WEBVTT 00:00:00.090 --> 00:00:02.430 The following content is provided under a Creative 00:00:02.430 --> 00:00:03.820 Commons license. 00:00:03.820 --> 00:00:06.030 Your support will help MIT OpenCourseWare 00:00:06.030 --> 00:00:10.120 continue to offer high quality educational resources for free. 00:00:10.120 --> 00:00:12.690 To make a donation, or to view additional materials 00:00:12.690 --> 00:00:16.620 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.620 --> 00:00:17.830 at ocw.mit.edu. 00:00:21.522 --> 00:00:23.980 ANDREW LO: Well, let me pick up where we left off last time 00:00:23.980 --> 00:00:28.030 and give you just a very quick overview of where we're at now, 00:00:28.030 --> 00:00:32.350 because we're on the brink of a very important set of results 00:00:32.350 --> 00:00:36.290 that I think will change your perspective permanently 00:00:36.290 --> 00:00:38.410 on risk and expected return. 00:00:38.410 --> 00:00:41.590 Last time, remember, we looked at this trade-off 00:00:41.590 --> 00:00:45.430 between expected return and volatility. 00:00:45.430 --> 00:00:47.440 And we made the argument that when 00:00:47.440 --> 00:00:50.290 you combined a bunch of different securities 00:00:50.290 --> 00:00:53.200 that are not all perfectly correlated, 00:00:53.200 --> 00:00:56.890 what you get is this bullet-shaped curve in terms 00:00:56.890 --> 00:01:01.900 of the possible trade-offs between that expected return 00:01:01.900 --> 00:01:05.590 and riskiness of various different portfolios. 00:01:05.590 --> 00:01:10.570 So every single dot on this bullet-shaped curve 00:01:10.570 --> 00:01:15.160 corresponds to a specific portfolio, or weighting, 00:01:15.160 --> 00:01:18.100 or vector of portfolio weights, omega. 00:01:20.680 --> 00:01:24.520 So now what I want to ask you to do for the next lecture or two 00:01:24.520 --> 00:01:28.780 is to exhibit a little bit of a split personality 00:01:28.780 --> 00:01:30.910 kind of a perspective. 00:01:30.910 --> 00:01:33.860 I'm going to ask you to look at the geometry of risk 00:01:33.860 --> 00:01:36.400 and expected return, but at the same time, 00:01:36.400 --> 00:01:39.010 in the back of your brain, I want 00:01:39.010 --> 00:01:42.100 you to keep in mind the analytics 00:01:42.100 --> 00:01:44.650 of that set of geometries. 00:01:44.650 --> 00:01:46.960 In other words, I want you to keep in mind how we 00:01:46.960 --> 00:01:50.080 got this bullet-shaped curve. 00:01:50.080 --> 00:01:51.850 The way we got it was from taking 00:01:51.850 --> 00:01:55.720 different weighted averages of the securities 00:01:55.720 --> 00:01:59.320 that we have access to as investments. 00:01:59.320 --> 00:02:01.810 So every one of these points on the bullet 00:02:01.810 --> 00:02:03.970 corresponds to a specific weighting. 00:02:03.970 --> 00:02:05.680 As you change those weightings, you 00:02:05.680 --> 00:02:08.740 change the risk and return characteristics 00:02:08.740 --> 00:02:10.160 of your portfolio. 00:02:10.160 --> 00:02:13.150 So the example that I gave after showing you this 00:02:13.150 --> 00:02:17.440 curve where I argued that the upper branch of this bullet 00:02:17.440 --> 00:02:21.160 is where any rational person would want to be. 00:02:21.160 --> 00:02:25.030 And by rational, I've defined that as somebody 00:02:25.030 --> 00:02:30.100 who prefers more expected return to less, and somebody 00:02:30.100 --> 00:02:35.930 who prefers less risk to more, other things equal. 00:02:35.930 --> 00:02:38.380 So if you've got those kind of preferences, 00:02:38.380 --> 00:02:40.600 then you want to be in the Northeast. 00:02:40.600 --> 00:02:44.290 You want to be as north, sorry, Northwest as possible. 00:02:44.290 --> 00:02:48.160 And you would never want to be down in this lower branch when 00:02:48.160 --> 00:02:50.500 you could be in the upper branch because you'd 00:02:50.500 --> 00:02:55.570 have a higher expected return for the same level of risk. 00:02:55.570 --> 00:02:59.590 So after we developed this basic idea, 00:02:59.590 --> 00:03:01.750 I gave you this numerical example 00:03:01.750 --> 00:03:04.150 where you've got three stocks in your universe. 