WEBVTT 00:01.700 --> 00:05.530 Professor Robert Shiller: Today's lecture is about 00:05.528 --> 00:09.288 portfolio diversification and about supporting financial 00:09.287 --> 00:13.917 institutions, notably mutual funds. 00:13.920 --> 00:19.410 It's actually kind of a crusade of mine--I believe that the 00:19.410 --> 00:23.670 world needs more portfolio diversification. 00:23.670 --> 00:27.090 That might sound to you a little bit odd, 00:27.086 --> 00:32.116 but I think it's absolutely true that the same kind of cause 00:32.124 --> 00:35.374 that Emmett Thompson goes through, 00:35.370 --> 00:38.490 which is to help the poor people of the world, 00:38.493 --> 00:42.033 can be advanced through portfolio diversification--I 00:42.034 --> 00:43.704 seriously mean that. 00:43.700 --> 00:47.920 There are a lot of human hardships that can be solved by 00:47.916 --> 00:49.906 diversifying portfolios. 00:49.910 --> 00:53.850 What I'm going to talk about today applies not just to 00:53.851 --> 00:58.091 comfortable wealthy people, but it applies to everyone. 00:58.090 --> 01:03.050 01:03.050 --> 01:05.610 It's really about risk. 01:05.609 --> 01:09.509 When there's a bad outcome for anyone, that's the outcome of 01:09.513 --> 01:10.773 some random draw. 01:10.769 --> 01:14.599 When people get into real trouble in their lives, 01:14.595 --> 01:19.295 it's because of a sequence of bad events that push them into 01:19.298 --> 01:23.768 unfortunate positions and, very often, financial risk 01:23.767 --> 01:28.077 management is part of the thing that prevents that from 01:28.081 --> 01:32.991 happening. The first--let me go--I want to 01:32.988 --> 01:38.228 start this lecture with some mathematics. 01:38.230 --> 01:42.630 It's a continuation of the second lecture, 01:42.632 --> 01:48.862 where I talked about the principle of dispersal of risk. 01:48.860 --> 01:53.540 I want now to carry that forward into something a little 01:53.543 --> 01:57.293 bit more focused on the portfolio problem. 01:57.290 --> 02:01.710 I'm going to start this lecture with a discussion of how one 02:01.707 --> 02:06.197 constructs a portfolio and what are the mathematics of it. 02:06.200 --> 02:10.720 That will lead us into the capital asset pricing model, 02:10.720 --> 02:15.660 which is the cornerstone of a lot of thinking in finance. 02:15.659 --> 02:20.799 I'm going to go through this rather quickly because there are 02:20.797 --> 02:26.017 other courses at Yale that will cover this more thoroughly, 02:26.020 --> 02:30.950 notably, John Geanakoplos's Econ 251. 02:30.949 --> 02:33.569 I think we can get the basic points here. 02:33.570 --> 02:38.690 02:38.690 --> 02:40.810 Let's start with the basic idea. 02:40.810 --> 02:44.220 I want to just say it in the simplest possible terms. 02:44.220 --> 02:48.010 What is it that--First of all, a portfolio, 02:48.013 --> 02:49.823 let's define that. 02:49.819 --> 02:53.239 A portfolio is the collection of assets that you 02:53.241 --> 02:56.811 have--financial assets, tangible assets--it's your 02:56.809 --> 02:59.889 wealth. The first and fundamental 02:59.888 --> 03:03.718 principle is: you care only about the total 03:03.715 --> 03:06.925 portfolio. You don't want to be someone 03:06.926 --> 03:10.906 like the fisherman who boasts about one big fish that he 03:10.907 --> 03:15.247 caught because it's not--we're talking about livelihoods. 03:15.250 --> 03:18.680 It's all the fish that you caught, so there's nothing to be 03:18.681 --> 03:20.931 proud of if you had one big success. 03:20.930 --> 03:24.630 That's the first very basic principle. 03:24.630 --> 03:26.510 Do you agree with me on that? 03:26.509 --> 03:29.729 So, when we say portfolio management, we mean managing 03:29.733 --> 03:32.473 everything that gives you economic benefit. 03:32.470 --> 03:37.220 Now, underlying our theory is the idea that we measure the 03:37.222 --> 03:42.062 outcome of your investment in your portfolio by the mean of 03:42.059 --> 03:47.059 the return on the portfolio and the variance of the return on 03:47.062 --> 03:50.082 the portfolio. The return, of course, 03:50.083 --> 03:53.593 in any given time period is the percentage increase in the 03:53.590 --> 03:56.180 portfolio; or, it could be a negative 03:56.178 --> 03:58.378 number, it could be a decrease. 03:58.379 --> 04:03.639 The principle is that you want the expected value of the return 04:03.641 --> 04:08.731 to be as high as possible given its variance and you want the 04:08.732 --> 04:13.312 variance of the return on the portfolio to be as low as 04:13.314 --> 04:20.004 possible given the return, because high expected return is 04:19.997 --> 04:22.657 a good thing. You could say, 04:22.658 --> 04:26.068 I think my portfolio has an expected return of 12%--that 04:26.069 --> 04:29.789 would be better than if it had an expected return of 10%. 04:29.790 --> 04:32.690 But, on the other hand, you don't want high variance 04:32.689 --> 04:36.989 because that's risk; so, both of those matter. 04:36.990 --> 04:41.230 In fact, different people might make different choices about how 04:41.231 --> 04:45.071 much risk they're willing to bear to get a higher expected 04:45.069 --> 04:47.179 return. But ultimately, 04:47.179 --> 04:51.009 everyone agrees I--that's the premise here, 04:51.013 --> 04:55.763 that for the--if you're comparing two portfolios with 04:55.760 --> 04:59.230 the same variance, then you want the one with the 04:59.226 --> 05:00.316 higher expected return. 05:00.319 --> 05:03.029 If you're comparing two portfolios with the same 05:03.030 --> 05:05.680 expected return, then you want the one with the 05:05.684 --> 05:09.914 lower variance. All right is that clear 05:09.912 --> 05:13.022 and--okay. So let's talk about--why don't 05:13.015 --> 05:15.205 I just give it in a very intuitive term. 05:15.209 --> 05:18.849 Suppose we had a lot of different stocks that we could 05:18.850 --> 05:21.960 put into a portfolio, and suppose they're all 05:21.964 --> 05:26.104 independent of each other--that means there's no correlation. 05:26.100 --> 05:28.140 We talked about that in Lecture 2. 05:28.139 --> 05:34.269 There's no correlation between them and that means that the 05:34.268 --> 05:39.548 variance--and I want to talk about equally-weighted 05:39.550 --> 05:44.040 portfolio. So, we're going to have 05:44.037 --> 05:47.827 n independent assets; 05:47.830 --> 05:52.530 05:52.530 --> 05:55.000 they could be stocks. 05:55.000 --> 06:00.580 Each one has a standard deviation of return, 06:00.576 --> 06:03.036 call that σ. 06:03.040 --> 06:09.560 06:09.560 --> 06:12.860 Let's suppose that all of them are the same--they all have the 06:12.862 --> 06:14.272 same standard deviation. 06:14.269 --> 06:21.689 We're going to call r the expected return of these 06:21.687 --> 06:22.877 assets. 06:22.880 --> 06:27.000 06:27.000 --> 06:32.980 Then, we have something called the square root rule, 06:32.979 --> 06:39.309 which says that the standard deviation of the portfolio 06:39.311 --> 06:45.761 equals the standard deviation of one of the assets, 06:45.759 --> 06:48.869 divided by the square root of n. 06:48.870 --> 06:49.820 Can you read this in the back? 06:49.820 --> 06:51.420 Am I making that big enough? 06:51.420 --> 06:54.220 Just barely, okay. 06:54.220 --> 06:58.560 This is a special case, though, because I've assumed 06:58.556 --> 07:02.376 that the assets are independent of each other, 07:02.383 --> 07:05.193 which isn't usually the case. 07:05.189 --> 07:08.859 It's like an insurance where people imagine they're insuring 07:08.858 --> 07:12.218 people's lives and they think that their deaths are all 07:12.215 --> 07:15.175 independent. I'm transferring this to the 07:15.179 --> 07:19.749 portfolio management problem and you can see it's the same idea. 07:19.750 --> 07:31.060 I've made a very special case that this is the case of an 07:31.055 --> 07:36.905 equally-weighted portfolio. 07:36.910 --> 07:40.960 It's a very important point, if you see the very simple math 07:40.957 --> 07:42.807 that I'm showing up here. 07:42.810 --> 07:50.860 The return on the portfolio is r, but the standard 07:50.863 --> 07:59.063 deviation of the portfolio is σ/√(n). 