a 1 although with my handwriting it looked like a 1 anyway, 1:02:31.976 --> 1:02:35.656 what's the aggregate endowment in state 1? 1:02:35.659 --> 1:02:39.459 The aggregate endowment in state 1 is 200. 1:02:39.460 --> 1:02:41.550 This is a 1 now. 1:02:41.550 --> 1:02:48.300 That's 200, so that means P_1 over 1:02:48.295 --> 1:02:53.155 gamma_1 = 1, right? 1:02:53.159 --> 1:02:56.709 Because 400 and 400 so it's 1. 1:02:56.710 --> 1:02:59.220 So you're not discounting the first state at all. 1:02:59.219 --> 1:03:01.239 You're looking at the probability of it. 1:03:01.239 --> 1:03:02.979 But what if I go to P_2 over 1:03:02.978 --> 1:03:03.918 gamma_2? 1:03:03.920 --> 1:03:06.030 Well, the demand is going to be the same. 1:03:06.030 --> 1:03:11.550 It's the price that's going to change to make up for the fact 1:03:11.547 --> 1:03:15.317 that the supply is much different, namely, 1:03:15.318 --> 1:03:17.248 namely what, 280. 1:03:17.250 --> 1:03:22.390 So now if I subtract I get 400,280 minus that is 320 1:03:22.394 --> 1:03:27.444 divided by 400 which looks like 4 fifths, maybe. 1:03:27.440 --> 1:03:34.050 320 over 400 is 4 fifths, right? 1:03:34.050 --> 1:03:36.760 Because 320 divided by 400 is 4 fifths, so P_2 over 1:03:36.762 --> 1:03:38.212 gamma_2 is 4 fifths. 1:03:38.210 --> 1:03:40.340 So they're not proportional anymore. 1:03:40.340 --> 1:03:50.080 And then P_3 over gamma_3 equals--now 1:03:50.077 --> 1:03:58.597 the outcome is 440, so if I subtract 440 from this 1:03:58.597 --> 1:04:05.897 I get 160 divided by 400, what's that? 1:04:05.900 --> 1:04:06.570 Student: 2 fifths. 1:04:06.570 --> 1:04:07.990 Prof: 2 fifths, thank you. 1:04:07.989 --> 1:04:14.979 P_3 over gamma_3 = 2 fifths. 1:04:14.980 --> 1:04:19.440 So the prices turned out to be quite different. 1:04:19.440 --> 1:04:22.650 Now, the reason why they're slightly higher on average, 1:04:22.648 --> 1:04:25.918 of course, than they were before is because there's less 1:04:25.916 --> 1:04:27.696 consumption in the future. 1:04:27.699 --> 1:04:30.589 We've suddenly made our future much worse off. 1:04:30.590 --> 1:04:33.000 So people are more desperate to consume in the future, 1:04:33.001 --> 1:04:35.641 so that means the prices of future consumption are going to 1:04:35.641 --> 1:04:36.281 be higher. 1:04:36.280 --> 1:04:37.710 So we have two effects here. 1:04:37.710 --> 1:04:40.670 These prices instead of being 1 quarter everywhere are higher, 1:04:40.670 --> 1:04:43.480 much higher than 1 quarter because the future looks so much 1:04:43.483 --> 1:04:43.973 worse. 1:04:43.969 --> 1:04:47.069 The interest rate is going to go down. 1:04:47.070 --> 1:04:49.080 It's not going to be 300 percent anymore. 1:04:49.079 --> 1:04:52.769 But more interesting is that the prices are no longer 1:04:52.766 --> 1:04:55.316 proportional to the probabilities. 1:04:55.320 --> 1:04:59.440 Just as he said over there the price in state 1 is going to be 1:04:59.443 --> 1:05:02.353 much higher relative to the probability, 1:05:02.349 --> 1:05:06.909 namely 100 percent of it, than the price in state 3 which 1:05:06.909 --> 1:05:09.109 is only 40 percent of it. 1:05:09.110 --> 1:05:12.310 So that's the conclusion. 1:05:12.309 --> 1:05:14.279 So now what do we do for our price? 