00:03:04.150 --> 00:03:07.240 General Motors, IBM, and Motorola. 00:03:07.240 --> 00:03:10.330 And these are the parameters that we've estimated 00:03:10.330 --> 00:03:11.500 using historical data. 00:03:11.500 --> 00:03:13.150 Now there's going to be a question, 00:03:13.150 --> 00:03:16.300 and we've already raised that question, of how stable 00:03:16.300 --> 00:03:17.200 are these parameters. 00:03:17.200 --> 00:03:19.533 Are they really parameters, or do they change over time. 00:03:19.533 --> 00:03:22.610 And I told you, in reality of course, they change over time. 00:03:22.610 --> 00:03:26.170 But for now, let's play the game and assume 00:03:26.170 --> 00:03:27.790 that they are constant over time, 00:03:27.790 --> 00:03:30.320 and see what we can do with those parameters. 00:03:30.320 --> 00:03:33.820 So with the means, the standard deviations, 00:03:33.820 --> 00:03:38.090 and most importantly, the covariance matrix-- 00:03:38.090 --> 00:03:41.290 So this is the matrix of variances and covariances-- 00:03:41.290 --> 00:03:45.010 With these data as inputs, we can now 00:03:45.010 --> 00:03:48.190 construct that bullet-shaped curve. 00:03:48.190 --> 00:03:51.310 The way we do it is of course, to recognize 00:03:51.310 --> 00:03:53.830 that the expected return of the portfolio 00:03:53.830 --> 00:03:56.830 is just a weighted average of the expected returns 00:03:56.830 --> 00:04:00.820 of the component securities, where the weights are 00:04:00.820 --> 00:04:02.180 our choice variables. 00:04:02.180 --> 00:04:04.060 That's what we are getting to pick, 00:04:04.060 --> 00:04:08.560 is how we allocate the 100% of our wealth 00:04:08.560 --> 00:04:11.860 to these three different securities. 00:04:11.860 --> 00:04:14.410 And the variance, of course, is going 00:04:14.410 --> 00:04:17.680 to be given by a somewhat more complicated expression where 00:04:17.680 --> 00:04:21.519 you have the individual security variances entering here 00:04:21.519 --> 00:04:23.230 from the diagonals. 00:04:23.230 --> 00:04:27.320 But you also have the off diagonal terms 00:04:27.320 --> 00:04:30.640 entering in that same equation for that variance 00:04:30.640 --> 00:04:32.320 of the portfolio. 00:04:32.320 --> 00:04:35.210 And when we put these two equations together, 00:04:35.210 --> 00:04:37.819 the mean and the variance, and we take the square root 00:04:37.819 --> 00:04:39.610 the variance to get the standard deviation, 00:04:39.610 --> 00:04:44.060 and we plot it on a graph, we get this. 00:04:44.060 --> 00:04:46.790 This is the curve, the bullet-shaped curve, 00:04:46.790 --> 00:04:49.430 that we generate just from three securities, 00:04:49.430 --> 00:04:51.970 and from their covariances. 00:04:51.970 --> 00:04:54.650 And where we left off last time is 00:04:54.650 --> 00:04:56.630 that I pointed out a couple of things that was 00:04:56.630 --> 00:04:58.370 interesting about this curve. 00:04:58.370 --> 00:05:03.710 One is that unlike the two asset example, where when you start 00:05:03.710 --> 00:05:05.990 with two assets, the curve, the bullet 00:05:05.990 --> 00:05:08.600 goes through the two assets. 00:05:08.600 --> 00:05:12.380 In this case, with three or more assets, 00:05:12.380 --> 00:05:16.430 it's going to turn out that the bullet is actually 00:05:16.430 --> 00:05:20.630 going to include these assets as special cases, 00:05:20.630 --> 00:05:22.950 but they won't be on the curve. 00:05:22.950 --> 00:05:25.430 In other words, what this curve suggests 00:05:25.430 --> 00:05:28.460 is that any rational person is going to want 00:05:28.460 --> 00:05:31.490 to be on this upper branch. 00:05:31.490 --> 00:05:35.120 What that means is that it never makes sense 00:05:35.120 --> 00:05:40.430 to put all your money in one single security. 