07:59.060 --> 08:02.550 So, the optimal thing to do if you live in a world like this is 08:02.552 --> 08:05.652 to get n as large possible and you can reduce the 08:05.650 --> 08:08.800 standard deviation of the portfolio very much and there's 08:08.804 --> 08:11.174 no cost in terms of expected return. 08:11.170 --> 08:15.570 In this simple world, you'd want to make n 100 08:15.574 --> 08:18.374 or 1,000 or whatever you could. 08:18.370 --> 08:22.140 Suppose you could find 10,000 independent assets, 08:22.144 --> 08:26.474 then you could drive the uncertainty about the portfolio 08:26.468 --> 08:28.118 practically to 0. 08:28.120 --> 08:30.390 Because the square root of 10,000 is 100, 08:30.392 --> 08:33.462 whatever the standard deviation of the portfolio is, 08:33.460 --> 08:37.590 you would divide it by 100 and it would become really small. 08:37.590 --> 08:42.410 If you can find assets that all have--that are all independent 08:42.405 --> 08:45.875 of each other, you can reduce the variance of 08:45.878 --> 08:48.008 the portfolio very far. 08:48.009 --> 08:54.529 That's the basic principle of portfolio diversification. 08:54.529 --> 08:59.349 That's what portfolio managers are supposed to be doing all the 08:59.346 --> 08:59.886 time. 08:59.890 --> 09:03.950 09:03.950 --> 09:07.930 Now, I want to be more general than this and talk about the 09:07.927 --> 09:11.577 real case. In the real world we don't have 09:11.578 --> 09:15.318 the problem that assets are independent. 09:15.320 --> 09:18.910 The different stocks tend to move up and down together. 09:18.909 --> 09:21.839 We don't have the ideal world that I just described, 09:21.838 --> 09:24.708 but to some extent we do, so we want to think about 09:24.709 --> 09:26.489 diversifying in this world. 09:26.490 --> 09:33.480 09:33.480 --> 09:39.010 Now, I want to talk about forming a portfolio where the 09:39.009 --> 09:45.049 assets are not independent of each other, but are correlated 09:45.051 --> 09:49.761 with each other. What I'm going to do now--let's 09:49.761 --> 09:54.311 start out with the case where--now it's going to get a 09:54.310 --> 09:58.950 little bit more complicated if we drop the independence 09:58.946 --> 10:02.026 assumption. I'm going to drop more than the 10:02.026 --> 10:04.886 independence assumption, I'm going to assume that the 10:04.889 --> 10:07.969 assets don't have the same expected return and they don't 10:07.972 --> 10:10.012 have the same expected variance. 10:10.009 --> 10:13.509 I'm going to--let's do the two-asset case. 10:13.510 --> 10:21.150 10:21.149 --> 10:33.769 There's n = 2, but not independent or not 10:33.769 --> 10:40.749 necessarily independent. 10:40.750 --> 10:49.940 Asset 1 has expected return r_1. 10:49.940 --> 10:54.260 This is different--I was assuming a minute ago that 10:54.258 --> 10:58.488 they're all the same--it has standard--this is the 10:58.490 --> 11:03.670 expectation of the return of Asset 1 and r_2 11:03.671 --> 11:07.991 is the expectation of the return--I'm sorry, 11:07.990 --> 11:21.520 σ_1 is the standard deviation of the return 11:21.515 --> 11:25.965 on Asset 1. We have the same for Asset 2; 11:25.970 --> 11:30.000 11:30.000 --> 11:34.260 it has an expected return of r_2, 11:34.260 --> 11:39.390 it has a standard deviation of return of σ_2. 11:39.390 --> 11:43.130 Those are the inputs into our analysis. 11:43.129 --> 11:50.139 One more thing, I said they're not independent, 11:50.144 --> 11:59.604 so we have to talk about the covariance between the returns. 11:59.600 --> 12:03.060 So, we're going to have the covariance between 12:03.059 --> 12:06.749 r_1 and r_2, 12:06.750 --> 12:11.090 which you can also call σ_12 and those 12:11.093 --> 12:13.853 are the inputs to our analysis. 12:13.850 --> 12:18.350 What we want to do now is compute the mean and variance of 12:18.347 --> 12:22.447 the portfolio--or the mean and standard deviation, 12:22.450 --> 12:27.370 since standard deviation is the square root of the variance--for 12:27.365 --> 12:30.715 different combinations of the portfolios. 12:30.720 --> 12:35.280 I'm going to generalize from our simple story even more by 12:35.284 --> 12:40.254 saying that, let's not assume that we have equally-weighted. 12:40.250 --> 12:42.530 We're going to put x_1 12:42.527 --> 12:46.237 dollars--let's say we have $1 to invest, we can scale it up and 12:46.242 --> 12:47.922 down, it doesn't matter. 12:47.919 --> 12:53.109 Let's say it's $1 and we're going to put 12:53.113 --> 13:00.173 x_1 in asset 1 and that leaves behind 13:00.170 --> 13:05.230 1-x_1 in asset 2, 13:05.230 --> 13:08.270 because we have $1 total. 13:08.269 --> 13:10.879 We're not going to restrict x_1 to be a 13:10.877 --> 13:13.807 positive number because, as you know or you should know, 13:13.812 --> 13:16.232 you can hold negative quantities of assets, 13:16.230 --> 13:17.730 that's called shorting them. 13:17.730 --> 13:21.630 You can call your broker and say, I'd like to short stock 13:21.625 --> 13:25.865 number one and what the broker will do is borrow the shares on 13:25.868 --> 13:30.248 your behalf and sell them and then you own negative shares. 13:30.250 --> 13:35.470 So, we're not going to--x_1 can be 13:35.474 --> 13:40.914 anything and x--this is x_2 = 13:40.908 --> 13:45.028 1-x_1, so x_1 + 13:45.029 --> 13:46.379 x_2 = 1. 13:46.380 --> 13:53.700 13:53.700 --> 14:00.060 Now, we just want to compute what is the mean and variance of 14:00.060 --> 14:06.320 the portfolio and that's simple arithmetic, based on what we 14:06.315 --> 14:08.855 talked about before. 14:08.860 --> 14:09.770 I'm going to erase this. 14:09.770 --> 14:16.620 14:16.620 --> 14:20.100 The portfolio mean variance will depend on 14:20.103 --> 14:25.123 x_1 in the way that if you put--if you made 14:25.116 --> 14:29.936 x_1 = 1, it would be asset 1 and if you 14:29.943 --> 14:34.513 made x_1 = 0, then it would be the same as 14:34.513 --> 14:36.953 asset 2 returns. But, in between, 14:36.953 --> 14:39.493 if some other number, it'll be some blend of 14:39.489 --> 14:43.089 the--mean and variance of--the portfolio will be some blend of 14:43.086 --> 14:45.736 the mean and variance of the two assets. 14:45.740 --> 14:50.440 14:50.440 --> 15:00.240 The portfolio expected return is going to be given by the 15:00.235 --> 15:10.025 summation i = 1 to n, of x_i*r 15:10.031 --> 15:13.881 _i,. 15:13.879 --> 15:16.409 In this case, since n = 2 that's 15:16.407 --> 15:19.197 x_1 r_1 + 15:19.200 --> 15:22.260 x_2 r_2, 15:22.259 --> 15:27.039 or that's x_1 r_1 + (1 - 15:27.043 --> 15:30.823 x_1) r_2; 15:30.820 --> 15:37.220 15:37.220 --> 15:41.030 that's the expected return on the portfolio. 15:41.029 --> 15:48.529 The variance of the portfolio σ²--this is the 15:48.532 --> 15:53.532 portfolio variance--is σ² = 15:53.534 --> 16:00.484 x_1² σ_1² 16:00.482 --> 16:07.712 + x_2² σ_2² 16:07.707 --> 16:13.677 + 2x_1 x_2 16:13.681 --> 16:21.331 σ_12; that's just the formula for the 16:21.333 --> 16:26.843 variance of the portfolio as a function of--Now, 16:26.840 --> 16:31.310 since they have to sum to 1, I can write this as 16:31.307 --> 16:36.057 x_1² σ_1² 16:39.196 --> 16:44.136 σ_2² + 2x_1 (1 - 16:44.138 --> 16:49.078 x_1) σ_12 and so that 16:49.081 --> 16:53.361 together traces out--I can choose any value of 16:53.358 --> 16:58.838 x_1 I want, it can be number from minus 16:58.840 --> 17:00.520 infinity to plus infinity. 17:00.519 --> 17:05.609 That shows me then for any value of x_1, 17:05.605 --> 17:10.235 I can compute what r is and what σ² 17:10.236 --> 17:15.226 is and I can then describe the opportunities I have from 17:15.230 --> 17:18.590 investing that depend on these. 17:18.589 --> 17:26.839 Now, one thing to do is to solve the equation for r 17:26.836 --> 17:35.656 and x_1 and I can then recast the variance in 17:35.662 --> 17:40.912 terms of r ; that gives us the variance of 17:40.913 --> 17:44.633 the portfolio as a function of the expected return of the 17:44.627 --> 17:48.327 portfolio. Let me just solve this 17:48.328 --> 17:53.488 for--let's solve x_1 for 17:53.489 --> 18:00.709 r. I've got--this should be x--did I make a 18:00.714 --> 18:07.424 mistake there--so it says that r_1 - 18:07.423 --> 18:12.843 r_2 = x_1 18:12.842 --> 18:19.682 r_1 - r_2, 18:19.680 --> 18:22.900 so x_1 = (r - 18:22.901 --> 18:26.931 r_2)/( r_1 - 18:26.928 --> 18:31.938 r_2) and I can substitute this into this 18:31.940 --> 18:37.400 equation and I get the portfolio variance as a function of the 18:37.399 --> 18:40.889 portfolio expected return r. 18:40.890 --> 18:47.170 That's all the basic math that we need. 18:47.170 --> 18:51.600 If I do that, then I get what's called the 18:51.603 --> 18:54.743 frontier for the portfolio. 