1:05:14.280 --> 1:05:17.860 What's the price of A? 1:05:17.860 --> 1:05:20.450 What's the price of A? 1:05:20.449 --> 1:05:22.179 What do I plug in here? 1:05:22.179 --> 1:05:33.769 That, so that equals 1 quarter times 50 4 fifths times 1 1:05:33.771 --> 1:05:41.161 quarter-- 1 fifth times 100 P_3 1:05:41.161 --> 1:05:50.001 was 2 fifths times 1 half 1 fifth times 75 which equals 1:05:49.998 --> 1:05:56.228 something, 20,35 and 12 and 1 half, 1:05:56.233 --> 1:05:58.703 47 and 1 half. 1:05:58.699 --> 1:06:00.709 Student: Why would you >? 1:06:00.710 --> 1:06:01.760 Prof: Why did I what? 1:06:01.760 --> 1:06:03.550 Student: Why would you >? 1:06:03.550 --> 1:06:05.570 Prof: So what is P_1? 1:06:05.570 --> 1:06:09.140 P_1 is equal to gamma_1 and 1:06:09.139 --> 1:06:11.699 gamma_1 is 1 quarter. 1:06:11.699 --> 1:06:12.969 So that's how I got 1 quarter here. 1:06:12.969 --> 1:06:16.779 So that's 1 times 1 quarter. 1:06:16.780 --> 1:06:20.980 P_2 was 3 fifths. 1:06:20.980 --> 1:06:21.750 What was P_2? 1:06:21.750 --> 1:06:23.550 Maybe I did it wrong anyway. 1:06:23.550 --> 1:06:29.350 P_2 was 4 fifths times 1 quarter which is equal 1:06:29.346 --> 1:06:35.556 to 1 fifth, and P_3 was 2 fifths times 1 half which 1:06:35.556 --> 1:06:38.036 is equal to 1 fifth. 1:06:38.039 --> 1:06:39.809 So that's how I got the prices. 1:06:39.809 --> 1:06:43.149 So all right, so you see that things changed, 1:06:43.150 --> 1:06:46.340 and we've captured the idea that people can't hedge fully by 1:06:46.336 --> 1:06:49.576 making the price of the Arrow security in the state where the 1:06:49.579 --> 1:06:53.269 economy's worse off, much smaller than it was 1:06:53.271 --> 1:06:56.401 before, I mean, much higher than it was 1:06:56.402 --> 1:06:59.032 relative to the probability than before. 1:06:59.030 --> 1:07:03.590 So we haven't gotten close to the punch line, 1:07:03.590 --> 1:07:04.420 sorry. 1:07:04.420 --> 1:07:05.610 Yes? 1:07:05.610 --> 1:07:08.790 Student: Can you repeat the part where you said stuff 1:07:08.786 --> 1:07:11.856 about the future looks so much worse they need to increase 1:07:11.858 --> 1:07:12.718 consumption? 1:07:12.719 --> 1:07:16.679 Prof: Two things happened to the prices compared 1:07:16.679 --> 1:07:17.559 to before. 1:07:17.559 --> 1:07:20.449 One is that we no longer have the prices proportional to the 1:07:20.452 --> 1:07:21.582 probabilities, right? 1:07:21.579 --> 1:07:24.899 Their proportion is 1,4 fifths, 2 fifths instead of the same 1:07:24.902 --> 1:07:28.002 constant 1 quarter everywhere, and that's because of the 1:07:28.000 --> 1:07:29.240 relative scarcity. 1:07:29.239 --> 1:07:31.769 People are much more worried about the first state than the 1:07:31.768 --> 1:07:34.908 fourth state and that's why, relative to the probability, 1:07:34.907 --> 1:07:37.907 the price is much higher than the third state. 1:07:37.909 --> 1:07:39.279 You agree with that, right? 1:07:39.280 --> 1:07:40.480 Student: Yes. 1:07:40.480 --> 1:07:42.320 Prof: But there's a second effect which is that all 1:07:42.320 --> 1:07:44.000 these numbers, 1,4 fifths, and 2 fifths 1:07:44.001 --> 1:07:45.701 they're bigger than the 1 quarter, 1:07:45.699 --> 1:07:47.349 1 quarter, 1 quarter we had before, 1:07:47.349 --> 1:07:48.349 but that's obvious. 