00:05:40.430 --> 00:05:41.630 You see that? 00:05:41.630 --> 00:05:45.620 In other words, if we agree that any rational investor is going 00:05:45.620 --> 00:05:49.730 to want to be on that efficient frontier, that upper branch, 00:05:49.730 --> 00:05:52.340 why would you ever want to be off of that branch? 00:05:52.340 --> 00:05:54.740 You'd like to be Northwest of that, but you can't. 00:05:54.740 --> 00:05:56.900 You'd never want to be below that branch, 00:05:56.900 --> 00:06:00.050 or to the right of that branch because you could do better 00:06:00.050 --> 00:06:02.310 by being on that branch. 00:06:02.310 --> 00:06:05.900 So what this suggests is that we never 00:06:05.900 --> 00:06:08.510 are going to want to hold 100% of IBM, 00:06:08.510 --> 00:06:12.470 or 100% of General Motors, or 100% of Motorola. 00:06:12.470 --> 00:06:14.360 If we did, we'd be on those dots, 00:06:14.360 --> 00:06:18.080 and those dots would lie on that efficient frontier. 00:06:18.080 --> 00:06:21.200 But in fact, they don't. 00:06:21.200 --> 00:06:23.660 So right away, we have now departed 00:06:23.660 --> 00:06:26.930 from Warren Buffett's world of, I want to pick a few stocks 00:06:26.930 --> 00:06:28.845 and watch them very, very carefully. 00:06:28.845 --> 00:06:29.420 Yeah, Brian? 00:06:29.420 --> 00:06:31.711 AUDIENCE: Would you expand that to say that you'd never 00:06:31.711 --> 00:06:35.690 want to invest in less than three stocks at a given time? 00:06:35.690 --> 00:06:37.500 ANDREW LO: That's not necessarily true. 00:06:37.500 --> 00:06:41.120 There are points on this line where-- 00:06:41.120 --> 00:06:43.890 and they may be pathological, so in other words, 00:06:43.890 --> 00:06:44.960 they may be very rare-- 00:06:44.960 --> 00:06:46.670 but there may be points on the line 00:06:46.670 --> 00:06:50.070 where you are holding two stocks, but not the third. 00:06:50.070 --> 00:06:51.850 So you've got to be careful about that. 00:06:51.850 --> 00:06:53.720 But those are exceptions. 00:06:53.720 --> 00:06:57.170 As a generic statement, you're absolutely right. 00:06:57.170 --> 00:07:00.140 The typical portfolio is going to have some of all three 00:07:00.140 --> 00:07:00.800 of them. 00:07:00.800 --> 00:07:03.080 And if you had four stocks, the typical portfolio 00:07:03.080 --> 00:07:04.982 would have some of all four. 00:07:04.982 --> 00:07:05.732 Yeah, [INAUDIBLE]. 00:07:05.732 --> 00:07:07.190 AUDIENCE: You answered my question, 00:07:07.190 --> 00:07:09.370 which is if you take one more stock, 00:07:09.370 --> 00:07:13.825 you'll always have your package [INAUDIBLE] n stocks include 00:07:13.825 --> 00:07:17.785 all the [INAUDIBLE], all the n stocks so, at the limit 00:07:17.785 --> 00:07:20.270 you should have an infinite number of stocks [INAUDIBLE] 00:07:20.270 --> 00:07:22.639 ANDREW LO: Well, let me put it another way that may 00:07:22.639 --> 00:07:23.930 be a little bit more intuitive. 00:07:23.930 --> 00:07:27.830 What this diagram suggests-- you guys are already groping 00:07:27.830 --> 00:07:28.350 towards-- 00:07:28.350 --> 00:07:32.870 is the insight that the more, the merrier. 00:07:32.870 --> 00:07:37.880 As you add more stocks, you cannot make this investor worse 00:07:37.880 --> 00:07:39.830 off. 00:07:39.830 --> 00:07:42.380 So in other words, I've now shown you an example with three 00:07:42.380 --> 00:07:43.520 stocks, we used to do two. 00:07:46.820 --> 00:07:49.720 Is it possible that by giving you an extra stock 00:07:49.720 --> 00:07:53.365 to invest in, I've made you worse off? 00:07:53.365 --> 00:07:53.865 Yeah? 00:07:53.865 --> 00:07:54.420 AUDIENCE: No 00:07:54.420 --> 00:07:54.690 ANDREW LO: Why 00:07:54.690 --> 00:07:57.023 AUDIENCE: Because you can just not invest in that stock. 00:07:57.023 --> 00:07:57.930 ANDREW LO: Exactly. 