18:54.740 --> 18:59.200 I have an example on the screen here, but it shows other things. 18:59.200 --> 19:05.460 Let me just--rather than--maybe I'm showing too many things at 19:05.461 --> 19:07.081 once. Let me just draw it. 19:07.079 --> 19:09.739 I'll leave that up for now but we're moving to that. 19:09.740 --> 19:18.260 What we're doing here is the--with two assets, 19:18.256 --> 19:29.796 if I plot the expected annual return r on this axis and 19:29.800 --> 19:40.020 I plot the variance of the portfolio on this axis, 19:40.019 --> 19:47.119 what we have--I'm sorry, the standard deviation of the 19:47.123 --> 19:49.673 portfolio return. 19:49.670 --> 19:56.370 19:56.369 --> 20:00.729 It tends to look--it looks something like this--it's a 20:00.732 --> 20:05.642 hyperbola; there's a minimum variance 20:05.637 --> 20:13.767 portfolio where this σ is as small as possible and 20:13.773 --> 20:22.663 there are many other possible portfolios that lie along this 20:22.662 --> 20:27.082 curve. The curve includes points on 20:27.078 --> 20:31.608 it, which would represent the initial assets. 20:31.609 --> 20:36.569 For example, we might have--this is asset 20:36.568 --> 20:44.128 1--and we might have something here--this could be asset 2. 20:44.130 --> 20:51.590 Depending on where the assets expected returns are and the 20:51.585 --> 20:57.705 assets' standard deviations, we can see that we might be 20:57.711 --> 21:02.281 able to do better than--have a lower variance than either 21:02.276 --> 21:05.146 asset. The equally-weighted case that 21:05.147 --> 21:09.207 I gave a minute ago was one where the two assets had--were 21:09.207 --> 21:12.977 at the same--had the same expected return and the same 21:12.982 --> 21:16.582 variance; but this is quite a bit more 21:16.575 --> 21:19.325 general. So that's the expected return 21:19.325 --> 21:22.095 and efficient portfolio frontier problem. 21:22.099 --> 21:26.759 I wanted to show an example with real data that I computed 21:26.757 --> 21:29.777 and that's what's up on the screen. 21:29.779 --> 21:36.669 The pink line takes two assets, one is stocks and the other is 21:36.667 --> 21:40.617 bonds, actually government bonds. 21:40.619 --> 21:45.209 I computed the efficient portfolio frontier for 21:45.212 --> 21:50.702 various--it's the efficient portfolio frontier using the 21:50.704 --> 21:53.504 formula I just gave you. 21:53.500 --> 22:01.770 The pink line here is the efficient portfolio frontier 22:01.773 --> 22:09.113 when we have only stocks and bonds to invest. 22:09.109 --> 22:13.019 You can see the different points--I've calculated this 22:13.015 --> 22:17.215 using data from 1983 until 2006--and I computed all of the 22:17.216 --> 22:20.676 inputs to those equations that we just saw. 22:20.680 --> 22:25.260 I computed the average return on stocks over that time period 22:25.262 --> 22:29.542 and I computed the average return on bonds over that time 22:29.539 --> 22:32.549 period. These are long-term government 22:32.546 --> 22:36.226 bonds and I--now these are--since they're long-term, 22:36.230 --> 22:40.130 they have some uncertainty and variability to them. 22:40.130 --> 22:41.710 I computed the σ_1, 22:41.706 --> 22:43.756 σ_2, r_1, 22:43.759 --> 22:46.059 and r_2 for those and I plugged it into that 22:46.062 --> 22:47.292 formula, which we just showed. 22:47.290 --> 22:49.640 That's the curve that I got out. 22:49.640 --> 22:54.440 It shows the standard deviation of the return on the portfolio 22:54.436 --> 22:58.836 as a function of the expected return on the portfolio. 22:58.839 --> 23:02.979 I can achieve any combination--I can achieve any 23:02.976 --> 23:08.076 point on that by choosing an allocation of my portfolio. 23:08.079 --> 23:12.449 This point right here is, on the pink line, 23:12.453 --> 23:15.373 is a portfolio 100% bonds. 23:15.369 --> 23:19.569 Over this time period, that portfolio had an expected 23:19.565 --> 23:23.835 return of something like a little over 9% and it had a 23:23.840 --> 23:27.310 standard deviation of a little over 9%. 23:27.309 --> 23:32.469 This is a portfolio, which is 100% stocks, 23:32.467 --> 23:39.127 and that portfolio had a much higher average return or 23:39.133 --> 23:46.563 expected return--13%--but it also had a much higher standard 23:46.555 --> 23:52.085 deviation of return--it was about 16%. 23:52.089 --> 23:56.109 So, you can see that those are the two raw portfolios. 23:56.109 --> 23:59.839 That could be investor only in bonds or an investor only in 23:59.844 --> 24:03.904 stocks, but I also show on here what some other returns are that 24:03.901 --> 24:07.771 are available. The minimum variance portfolio 24:07.766 --> 24:11.466 is down here. That's got the lowest possible 24:11.469 --> 24:16.509 standard deviation of expected return and that's 25% stocks and 24:16.511 --> 24:19.521 75% bonds with this sample period. 24:19.520 --> 24:24.890 I can try other portfolios; this one right here--I'm 24:24.893 --> 24:28.653 pointing to a point on the pink line--that point right there, 24:28.653 --> 24:30.223 50% stocks, 50% bonds. 24:30.220 --> 24:35.810 You can see–You can also go up here, you can go beyond 24:35.807 --> 24:41.297 100% stocks, you can have 150% stocks in your portfolio. 24:41.299 --> 24:43.669 That means you'd have a leveraged portfolio, 24:43.671 --> 24:45.051 you would be borrowing. 24:45.049 --> 24:50.639 If you had $1 to invest you can borrow $.50 and invest in a 24:50.644 --> 24:52.964 $1.50 worth of stocks. 24:52.960 --> 24:57.750 That would put you out here; you would have very much more 24:57.753 --> 25:00.563 return, but you'd have more risk. 25:00.559 --> 25:03.659 Borrowing to buy stocks is going to be risky. 25:03.660 --> 25:08.130 You could also pick a point down here, which is more than 25:08.133 --> 25:11.013 100% bonds--how would you do that? 25:11.009 --> 25:14.229 Well, you could short in the stock market, 25:14.229 --> 25:18.779 you could short $.50 worth of stocks and buy $1.50 worth of 25:18.783 --> 25:22.163 bonds and that would put you down here. 25:22.160 --> 25:25.880 Any one of those things is possible it's just the simple 25:25.880 --> 25:27.910 math that I just showed you. 25:27.910 --> 25:30.510 Do you have any idea what you would like to do, 25:30.506 --> 25:33.566 assuming this? Well if you're an investor, 25:33.569 --> 25:35.569 you don't like variance. 25:35.569 --> 25:39.829 So, you probably don't want to pick any point down here, 25:39.825 --> 25:44.305 because you're not getting anything by picking a point down 25:44.313 --> 25:48.963 there because you could have a better point by just moving it 25:48.956 --> 25:51.646 up here. You'd have a higher expected 25:51.652 --> 25:53.382 return with no more variance. 25:53.380 --> 25:55.110 It's getting kind of complicated, isn't it? 25:55.109 --> 25:58.459 We started out with just a simple idea: that you don't want 25:58.461 --> 26:01.761 to put all your eggs in one basket and if you had a lot of 26:01.755 --> 26:04.755 independent stocks you would want to just weight them 26:04.760 --> 26:07.450 equally. But now, you see there are a 26:07.450 --> 26:11.480 lot of possibilities and the outcome of your portfolio choice 26:11.479 --> 26:13.829 can be anything along this line. 26:13.829 --> 26:17.859 I'm not going to tell you what you want to do except to say, 26:17.863 --> 26:21.553 you would never pick a point below the minimum variance 26:21.554 --> 26:22.994 portfolio, right? 26:22.990 --> 26:27.470 Because, if you did, then you would always be 26:27.468 --> 26:29.898 dominated. You could always find a 26:29.899 --> 26:32.789 portfolio that had a higher expected return for the same 26:32.792 --> 26:33.952 standard deviation. 