1:07:48.349 --> 1:07:50.899 That's because we wiped out the future. 1:07:50.900 --> 1:07:52.890 Half the endowment in the future disappeared, 1:07:52.893 --> 1:07:55.473 so naturally people are willing to pay more for the future 1:07:55.474 --> 1:07:56.974 because they're poorer there. 1:07:56.969 --> 1:08:00.139 In the first day of class we said that the interest rate, 1:08:00.139 --> 1:08:03.689 or the third week, the interest rate according to 1:08:03.690 --> 1:08:07.760 Fisher would go down if you got poorer in the future. 1:08:07.760 --> 1:08:09.250 So that's part of the reason that's happened. 1:08:09.250 --> 1:08:11.450 By the way, what is the riskless rate of interest? 1:08:11.449 --> 1:08:17.499 So P_1 P_2 P_3 equals what now? 1:08:17.500 --> 1:08:24.350 It's equal to 1 quarter 1 fifth 1 fifth, so 1 quarter 1 fifth 1 1:08:24.345 --> 1:08:25.225 fifth. 1:08:25.229 --> 1:08:31.299 These are the prices, 1 quarter 1 fifth 1 fifth and 1:08:31.298 --> 1:08:36.688 that's equal to 20, 10,14 over 20, 1:08:36.692 --> 1:08:43.612 so that's 7 over 10, so therefore the interest rate 1:08:43.605 --> 1:08:47.905 1 r = 10 over 7 so r = 3 sevenths which is like 40 1:08:47.912 --> 1:08:48.882 percent. 1:08:48.880 --> 1:08:52.190 So the interest rate went from 300 percent to 40 percent, 1:08:52.189 --> 1:08:55.559 but that's because we lost all this future consumption. 1:08:55.560 --> 1:08:57.330 But that's not what I'm concentrating on. 1:08:57.328 --> 1:08:59.248 Fisher would have already known that. 1:08:59.250 --> 1:09:02.360 What I'm concentrating on is the fact that the prices are no 1:09:02.360 --> 1:09:04.630 longer proportional to the probabilities. 1:09:04.630 --> 1:09:07.070 You're discounting every probability, but adjusting the 1:09:07.067 --> 1:09:09.637 probability because people are much more worried about the 1:09:09.640 --> 1:09:11.310 first state than the third state. 1:09:11.310 --> 1:09:15.380 Student: So people are much more worried about A? 1:09:15.380 --> 1:09:17.220 Prof: Not A, they're more worried about the 1:09:17.222 --> 1:09:17.752 first state. 1:09:17.750 --> 1:09:19.900 The firms are A and B. 1:09:19.899 --> 1:09:22.619 The states are 1,2 and 3, so they're much more worried 1:09:22.622 --> 1:09:25.142 about the first state where the payoff is 200, 1:09:25.140 --> 1:09:27.430 than they are about the third state where the payoff, 1:09:27.430 --> 1:09:34.550 total dividends in the economy are 440. 1:09:34.550 --> 1:09:35.390 Are you with me? 1:09:35.390 --> 1:09:37.470 Student: Yeah. 1:09:37.470 --> 1:09:38.960 Prof: Oh boy. 1:09:38.960 --> 1:09:41.510 That sounded so unconvincing. 1:09:41.510 --> 1:09:42.530 I want to say the punch line. 1:09:42.529 --> 1:09:43.929 So I've got three more minutes to go. 1:09:43.930 --> 1:09:45.110 There are two punch lines. 1:09:45.109 --> 1:09:48.159 I haven't gotten to the stunning conclusion. 1:09:48.158 --> 1:09:52.908 So far I've said stuff which Arrow and Debreu had already 1:09:52.908 --> 1:09:55.498 figured out, but now I want to go to the 1:09:55.502 --> 1:09:57.712 thing that Tobin and Markowitz figured out, 1:09:57.710 --> 1:10:00.870 which is one more step we haven't noticed yet. 1:10:00.868 --> 1:10:04.798 Arrow has already figured out that because not everybody could 1:10:04.802 --> 1:10:08.542 hedge that means that the price of an Arrow security is not 1:10:08.542 --> 1:10:10.932 exactly equal to the probability. 