00:07:57.930 --> 00:08:01.170 I can never make you worse off in a world 00:08:01.170 --> 00:08:03.420 where you're free to choose, that is. 00:08:03.420 --> 00:08:06.900 Because you always have the option of getting 00:08:06.900 --> 00:08:09.630 rid of the stock that you don't like. 00:08:09.630 --> 00:08:11.490 You can always put 0 on it. 00:08:11.490 --> 00:08:14.350 So to your point, [INAUDIBLE], as I add more stocks, 00:08:14.350 --> 00:08:16.620 first of all my risk-reward trade-off 00:08:16.620 --> 00:08:18.250 curve will get better. 00:08:18.250 --> 00:08:20.032 What does it mean to get better? 00:08:20.032 --> 00:08:21.990 What does it mean for the risk-reward trade-off 00:08:21.990 --> 00:08:22.680 to be better? 00:08:24.990 --> 00:08:25.490 Yes? 00:08:25.490 --> 00:08:29.205 AUDIENCE: It means you get a higher return 00:08:29.205 --> 00:08:30.330 for the same level of risk. 00:08:30.330 --> 00:08:30.870 ANDREW LO: That's right. 00:08:30.870 --> 00:08:32.661 A higher return for the same level of risk, 00:08:32.661 --> 00:08:34.770 or a lower risk for the same level of return. 00:08:34.770 --> 00:08:39.570 In other words, your upper branch actually 00:08:39.570 --> 00:08:41.970 moves to the Northwest. 00:08:41.970 --> 00:08:43.559 That's what it means to get better. 00:08:43.559 --> 00:08:47.610 As I add more stocks, this will move to the Northwest. 00:08:47.610 --> 00:08:50.640 And therefore, you have available 00:08:50.640 --> 00:08:54.090 all of the opportunities to the south and to the east, 00:08:54.090 --> 00:08:56.400 but you would never take those because you're 00:08:56.400 --> 00:08:58.110 rational in the sense that you always 00:08:58.110 --> 00:09:03.304 prefer less risk to more, and more return to less. 00:09:03.304 --> 00:09:03.804 Yeah? 00:09:03.804 --> 00:09:06.790 AUDIENCE: If we put all of the stocks on the index, 00:09:06.790 --> 00:09:07.880 on [INAUDIBLE]. 00:09:07.880 --> 00:09:11.618 And if we looked at all the possible combinations that-- 00:09:11.618 --> 00:09:13.326 we can look at them all at the same time, 00:09:13.326 --> 00:09:15.284 but then all the subsets that you can think of, 00:09:15.284 --> 00:09:19.290 then you must come up with some most efficient frontier 00:09:19.290 --> 00:09:20.959 in that market. 00:09:20.959 --> 00:09:23.000 ANDREW LO: Hold onto that thought for 10 minutes. 00:09:23.000 --> 00:09:24.375 We're going to come back to that. 00:09:24.375 --> 00:09:26.930 Let's do three first, and then we could do all of them. 00:09:26.930 --> 00:09:27.431 Yeah, Chris? 00:09:27.431 --> 00:09:29.763 AUDIENCE: Trying to better understand Buffett's strategy 00:09:29.763 --> 00:09:30.740 relative to this one. 00:09:30.740 --> 00:09:33.030 Is the correlation of these stocks due primarily 00:09:33.030 --> 00:09:35.370 to just psychological factors of the market, 00:09:35.370 --> 00:09:39.430 or is it due to intrinsic correlation? 00:09:39.430 --> 00:09:42.290 And then the follow on is when Buffet says invest in one stock 00:09:42.290 --> 00:09:44.623 and just watch it carefully, isn't that sort of assuming 00:09:44.623 --> 00:09:46.507 that the market will determine at some point 00:09:46.507 --> 00:09:49.069 that the stock is undervalued, and what was he-- 00:09:49.069 --> 00:09:50.860 ANDREW LO: So those are two good questions. 00:09:50.860 --> 00:09:52.360 Let me take each of them separately. 00:09:52.360 --> 00:09:54.270 Let's first talk about the correlation. 00:09:54.270 --> 00:09:55.440 Why is there correlation? 00:09:55.440 --> 00:09:57.270 We haven't really talked much about it, 00:09:57.270 --> 00:09:59.910 but it turns out that there are many different arguments 00:09:59.910 --> 00:10:02.370 for why there is correlation. 00:10:02.370 --> 00:10:06.630 Probably the most compelling is that a rising tide lifts 00:10:06.630 --> 00:10:08.610 all boats, and vice versa. 00:10:08.610 --> 00:10:10.800 In other words, when business conditions are good, 00:10:10.800 --> 00:10:12.630 then that helps all companies. 