26:33.950 --> 26:37.690 But beyond that, if you were confined to just 26:37.685 --> 26:41.585 stocks and bonds, it would be a matter of taste 26:41.589 --> 26:45.239 where along this frontier you would be. 26:45.240 --> 26:47.280 You would call it an efficient portfolio frontier. 26:47.279 --> 26:52.269 It would be anywhere from here to here, depending on how much 26:52.266 --> 26:57.416 you're afraid of risk and how much you want expected return. 26:57.420 --> 27:02.360 Now, we can also move to three assets and, in fact, 27:02.360 --> 27:04.930 to any number of assets. 27:04.930 --> 27:08.370 The same formula extends to more assets. 27:08.369 --> 27:14.019 In fact, I have it--suppose we have three assets and we want to 27:14.018 --> 27:19.668 compute the efficient portfolio frontier, the mean and variance 27:19.666 --> 27:21.576 of the portfolio. 27:21.579 --> 27:26.879 What I have up there on the diagram are calculations I made 27:26.875 --> 27:32.075 for the efficient portfolio frontier with three assets. 27:32.079 --> 27:37.889 So, now we have n = 3 and in the chart are stocks, 27:37.887 --> 27:42.967 bonds, and oil. Oil is a very important asset, 27:42.972 --> 27:48.952 so we want to compute what that--so now we have lots of 27:48.952 --> 27:50.732 inputs. Let's put the 27:50.733 --> 27:53.333 inputs--r_1, r_2, 27:53.325 --> 27:56.175 and r_3 are the expected returns on the 27:56.176 --> 27:59.126 three assets. Then, we have the standard 27:59.134 --> 28:03.814 deviations of the returns of the three assets and we have the 28:03.805 --> 28:08.005 covariance between the returns on the three assets. 28:08.009 --> 28:11.909 There are three of them-- σ_12, 28:11.908 --> 28:15.978 σ_13, and σ_23. 28:15.980 --> 28:22.480 That's what we have to know to compute the efficient portfolio 28:22.479 --> 28:25.569 frontier with three assets. 28:25.569 --> 28:27.949 To make this picture, I did that. 28:27.950 --> 28:32.550 I computed the returns on the stocks, bonds, 28:32.546 --> 28:38.746 and oil for every year from 1983 and I computed the average 28:38.747 --> 28:41.767 returns, which I take as the expected 28:41.765 --> 28:45.105 returns, I took the standard deviations, and I took the 28:45.112 --> 28:47.122 covariance. These are all formulas, 28:47.124 --> 28:49.914 I just plugged it into formulas that we did in the second 28:49.912 --> 28:54.392 lecture. What is the portfolio expected 28:54.391 --> 28:56.811 return? The portfolio expected 28:56.806 --> 28:59.416 return--we have to choose three things now: 28:59.416 --> 29:02.026 x_1, x_2, 29:02.026 --> 29:03.886 and x_3. 29:03.890 --> 29:07.100 x_1 is the amount that I put into the first 29:07.095 --> 29:10.355 asset, x_2 is the amount that I put into the 29:10.356 --> 29:12.996 second asset, and x3 is the amount I 29:13.004 --> 29:14.554 put into the third asset. 29:14.549 --> 29:16.829 I'm going to constrain them to sum to one. 29:16.829 --> 29:21.779 The return on the portfolio is x_1 29:21.783 --> 29:25.943 r_1 + x_2 29:25.944 --> 29:30.114 r_2 + x_3 29:30.105 --> 29:32.775 r_3. 29:32.779 --> 29:38.199 The variance of the portfolio, σ², 29:38.195 --> 29:44.865 is x_1² σ_1^(2) + 29:44.869 --> 29:51.169 x_2² σ_2² 29:51.166 --> 29:57.716 + x_3² σ_3² 29:57.715 --> 30:05.645 -- then we have to the count of all of the covariance terms -- + 30:05.649 --> 30:09.929 2x_1x _2 30:09.931 --> 30:16.861 σ_12 + 2x_1x_3 30:16.858 --> 30:23.778 σ_13 + 2x_2x_3 30:23.784 --> 30:29.204 σ_23. 30:29.200 --> 30:30.990 Is that clear enough? 30:30.990 --> 30:34.900 It seems like a logical extension of that formula to 30:34.903 --> 30:39.663 three assets and you can easily see how to extend it to four or 30:39.662 --> 30:42.702 more assets, it's just the logical extension 30:42.703 --> 30:46.123 of that. What I did in this diagram is I 30:46.118 --> 30:52.078 computed the efficient portfolio frontier--now it's the blue line 30:52.083 --> 30:54.043 with three assets. 30:54.039 --> 31:00.399 Now, once you have more than three--more than two assets--it 31:00.399 --> 31:06.219 might be possible to get points inside the frontier. 31:06.220 --> 31:09.870 But I'm talking here--this is the actually the frontier--the 31:09.870 --> 31:13.150 best possible portfolio consisting of three assets. 31:13.150 --> 31:17.160 You can see that it dominates the pink line. 31:17.160 --> 31:21.000 When you add another asset, you do better when you have 31:21.003 --> 31:24.063 three assets, you do better than if you just 31:24.064 --> 31:28.124 had two because there's more diversification possible with 31:28.121 --> 31:30.471 three assets than with two. 31:30.470 --> 31:34.060 Oil, bonds, and stocks are all independent--somewhat 31:34.060 --> 31:37.510 independent--they're not perfectly independent, 31:37.509 --> 31:39.289 but they're somewhat independent and, 31:39.294 --> 31:41.974 to the extent that they are, it lowers the variance. 31:41.970 --> 31:46.020 You should see that the blue line is better than the pink 31:46.016 --> 31:48.686 line because, for any expected return, 31:48.690 --> 31:51.940 the blue line is to the left of the pink line, 31:51.942 --> 31:53.982 right? So, for example, 31:53.977 --> 31:58.277 at an annual expected return of 12% if I have a portfolio of 31:58.280 --> 32:01.010 stocks, bonds, and oil I can get a 32:01.014 --> 32:05.414 standard deviation of something like 8% on my portfolio. 32:05.410 --> 32:09.010 But if I would confine myself just to stocks and bonds, 32:09.011 --> 32:12.481 then I would get a much higher standard deviation. 32:12.480 --> 32:13.860 Are you following this? 32:13.859 --> 32:17.589 The general principle of portfolio management is: 32:17.589 --> 32:21.319 you want to include as many assets as you can. 32:21.319 --> 32:24.099 You want to get it--if you keep adding assets, 32:24.099 --> 32:27.439 you can do better and better on your portfolio standard 32:27.435 --> 32:31.295 deviation. You can see some of the points 32:31.304 --> 32:34.914 I've made along the blue line here. 32:34.910 --> 32:39.080 This is--let's see if I have it. 32:39.079 --> 32:45.439 This is a portfolio, which has all oil and stocks 32:45.436 --> 32:48.346 and it has no bonds. 32:48.349 --> 32:53.639 This portfolio, the minimum variance portfolio, 32:53.636 --> 32:59.036 is 9% oil, 27% stocks, and 64% bonds and most of 32:59.037 --> 33:03.057 the--many choices you can make. 33:03.059 --> 33:08.279 The first--you see, the idea here is that in order 33:08.275 --> 33:14.335 to manage portfolios what we want to do is calculate these 33:14.343 --> 33:18.173 statistics, which are the expected returns 33:18.174 --> 33:22.054 on the various assets, the standard deviations of the 33:22.048 --> 33:25.448 various assets, and you've got to know their 33:25.450 --> 33:30.160 covariances because that affects the variance of the portfolio. 33:30.160 --> 33:34.140 The more they covary--they move together--the less they cancel 33:34.138 --> 33:37.038 out. So, the higher the covariance 33:37.044 --> 33:40.694 is, generally, the higher--you can see from 33:40.687 --> 33:45.367 here--the higher the σ² of the portfolio. 33:45.370 --> 33:51.990 33:51.990 --> 33:57.200 Is that clear? There's one more thing that we 33:57.202 --> 33:59.612 can do. I have three assets shown here. 33:59.609 --> 34:03.339 I have stocks, bonds, and oil but I want also 34:03.341 --> 34:07.921 to add one more final asset, we'll call it the riskless 34:07.921 --> 34:09.841 asset, which is the 34:09.839 --> 34:15.369 asset–long-term bonds are somewhat uncertain and variable 34:15.365 --> 34:17.945 because they're long-term. 34:17.949 --> 34:21.009 If we have an annual return that we're looking at, 34:21.011 --> 34:24.321 we can find a completely riskless asset with an annual 34:24.323 --> 34:28.513 return--it would be a government bond that matures in one year. 34:28.510 --> 34:33.010 Now, assuming that we trust the government--I think the U.S. 34:33.010 --> 34:36.130 Government has never defaulted on its debt--we'd take that as a 34:36.125 --> 34:38.725 riskless return. It probably has some risk, 34:38.734 --> 34:41.624 but the way we approximate things in finance, 34:41.618 --> 34:44.108 we take the government as riskless. 34:44.110 --> 34:48.850 With the government expected return, we want to make that 34:48.850 --> 34:53.080 expected return as a fourth asset--we could call it 34:53.083 --> 34:56.133 r_4 but I'll call it 34:56.131 --> 35:00.281 r_f--it's a special asset. 