1:10:10.930 --> 1:10:14.670 It's relatively high if the economy's poor like in state 1 1:10:14.672 --> 1:10:18.682 and relatively lower if the economy's rich like in state 3. 1:10:18.680 --> 1:10:20.480 That's common sense. 1:10:20.479 --> 1:10:23.739 Now, what's not common sense is the extraordinary conclusion I'm 1:10:23.735 --> 1:10:24.765 about to show you. 1:10:24.770 --> 1:10:26.940 Let's look at what the consumption is; 1:10:26.939 --> 1:10:37.049 the final consumption of these two people. 1:10:37.050 --> 1:10:40.690 So if we look at the final consumption of these two people, 1:10:40.692 --> 1:10:43.772 what's her final consumption, so X_A. 1:10:43.770 --> 1:10:50.460 In the three states it's 200 - 100 times 1 which is 100. 1:10:50.460 --> 1:10:51.890 What is it in the second state? 1:10:51.890 --> 1:10:59.150 It's 200 - 100 times--what was P_s over--times 4 1:10:59.152 --> 1:11:03.742 fifths which is, help, 160, maybe. 1:11:03.738 --> 1:11:10.428 And the last step was 200 - 100 times 2 fifths. 1:11:10.430 --> 1:11:14.200 2 fifths is 20 so this is 180. 1:11:14.199 --> 1:11:15.249 That's hers. 1:11:15.250 --> 1:11:17.730 And his consumption in the future-- 1:11:17.729 --> 1:11:19.389 I'll put a tilde, I haven't talked about 1:11:19.386 --> 1:11:26.026 X_0 yet-- is 400 - 300 times 1 which 1:11:26.025 --> 1:11:32.415 equals 100, and here's it's 400 - 300 times 1:11:32.417 --> 1:11:36.467 4 fifths which is equal to, help! 1:11:36.470 --> 1:11:49.700 4 fifths of 300 is 160, and here it's 400 - 300 times 2 1:11:49.703 --> 1:11:57.303 fifths which is equal to 280. 1:11:57.300 --> 1:11:58.540 Is this right? 1:11:58.539 --> 1:12:02.099 100,160, who told me it was 160? 1:12:02.100 --> 1:12:04.780 Yes and what's that? 1:12:04.779 --> 1:12:05.969 2 fifths is 120. 1:12:05.970 --> 1:12:07.170 This is 280. 1:12:07.170 --> 1:12:10.870 Student: The number > 1:12:10.871 --> 1:12:14.211 , like 200 - 100 times 4 fifths is like 120. 1:12:14.210 --> 1:12:15.120 Prof: What? 1:12:15.118 --> 1:12:17.418 Student: The first > 1:12:17.420 --> 1:12:18.940 Prof: Which mistake is there here? 1:12:18.939 --> 1:12:20.279 Student: No, the second 1:12:20.280 --> 1:12:21.530 > 1:12:21.529 --> 1:12:23.509 Student: The second 120 > 1:12:23.510 --> 1:12:24.840 Prof: Is this the wrong one? 1:12:24.840 --> 1:12:25.190 Student: The wrong one. 1:12:25.189 --> 1:12:26.509 Student: 160 should be 120. 1:12:26.510 --> 1:12:28.360 Prof: Here. 1:12:28.359 --> 1:12:30.939 200 - 80 is 120. 1:12:30.939 --> 1:12:32.489 Thank you. 1:12:32.489 --> 1:12:35.059 So these are all right now? 1:12:35.060 --> 1:12:38.230 Student: 180 should be > 1:12:38.229 --> 1:12:41.969 Prof: 180 should be, okay, this is 40 so this should 1:12:41.969 --> 1:12:42.549 be 160. 1:12:42.550 --> 1:12:44.380 Thank you. 1:12:44.380 --> 1:12:46.050 That's it, great. 1:12:46.050 --> 1:12:48.160 So now what's so shocking about those numbers? 1:12:48.159 --> 1:12:49.419 That I finally got them right? 1:12:49.420 --> 1:12:50.350 Thank you. 1:12:50.350 --> 1:12:55.410 What's shocking is this consumption is just the sum of 1:12:55.405 --> 1:13:00.935 the aggregate endowment--what's the aggregate endowment? 