35:00.280 --> 35:02.960 So r_f is the riskless asset. 35:02.960 --> 35:11.610 35:11.610 --> 35:17.720 So, for it, σ_f = 0. 35:17.719 --> 35:22.209 It's like a fourth asset but we're using a special feature of 35:22.213 --> 35:24.763 this asset: that it has no risk. 35:24.760 --> 35:32.490 Moreover, the correlation--the covariance between any of these 35:32.491 --> 35:36.421 σ_1f = 0, etc. 35:36.420 --> 35:39.350 It just doesn't have any risk to it, it has no variability. 35:39.349 --> 35:43.199 If we want to add that asset to the portfolio, 35:43.196 --> 35:48.146 what it does is it produces an efficient portfolio frontier 35:48.154 --> 35:53.254 that is now a straight line; I show that on the diagram. 35:53.250 --> 35:58.780 The best possible portfolio that you can get would be points 35:58.779 --> 36:01.309 along this straight line. 36:01.309 --> 36:08.589 That is the final aspect of the efficient portfolio 36:08.586 --> 36:12.916 calculations. Again, I'm not able to give 36:12.921 --> 36:17.621 this as much of a discussion as I would like because I don't 36:17.622 --> 36:20.732 want to spend too much time on this. 36:20.730 --> 36:24.520 In the review sections I'm hopeful that your T.A.'s can 36:24.516 --> 36:26.266 elaborate on this more. 36:26.269 --> 36:30.499 There's a very important principle that finally comes out 36:30.497 --> 36:35.177 here, it is that you always want to reduce the variance of your 36:35.178 --> 36:37.668 portfolio as much as you can. 36:37.670 --> 36:40.770 That means that you want to pick, ultimately, 36:40.770 --> 36:44.500 a point on this--this line is tangent to the efficient 36:44.504 --> 36:48.384 portfolio frontier with all the other assets in it. 36:48.380 --> 36:52.570 Tangent means that it has the same slope, it just touches the 36:52.565 --> 36:56.815 efficient portfolio frontier for risky assets at one point, 36:56.820 --> 37:00.760 and the slope of the efficient portfolio frontier, 37:00.764 --> 37:04.854 including the riskless asset, is a straight line that goes 37:04.849 --> 37:06.779 through the tangency point, here. 37:06.780 --> 37:12.840 That is the--I think that's the end of my mathematics. 37:12.840 --> 37:19.920 What I have shown here is how you calculate your portfolio 37:19.916 --> 37:23.656 management. The way you would go about it, 37:23.659 --> 37:27.699 if you're a portfolio manager, is you have to come up with 37:27.704 --> 37:31.684 estimates of the inputs to these formulas--that means the 37:31.678 --> 37:34.658 expected returns, the standard deviations, 37:34.664 --> 37:35.914 and the covariances. 37:35.909 --> 37:41.009 You take all the risky assets and you analyze them first to 37:41.013 --> 37:45.763 get their--you have to do a statistical analysis to get 37:45.764 --> 37:48.774 their expected returns, their variances, 37:48.771 --> 37:49.681 and their covariances. 37:49.679 --> 37:54.069 Once you've got them together, then you can compute the 37:54.067 --> 37:58.777 efficient portfolio frontier without the riskless asset. 37:58.780 --> 38:02.610 Then finally, the final step is to find what 38:02.607 --> 38:07.857 is the tangency line that goes through the riskless rate. 38:07.860 --> 38:12.500 It doesn't show it on this chart, it goes through 5% at a 38:12.503 --> 38:18.463 standard deviation of 0, then it touches the risky asset 38:18.460 --> 38:25.820 efficient portfolio frontier at one point, then from there it 38:25.815 --> 38:33.285 goes up above in terms of higher expected returns for the same 38:33.293 --> 38:38.483 variance. That's the theory of efficient 38:38.479 --> 38:41.139 portfolio calculation. 38:41.139 --> 38:47.229 There's something that--a fundamental principle--and this 38:47.234 --> 38:52.794 is leading us now to the institutional topic of this 38:52.785 --> 38:59.205 course--is that there's only one tangency portfolio and that 38:59.206 --> 39:04.536 portfolio is called the tangency portfolio, 39:04.539 --> 39:09.239 where a line drawn from the risk--from the x-axis at the 39:09.242 --> 39:14.632 riskless rate is tangent to the efficient portfolio frontier. 39:14.630 --> 39:31.990 The tangency portfolio is the portfolio that one should hold. 39:31.989 --> 39:41.289 The tangency portfolio gives rise to what's called the mutual 39:41.285 --> 39:49.495 fund theorem in finance, which says that all investors 39:49.497 --> 39:54.607 need is a single mutual fund. 39:54.610 --> 39:56.870 Now, I haven't defined mutual fund yet. 39:56.869 --> 40:03.239 A mutual fund is an investment vehicle that allows investors to 40:03.238 --> 40:05.188 hold a portfolio. 40:05.190 --> 40:09.630 The theory of mutual funds is: nobody is supposed to be 40:09.627 --> 40:14.807 holding anything other than--the ideal theory of mutual funds is 40:14.805 --> 40:19.485 holding something other than this tangency portfolio. 40:19.489 --> 40:23.269 So, why don't we set up a company that creates a portfolio 40:23.269 --> 40:26.849 like that and investors can buy into that portfolio. 40:26.849 --> 40:30.609 What this--if my analysis is right--namely, 40:30.612 --> 40:35.362 if I've got all of the right estimates of the expected 40:35.359 --> 40:38.339 returns on stocks, bonds, and oil, 40:38.335 --> 40:42.355 and the standard deviations and covariances--and assuming the 40:42.358 --> 40:45.448 interest rate is 5%, which is what I've assumed 40:45.450 --> 40:48.670 here, that this line, if you go to 0 it hits that 5%. 40:48.670 --> 40:52.240 It hits the vertical access at 5%. 40:52.239 --> 40:56.629 Then everyone should be holding the tangency portfolio. 40:56.630 --> 40:58.720 What is the tangency portfolio in this case? 40:58.719 --> 41:06.719 It's 12% oil, 36% stocks, and 52% bonds. 41:06.719 --> 41:09.819 That's what I got using this sample period. 41:09.820 --> 41:12.540 Some people might disagree with that, they might not take my 41:12.537 --> 41:15.177 estimates. They might say my sample period 41:15.183 --> 41:19.033 was off, but that's what the theory--using my data for the 41:19.033 --> 41:22.413 sample period that I computed--the expected returns 41:22.410 --> 41:25.180 and co-variances says one should do. 41:25.179 --> 41:30.929 The theory says everybody should be investing in these 41:30.934 --> 41:36.694 proportions and this theory then--it doesn't leave any 41:36.689 --> 41:43.419 latitude for individual choice except that you can choose which 41:43.422 --> 41:50.482 mixture of the mutual fund and the riskless asset you want. 41:50.480 --> 41:54.350 Somebody who is very risk averse could say, 41:54.349 --> 41:59.229 I want to hold only the riskless assets because I just 41:59.233 --> 42:02.093 don't like any risk at all. 42:02.090 --> 42:05.170 That person--I should have maybe included that in the 42:05.171 --> 42:08.491 diagram--that person could get 5% return with no risk. 42:08.489 --> 42:11.549 Somebody else might say, well I want to just hold this 42:11.548 --> 42:14.258 point, I want to hold the tangency portfolio. 42:14.260 --> 42:18.400 That's attractive to me because I could then get a bigger 42:18.401 --> 42:21.731 expected return, I could get almost 12% return 42:21.728 --> 42:24.408 per year, and I'd sacrifice--I'd have 42:24.412 --> 42:26.712 some standard deviation of like 8%; 42:26.710 --> 42:30.600 but, if that's what I want, if I have different tastes 42:30.599 --> 42:34.709 about risk, then--and that's what I want--then that's the 42:34.708 --> 42:36.468 optimal thing to do. 42:36.469 --> 42:39.029 Other people might say, well you know, 42:39.028 --> 42:42.618 I'm really an adventurer--I don't care too much about 42:42.624 --> 42:45.394 risk--I want the much higher return. 42:45.389 --> 42:49.879 Such a person might pick a point up here and that would be 42:49.880 --> 42:53.190 a portfolio with--a leveraged portfolio. 42:53.190 --> 42:56.880 That would be a portfolio where you borrowed at the riskless 42:56.878 --> 43:00.748 rate and you put more than 100% of your money into the tangency 43:00.754 --> 43:03.914 portfolio. What you could do is, 43:03.911 --> 43:10.211 say, borrow $.50 on your $1 and put $1.50 into a portfolio, 43:10.214 --> 43:16.304 which consisted of 9% oil, 27% stocks, and 64% bonds. 