1:13:00.939 --> 1:13:11.729 Remember the aggregate endowment is just 200,280 and 1:13:11.729 --> 1:13:12.999 440. 1:13:13.000 --> 1:13:18.320 So let's say you take 1 quarter of this. 1:13:18.319 --> 1:13:19.369 Let's take 1 quarter of that. 1:13:19.368 --> 1:13:26.368 That's 50,70--that's 50,70 and 110. 1:13:26.368 --> 1:13:30.498 So 1 quarter of this plus if you add to that 150 you're going 1:13:30.501 --> 1:13:32.361 to get all these numbers. 1:13:32.359 --> 1:13:35.629 So this person, alpha, A, I claim, 1:13:35.627 --> 1:13:41.567 just holds 50 of the riskless bond, pays 50,50 plus 50,70 and 1:13:41.567 --> 1:13:42.357 110. 1:13:42.359 --> 1:13:46.609 No. 1:13:46.609 --> 1:13:53.599 Is this the right--let's just check the numbers. 1:13:53.600 --> 1:13:55.740 Sorry, only one more second. 1:13:55.738 --> 1:14:03.148 I should have--so 100,120 and 160, that's the right number and 1:14:03.154 --> 1:14:10.694 that's equal to 50 of the bond plus 1 quarter of this thing. 1:14:10.689 --> 1:14:15.439 So 50 50 is--1 quarter of this is 50,70 and 110, 1:14:15.440 --> 1:14:16.250 right? 1:14:16.250 --> 1:14:19.510 So if you hold 50 of the bond plus 1 quarter of this you get 1:14:19.507 --> 1:14:19.837 100. 1:14:19.840 --> 1:14:22.080 50 of the bond plus 70 is 120. 1:14:22.079 --> 1:14:25.439 50 of the bond plus 110 is 160. 1:14:25.439 --> 1:14:30.519 And this guy is going to hold 3 quarters of the aggregate 1:14:30.520 --> 1:14:34.060 endowment plus minus 50 of the bond, 1:14:34.060 --> 1:14:38.420 so 3 quarters of the aggregate endowment, 1:14:38.420 --> 1:14:42.400 3 quarters of this thing, 3 quarters of the aggregate 1:14:42.404 --> 1:14:44.784 endowment is 150 - 50 is 100. 1:14:44.779 --> 1:14:48.049 3 quarters of this is 210 - 50 is 160. 1:14:48.050 --> 1:14:51.650 3 quarters of that, is 330 - 50 is 280. 1:14:51.649 --> 1:14:55.219 So what they've done in equilibrium is everybody, 1:14:55.220 --> 1:14:58.240 despite having a million stocks to choose from and thousands of 1:14:58.238 --> 1:15:01.308 states and all that stuff, what everybody does is hold the 1:15:01.311 --> 1:15:04.051 riskless bond, puts money in the bank and 1:15:04.050 --> 1:15:06.220 holds the whole stock market. 1:15:06.220 --> 1:15:14.130 So the first theorem we're going to prove next time is 1:15:14.125 --> 1:15:22.025 called The Mutual Fund Theorem which is that everybody 1:15:22.030 --> 1:15:29.490 diversifies by holding the aggregate economy, 1:15:29.488 --> 1:15:36.918 all stocks in the same proportion, plus money in bank. 1:15:36.920 --> 1:15:39.430 So that theorem of Shakespeare of diversifying, 1:15:39.434 --> 1:15:40.914 what did it amount to do? 1:15:40.909 --> 1:15:42.569 We have a very concrete thing. 1:15:42.569 --> 1:15:44.649 You hold 10 percent. 1:15:44.649 --> 1:15:48.399 This person's holding 25 percent of every stock in the 1:15:48.404 --> 1:15:52.944 whole economy plus putting some money, 50 dollars in the bank. 1:15:52.939 --> 1:15:55.899 The other person is doing 3 quarters of every stock in the 1:15:55.903 --> 1:15:58.973 whole economy plus lending the money to the first person. 1:15:58.970 --> 1:16:02.160 So that's the first of the two amazing results and I'll start 1:16:02.164 --> 1:16:03.714 next time by explaining it. 1:16:03.710 --> 1:16:11.000