43:16.300 --> 43:19.850 Everyone would do that, no one would ever hold some 43:19.848 --> 43:23.818 other portfolio because you can see that this line is the 43:23.822 --> 43:27.662 lowest--it's far--you want to get to the left as far as 43:27.655 --> 43:29.295 possible. You want to, 43:29.302 --> 43:31.922 for any given expected return, you want to minimize the 43:31.922 --> 43:34.822 standard deviation, so it's the left-most line and 43:34.822 --> 43:38.552 that means that everyone will be holding the same portfolio. 43:38.550 --> 43:42.840 I don't find that my analysis is profound in the final answer, 43:42.836 --> 43:46.136 I just took some estimates using my data and, 43:46.139 --> 43:49.689 again, we could--if someone wanted to argue with us they 43:49.687 --> 43:53.487 could argue with my estimates of the expected returns of the 43:53.492 --> 43:56.332 standard deviations and the covariances, 43:56.330 --> 43:58.430 but not with this theory. 43:58.430 --> 44:00.000 This theory is very rigorous. 44:00.000 --> 44:05.000 If you agree with my estimates, then you should do this as an 44:04.998 --> 44:09.908 investor, you should hold only some mixture of this tangency 44:09.913 --> 44:12.803 portfolio, which is 9% oil, 44:12.795 --> 44:15.685 27% stocks, and 64% bonds. 44:15.690 --> 44:16.900 You see what we've got here? 44:16.900 --> 44:19.900 I started out with the equally-weighted--I was talking 44:19.902 --> 44:23.132 about stocks--about n stocks that all have the same 44:23.131 --> 44:26.021 variance and are all independent of each other. 44:26.019 --> 44:28.919 But, I've dropped that assumption and now I'm going on 44:28.924 --> 44:32.274 to assuming that they're taking account of their dependence on 44:32.267 --> 44:34.417 each other, taking account of their 44:34.424 --> 44:37.564 different expected returns, and taking account of their 44:37.557 --> 44:39.817 different covariances and variances; 44:39.820 --> 44:41.390 so that's what we've got. 44:41.390 --> 44:43.040 This is a famous framework. 44:43.039 --> 44:46.069 This diagram is, I think, the most famous 44:46.070 --> 44:50.240 diagram in all of financial theory and it's actually the 44:50.237 --> 44:52.507 first theoretical diagram. 44:52.510 --> 44:56.510 I did it myself using my data, but it would always look more 44:56.512 --> 45:00.312 or less like this--slightly different positions if people 45:00.310 --> 45:02.210 use different estimates. 45:02.210 --> 45:06.510 I actually showed this diagram--I went to Norway with 45:06.513 --> 45:11.483 my colleague--I actually have a couple more pictures here. 45:11.480 --> 45:18.110 45:18.110 --> 45:20.520 That's my colleague, Ronit Walny, 45:20.523 --> 45:25.503 and I, we're posing in front of the Parliament Building in Oslo. 45:25.500 --> 45:29.760 We went to Norway to discuss the--with the Norwegian 45:29.756 --> 45:32.256 Government--their portfolio. 45:32.260 --> 45:34.690 This is a slide that I showed them. 45:34.690 --> 45:38.100 I showed them the slide that I just showed you, 45:38.099 --> 45:42.099 showing the optimal portfolio, and then I looked at the 45:42.101 --> 45:44.771 Norwegian Government's position. 45:44.769 --> 45:51.359 The Norwegian Government has pension fund assets in the 45:51.358 --> 45:56.848 amount of, as of 2006, just under two trillion 45:56.847 --> 46:03.797 Norwegian Kroner; but they also own North Sea Oil. 46:03.800 --> 46:05.910 If you know that, it's kind of divided between 46:05.911 --> 46:06.851 the UK and Norway. 46:06.849 --> 46:11.699 Norway has a much smaller population than the UK and they 46:11.703 --> 46:15.693 have a lot of oil up there in the North Sea. 46:15.690 --> 46:23.120 As of then, I calculated the value of their oil in the North 46:23.117 --> 46:29.917 Sea and that's what I got--it's worth about 3.5 billion 46:29.916 --> 46:32.556 Norwegian Kroner. 46:32.560 --> 46:34.740 Do you see the difference? 46:34.739 --> 46:39.529 In fact, the assets that the Norwegian Government owns is 46:39.531 --> 46:43.981 about two-thirds oil and one-third government pension 46:43.980 --> 46:47.470 fund assets. This government pension fund, 46:47.467 --> 46:50.677 I guess, in dollars, is about $200 billion. 46:50.679 --> 46:54.969 It's a huge amount of money that they're managing, 46:54.968 --> 46:59.518 but I was trying to convince them that they should do 46:59.520 --> 47:04.250 something to manage their oil risks because they're way 47:04.246 --> 47:06.606 over-invested in oil. 47:06.610 --> 47:08.670 Where are they on the efficient portfolio frontier? 47:08.670 --> 47:20.900 47:20.900 --> 47:27.410 They have 64% oil in their portfolio. 47:27.410 --> 47:28.970 Where does that put them? 47:28.969 --> 47:31.089 Well it's not--it's really off the diagram. 47:31.090 --> 47:35.230 The furthest point that I recorded was 28% oil--that puts 47:35.233 --> 47:39.823 them there--so if they--if you wanted to, where would it be? 47:39.820 --> 47:42.170 It would be somewhere over there off the charts. 47:42.170 --> 47:46.750 What the Norwegian Government is doing wrong is--it's a little 47:46.753 --> 47:50.513 bit controversial, my pointing this out to them. 47:50.510 --> 47:55.600 I ended up in the newspaper the next day for having argued that 47:55.604 --> 48:00.294 they are way off on their investment opportunities because 48:00.288 --> 48:02.998 oil is such a volatile thing. 48:03.000 --> 48:05.460 They've got so much of their assets tied up in oil. 48:05.460 --> 48:08.260 I got a good hearing. 48:08.260 --> 48:11.540 I went to the Ministry of Finance and we went to the 48:11.537 --> 48:14.297 Norges Bank, which is their central bank, 48:14.300 --> 48:17.640 and I think the answer I got from them was, 48:17.639 --> 48:19.149 yes you're right. 48:19.150 --> 48:23.230 I never got it quite like that--something like that. 48:23.230 --> 48:26.570 There was a conditional agreement: yes, 48:26.568 --> 48:31.308 Norway should manage this oil risk but it's politically 48:31.312 --> 48:36.142 difficult and that's the problem--is that we're not able 48:36.144 --> 48:39.224 to do our optimal management. 48:39.219 --> 48:42.459 Maybe there are a number of structural problems that prevent 48:42.461 --> 48:44.551 them from doing that and they think, 48:44.550 --> 48:48.180 maybe, that--I think Norway may be moving that way, 48:48.182 --> 48:50.292 so we'll see in the future. 48:50.290 --> 48:55.950 I went to the Bank of Mexico; I tried to convince the--I met 48:55.952 --> 48:59.232 the president of the Bank of Mexico and tried to tell them 48:59.225 --> 49:01.345 that Mexico is too reliant on oil, 49:01.349 --> 49:03.859 too much oil--they have to get rid of their risk. 49:03.860 --> 49:06.870 I'm going to the Russian Stabilization Fund in--I think 49:06.865 --> 49:09.815 I've got an arrangement to meet with them in Moscow in 49:09.815 --> 49:12.315 March--that would be during the semester. 49:12.320 --> 49:16.740 I'll tell you what reaction I hope I get from the Russians. 49:16.739 --> 49:22.209 Oil is very important to the Russian economy and are they 49:22.208 --> 49:24.648 managing the risk well? 49:24.650 --> 49:28.470 I bet not. I'm going to do a diagram like 49:28.474 --> 49:31.604 this for Russia. The countries that really 49:31.603 --> 49:36.323 matter are the ones that--the Arabian--are the Persian Gulf 49:36.322 --> 49:39.092 countries; I was just talking to people at 49:39.089 --> 49:41.129 the World Economic Forum about that. 49:41.130 --> 49:44.550 Some of those countries are really reliant on oil, 49:44.550 --> 49:48.810 so they really have to do--they really should do a calculation 49:48.807 --> 49:51.387 of efficient portfolio frontiers. 49:51.389 --> 49:54.549 Well, one of the lessons of this course is we have a 49:54.554 --> 49:57.354 wonderful theory, but we don't manage it well, 49:57.347 --> 49:58.857 mostly. And I don't mean to be 49:58.860 --> 50:00.030 criticizing foreign countries. 50:00.030 --> 50:03.570 The same criticism applies to the United States as well. 50:03.570 --> 50:05.030 We're in a different position. 50:05.030 --> 50:08.070 Where are we on this frontier regarding oil in the United 50:08.071 --> 50:10.311 States? Well, we don't have much oil 50:10.311 --> 50:12.751 relative to the size of our portfolio. 50:12.750 --> 50:15.110 I don't know what percent; our oil reserves in the U.S. 50:15.110 --> 50:18.450 are pretty small, so we're kind of lying 50:18.450 --> 50:22.220 somewhere inside--maybe on this pink line. 50:22.220 --> 50:24.550 So, people in the U.S. 50:24.550 --> 50:26.860 don't have the optimal portfolio either. 50:26.860 --> 50:32.260 I set up a theoretical framework here and I wanted to 50:32.257 --> 50:38.067 give you--I mentioned oil because it seems to make it, 50:38.070 --> 50:41.680 to me, so clear what we're talking about here. 50:41.679 --> 50:47.579 It's talking about not getting tied up in risks and so--I was 50:47.578 --> 50:53.478 talking to someone at the World Economic Forum from a Persian 50:53.477 --> 50:59.817 Gulf country and I said, aren't you worried about 50:59.818 --> 51:04.178 reliance on oil? He said, of course we're 51:04.182 --> 51:07.902 worried on reliance, so much of our GDP and of our 51:07.904 --> 51:10.794 government revenues is oil-related. 51:10.789 --> 51:13.709 We've seen the price of oil, lately, move all over the map. 51:13.710 --> 51:17.330 It went up to $100 recently and it was just as late as late 51:17.332 --> 51:21.022 1990s that it was under $20 and people just don't know where 51:21.018 --> 51:22.328 it's going to go. 51:22.329 --> 51:27.149 I think that these countries are somewhat trying to manage 51:27.150 --> 51:31.970 this oil risk but they can't yet get onto the frontier. 51:31.969 --> 51:40.789 That's a sign to me that we're not there yet and we have a lot 51:40.789 --> 51:44.259 more to do in finance. 51:44.260 --> 51:50.860 There's one more equation that I wanted to write down and I'm 51:50.863 --> 51:57.473 going to not--I'm not going to spend a lot of time explaining 51:57.467 --> 52:02.197 this because it's going to take awhile. 52:02.199 --> 52:07.299 This is the equation that relates the expected return on 52:07.300 --> 52:11.390 an asset. It's so called capital asset 52:11.393 --> 52:16.943 pricing model. In the capital asset pricing 52:16.937 --> 52:25.427 model, in finance--this is the most famous model in finance. 52:25.430 --> 52:38.250 52:38.250 --> 52:42.130 It's abbreviated as the CAPM and I'm not going to do it 52:42.125 --> 52:44.995 justice here, I'm sorry, but there are so 52:44.996 --> 52:47.146 many ins and outs of this. 52:47.150 --> 52:51.630 You should really take ECON 251 to learn this more. 52:51.630 --> 52:59.190 It was--the model divided by James Tobin here at Yale, 52:59.186 --> 53:06.596 was the original--who got the original precursors, 53:06.599 --> 53:11.059 but it was more invented by William Sharpe, 53:11.061 --> 53:14.461 John Lintner, Harry Markowitz. 53:14.460 --> 53:21.260 53:21.260 --> 53:23.870 Everyone of these, except Lintner I think, 53:23.869 --> 53:25.269 won the Nobel Prize. 53:25.269 --> 53:29.209 I think he died too young--that's one of the 53:29.210 --> 53:31.410 misfortunes of living. 53:31.410 --> 53:34.770 Did I say that right? 53:34.769 --> 53:38.479 One of the misfortunes of scholarship, you have to live 53:38.478 --> 53:40.948 long enough to get your accolades. 53:40.949 --> 53:50.079 The asset pricing model--and this is critical--assumes 53:50.081 --> 53:59.731 everyone is rational and holds the tangency portfolio. 53:59.730 --> 54:11.080 54:11.079 --> 54:15.069 That is a wild assumption, but it's fun to make, 54:15.069 --> 54:19.819 because I know pretty well--I know very well--that people 54:19.822 --> 54:21.692 aren't doing this. 54:21.690 --> 54:24.790 They have lots of maybe good--maybe it's not because 54:24.785 --> 54:27.755 they're irrational, it's that they're political or 54:27.758 --> 54:31.398 they're constrained by tradition or laws or regulations, 54:31.400 --> 54:34.390 all sorts of things; but, they're not holding the 54:34.390 --> 54:35.670 tangency portfolio. 54:35.670 --> 54:40.380 It's a beautiful theory to assume, to see what would happen 54:40.382 --> 54:43.882 if they did. That would mean that everybody 54:43.884 --> 54:48.544 is holding that same portfolio of risky assets and nobody is 54:48.535 --> 54:51.385 different, they're only different in 54:51.388 --> 54:55.338 how--what proportions they hold the risky--the tangency 54:55.339 --> 55:00.379 portfolio. It implies that the tangency 55:00.384 --> 55:08.874 portfolio has to equal the actual market portfolio and that 55:08.866 --> 55:17.636 means then--it's a very simple implication of the theory. 55:17.639 --> 55:20.949 In my diagram, I said that the tangency 55:20.950 --> 55:26.180 portfolio--I estimated that the tangency portfolio is 9% oil, 55:26.176 --> 55:28.786 27% stocks, and 64% bonds. 55:28.789 --> 55:32.169 If we're all doing that, then that has to be what the 55:32.173 --> 55:35.133 outstanding is. If we're all holding the same 55:35.133 --> 55:38.903 portfolio, that has to be the total, so that would mean that 55:38.904 --> 55:44.494 9% of all wealth is oil, 9% of all--is oil--27% of all 55:44.493 --> 55:49.293 wealth is stocks, and 64% is bonds. 55:49.289 --> 55:53.579 If you accept my estimates and you accept the capital asset 55:53.583 --> 55:56.843 pricing model, that would have to be true. 55:56.840 --> 56:00.310 Again, I don't want to make too much of my estimates because 56:00.307 --> 56:03.947 different people would estimate these things in different ways, 56:03.951 --> 56:05.481 but that's the theory. 56:05.480 --> 56:09.750 The theory says that the tangency portfolio equals the 56:09.746 --> 56:14.816 optimal portfolio and that gives us the famous equation--I'm not 56:14.817 --> 56:18.937 going to derive this, but there is the most famous 56:18.937 --> 56:22.397 equation in finance--says--can you read that? 56:22.400 --> 56:26.860 That r_i, the expected return on the 56:26.859 --> 56:30.809 ith asset, is equal to the riskless rate 56:30.805 --> 56:35.685 plus something called the beta for the ith asset times the 56:35.693 --> 56:40.843 expected return on the market minus the riskless rate. 56:40.840 --> 56:46.010 56:46.010 --> 56:49.530 Again, I'm not going to spend much time on this, 56:49.530 --> 56:52.830 but the β of the ith asset is the 56:52.826 --> 56:57.016 regression coefficient when you regress the return on the 56:57.020 --> 57:01.590 ith asset on the return of the market portfolio. 57:01.590 --> 57:12.380 r_m is the expected return on the market 57:12.383 --> 57:22.023 portfolio, which is the portfolio of all assets. 57:22.019 --> 57:25.429 The market portfolio is, if you took all the stocks and 57:25.425 --> 57:27.565 bonds, and oil, and real estate, 57:27.570 --> 57:31.090 anything that's available to invest in, in the whole world, 57:31.091 --> 57:33.581 put them all together in one portfolio. 57:33.579 --> 57:38.019 It's the world portfolio, it's everything and we compute 57:38.019 --> 57:41.569 the expected return on that portfolio, that's 57:41.571 --> 57:43.591 r_m. 57:43.590 --> 57:48.510 Also, we need to know how much individual stocks are correlated 57:48.508 --> 57:51.758 with r_m; we measure that by the 57:51.763 --> 57:52.893 regression coefficient. 57:52.889 --> 57:58.079 The β of a stock is how much it reacts to movements in 57:58.081 --> 58:00.141 the market portfolio. 58:00.139 --> 58:03.819 If β = 1, it means that if the market 58:03.815 --> 58:09.365 portfolio goes up 10% in value then this asset also goes up 10% 58:09.374 --> 58:12.214 in value. If β is two, 58:12.213 --> 58:16.943 it means that if the market goes up 10% in value, 58:16.939 --> 58:21.959 the stock tends to go up 20% in value and so on. 58:21.960 --> 58:27.680 These are the basic theoretical structures that you incidentally 58:27.680 --> 58:32.220 need for the problem set, the first problem set. 58:32.219 --> 58:35.759 The first problem set--I guess you've--you can turn in your 58:35.764 --> 58:39.254 first problem set here before you leave because it was due 58:39.248 --> 58:42.058 today. The second problem set is about 58:42.058 --> 58:44.708 this model. I realize I've given you some 58:44.712 --> 58:47.822 difficult mathematics, but--it's not that difficult 58:47.820 --> 58:50.990 actually--but I kind of went through it quickly. 58:50.989 --> 58:57.819 So, we are setting up our--you have already gotten email and 58:57.822 --> 59:04.892 you've talked about setting up your review sessions--so I have 59:04.887 --> 59:08.127 a few more minutes here. 59:08.130 --> 59:13.350 What I want to do, I want to talk about Jeremy 59:13.347 --> 59:18.677 Siegel's book and the equity premium puzzle. 59:18.679 --> 59:23.989 Underlying this analysis, we have estimates of the 59:23.986 --> 59:30.046 expected returns on assets, notably, the expected returns 59:30.050 --> 59:32.650 on stocks and bonds. 59:32.650 --> 59:45.320 59:45.320 --> 59:48.090 Jeremy Siegel, in his book, 59:48.094 --> 59:53.964 which is assigned for this course, is really emphasizing 59:53.962 --> 59:57.912 this capital asset pricing model, 59:57.909 --> 1:00:00.659 emphasizing the kind of efficient portfolio frontier 1:00:00.655 --> 1:00:02.265 calculations that I've done. 1:00:02.269 --> 1:00:06.879 What Siegel emphasized--the book is really about this--he 1:00:06.883 --> 1:00:12.163 talks about what is the expected return of stocks and what is the 1:00:12.156 --> 1:00:15.366 expected return of bonds and so on. 1:00:15.369 --> 1:00:21.509 We're going to call asset one stocks in the U.S. 1:00:21.510 --> 1:00:25.390 and we're going to call asset two bonds. 1:00:25.389 --> 1:00:28.099 He estimates, for his purposes, 1:00:28.099 --> 1:00:33.069 and he shows you calculations of the efficient portfolio 1:00:33.066 --> 1:00:36.226 frontier. I want to just talk a little 1:00:36.226 --> 1:00:37.926 bit about his estimate. 1:00:37.929 --> 1:00:46.209 He has data for a very long time period, 1802 to 2006, 1:00:46.212 --> 1:00:48.402 for the U.S. 1:00:48.400 --> 1:01:00.280 1:01:00.280 --> 1:01:10.140 Over that very long time interval, the expected return 1:01:10.140 --> 1:01:19.820 that we got for stocks was 6.8% a year in real terms, 1:01:19.815 --> 1:01:26.695 this is real inflation corrected; 1:01:26.699 --> 1:01:33.989 whereas, for bonds, it was, over this whole time 1:01:33.994 --> 1:01:38.344 period, 2.8% a year, real. 1:01:38.340 --> 1:01:40.000 Then he also computes σ_1 and 1:01:39.996 --> 1:01:42.036 σ_2 and σ_12 but I'm not 1:01:42.037 --> 1:01:43.537 going to talk about that right now. 1:01:43.539 --> 1:01:47.859 Then he computes the efficient portfolio frontier. 1:01:47.860 --> 1:01:50.820 Now, he's using a much longer sample than I did, 1:01:50.817 --> 1:01:54.717 so he's not going to get this tangency portfolio that I did. 1:01:54.719 --> 1:02:00.119 The thing that is very interesting that he finds is the 1:02:00.116 --> 1:02:04.306 difference, 4%, between the historical real 1:02:04.313 --> 1:02:10.413 return on stocks and the historical real return on bonds. 1:02:10.409 --> 1:02:12.359 This is called the equity premium. 1:02:12.360 --> 1:02:18.520 1:02:18.519 --> 1:02:21.519 Actually, I should take that back, this is really 1:02:21.521 --> 1:02:24.211 r_f--he shows all three. 1:02:24.210 --> 1:02:27.520 r_f that I'm reporting is the riskless rate. 1:02:27.519 --> 1:02:31.919 There's also--if you look on Table One in Chapter One, 1:02:31.915 --> 1:02:35.145 it shows r_1 stocks, 1:02:35.150 --> 1:02:40.120 r_2 long-term bonds, and these are short-term 1:02:40.121 --> 1:02:42.241 that I'm recording here. 1:02:42.239 --> 1:02:49.269 The equity premium is the--this short-term 2.8% is the riskless 1:02:49.268 --> 1:02:54.598 return, historically, for a period of almost 200 1:02:54.596 --> 1:02:58.146 years. This is the return on stocks 1:02:58.150 --> 1:03:01.090 for a period of almost 200 years. 1:03:01.090 --> 1:03:07.850 Stocks have paid 4% more a year over this incredibly long time 1:03:07.853 --> 1:03:11.293 period than short-term bonds. 1:03:11.289 --> 1:03:14.759 Also, they've paid much more than long-term--well it's not so 1:03:14.764 --> 1:03:17.284 different. I don't have the data in front 1:03:17.281 --> 1:03:20.141 of me, so let's emphasize the difference between 1:03:20.135 --> 1:03:23.105 r_1 and r_f. 1:03:23.110 --> 1:03:27.980 It surprises many people that stocks have paid such an 1:03:27.979 --> 1:03:32.939 enormous premium over the return on short-term debt. 1:03:32.940 --> 1:03:41.230 The theme of Siegel's book is, can we believe this? 1:03:41.230 --> 1:03:44.740 I mean, do you really--you might wonder, 1:03:44.742 --> 1:03:50.332 aren't people missing something if the excess return is so high 1:03:50.327 --> 1:03:53.567 of stocks over short-term bonds? 1:03:53.570 --> 1:03:58.320 Why would anyone not just hold a lot of--a really large number 1:03:58.316 --> 1:04:00.936 of stocks? That's the theme of his book 1:04:00.940 --> 1:04:04.080 and his book is at--his conclusion is that he largely 1:04:04.079 --> 1:04:08.929 believes that this is true, that the returns that we have 1:04:08.928 --> 1:04:13.138 seen in the U.S. in the stock market have 1:04:13.137 --> 1:04:19.407 exceeded those of other assets by quite a substantial margin. 1:04:19.409 --> 1:04:22.929 That means--his calculations are very different than the ones 1:04:22.929 --> 1:04:25.979 I have because he has a much longer sample period. 1:04:25.980 --> 1:04:28.730 I was using only from 1983 to the present. 1:04:28.730 --> 1:04:33.240 For him, the optimal portfolio should be very heavily into 1:04:33.238 --> 1:04:35.548 stocks. Now, this is a controversial 1:04:35.550 --> 1:04:38.980 view, but that is the view that he advances in the book and I 1:04:38.979 --> 1:04:41.379 think it's a very interesting analysis. 1:04:41.380 --> 1:04:46.900 So, what it means then is that the optimal portfolio should not 1:04:46.902 --> 1:04:51.622 be the one I show there, but should be one that's very 1:04:51.622 --> 1:04:53.762 heavily into stocks. 1:04:53.760 --> 1:04:59.990 This is--I'm showing here U.S. 1:04:59.989 --> 1:05:07.279 data, but Siegel also argues in the latest edition that the 1:05:07.275 --> 1:05:14.555 equity premium is also high for advanced countries over the 1:05:14.560 --> 1:05:18.410 whole world. If you look at his book, 1:05:18.406 --> 1:05:21.846 again in the first chapter, he gives a list of 1:05:21.850 --> 1:05:26.290 countries--it's based on an analysis that some others--that 1:05:26.289 --> 1:05:31.329 Professors Dimson, Marsh, and Stanton used in 1:05:31.333 --> 1:05:37.053 their 2002 book. They show the expected return 1:05:37.050 --> 1:05:43.450 on stocks versus bonds or short-term debt for a whole 1:05:43.451 --> 1:05:46.161 range of countries. 1:05:46.159 --> 1:05:49.739 In every one of these countries, since 1901, 1:05:49.743 --> 1:05:53.413 there has been a very high equity premium. 1:05:53.409 --> 1:05:56.619 I'll let you read Siegel and his discussion of this 1:05:56.617 --> 1:06:00.147 possibility, but I think that--what I want to get now is 1:06:00.145 --> 1:06:04.245 not necessarily any agreement on whether you believe that there's 1:06:04.251 --> 1:06:07.011 such a high excess return for stocks. 1:06:07.010 --> 1:06:10.020 but I just want you to understand the basic framework. 1:06:10.019 --> 1:06:15.329 The first problem set asked you to manipulate the model that I 1:06:15.328 --> 1:06:20.458 just presented--the model of how you form portfolios and the 1:06:20.463 --> 1:06:24.383 model of the capital asset pricing model. 1:06:24.380 --> 1:06:27.410 I have one final question, just about the mutual fund 1:06:27.409 --> 1:06:30.089 industry. The mutual fund industry is 1:06:30.087 --> 1:06:34.387 supposedly, and according to theory, doing this for you. 1:06:34.389 --> 1:06:37.389 The ideal thing would be that the mutual fund does these 1:06:37.394 --> 1:06:40.184 calculations and it puts it all together for you. 1:06:40.179 --> 1:06:43.649 At least in some approximate sense that's what they are 1:06:43.646 --> 1:06:45.806 doing. I asked you in the final 1:06:45.814 --> 1:06:50.234 question to just get on a couple of websites, The Federal Reserve 1:06:50.226 --> 1:06:53.706 and the ICI, which is a mutual industry 1:06:53.712 --> 1:06:59.562 website, and write a little bit, a couple paragraphs about what 1:06:59.559 --> 1:07:05.029 this industry is really achieving or how it's trending. 1:07:05.030 --> 1:07:08.840 Next Wednesday–well, there are three lectures this 1:07:08.837 --> 1:07:12.987 week, Wednesday will be about the insurance industry and I'll 1:07:12.992 --> 1:07:15.002 see you in two days then.