00:10:12.630 --> 00:10:15.210 Just like when business conditions are bad, 00:10:15.210 --> 00:10:16.600 it hurts all companies. 00:10:16.600 --> 00:10:19.110 So there's some macroeconomic type 00:10:19.110 --> 00:10:23.250 of commonality among businesses that create correlation. 00:10:23.250 --> 00:10:24.345 That's one reason. 00:10:24.345 --> 00:10:26.470 But the second reason is something you pointed out, 00:10:26.470 --> 00:10:29.095 which is quite apt, particularly over the last few weeks, which 00:10:29.095 --> 00:10:30.510 is the psychological factor. 00:10:30.510 --> 00:10:33.690 When the entire economy is under stress, 00:10:33.690 --> 00:10:35.924 and people are scared to death about what's 00:10:35.924 --> 00:10:37.590 going to happen to the market, what they 00:10:37.590 --> 00:10:41.670 will do is withdraw money in mass from equities 00:10:41.670 --> 00:10:45.570 and put them into safer assets like cash, or treasury 00:10:45.570 --> 00:10:48.540 bills, or money market funds, or whatever 00:10:48.540 --> 00:10:51.240 they can do to get to safety. 00:10:51.240 --> 00:10:53.760 So I would say that the answer is both. 00:10:53.760 --> 00:10:58.170 There are good economic reasons where correlations should exist 00:10:58.170 --> 00:11:00.060 among different companies, but there are also 00:11:00.060 --> 00:11:03.030 psychological or behavior reasons that exacerbate 00:11:03.030 --> 00:11:04.530 those kinds of commonalities. 00:11:04.530 --> 00:11:09.000 Now your second question about Buffett versus this approach. 00:11:09.000 --> 00:11:10.620 There's one fundamental difference 00:11:10.620 --> 00:11:13.710 between what Buffett would say about a company 00:11:13.710 --> 00:11:18.150 that he decides to buy versus how we're approaching it. 00:11:18.150 --> 00:11:20.040 The fundamental difference is that Buffett 00:11:20.040 --> 00:11:25.670 would say that he's been able to identify a severe mispricing. 00:11:25.670 --> 00:11:28.800 In other words, he would argue that markets are not 00:11:28.800 --> 00:11:29.880 in equilibrium. 00:11:29.880 --> 00:11:32.580 He would argue that Goldman Sachs is dramatically 00:11:32.580 --> 00:11:35.580 undervalued where it is today. 00:11:35.580 --> 00:11:38.790 And seven years from now, he may be right. 00:11:38.790 --> 00:11:40.860 And that's the kind of time frame he has in mind, 00:11:40.860 --> 00:11:43.470 if not longer. 00:11:43.470 --> 00:11:46.890 So far, I've made no such argument at all 00:11:46.890 --> 00:11:49.110 about deriving these analyses. 00:11:49.110 --> 00:11:51.990 I've not made any argument about whether prices are good or bad. 00:11:51.990 --> 00:11:55.500 In fact, I'm arguing, in a way, that these prices 00:11:55.500 --> 00:11:57.090 I'm taking as given. 00:11:57.090 --> 00:11:58.560 And the question is, what can I do 00:11:58.560 --> 00:12:00.570 to construct a good portfolio irrespective 00:12:00.570 --> 00:12:04.270 of whether markets are crazy or markets are rational. 00:12:04.270 --> 00:12:06.490 In a few minutes, I'm going to argue 00:12:06.490 --> 00:12:12.700 that when markets are rational and in equilibrium, then 00:12:12.700 --> 00:12:15.070 there is something that we can say 00:12:15.070 --> 00:12:17.710 about the relationship between risk and reward 00:12:17.710 --> 00:12:21.790 that's extraordinarily sharp and meaningful from the perspective 00:12:21.790 --> 00:12:23.530 of financial decision making. 00:12:23.530 --> 00:12:25.056 And then at the end of the course, 00:12:25.056 --> 00:12:26.680 I'm going to try to explain to you what 00:12:26.680 --> 00:12:30.037 the limitations of that set of assumptions are. 00:12:30.037 --> 00:12:31.767 Dennis? 00:12:31.767 --> 00:12:33.516 AUDIENCE: Just as we wouldn't put anything 00:12:33.516 --> 00:12:37.492 in bond's half of this frontier, does this graph 00:12:37.492 --> 00:12:39.977 imply that we strictly prefer IBM over GM? 00:12:39.977 --> 00:12:42.970 That we pretty much never weigh anything for GM? 00:12:42.970 --> 00:12:46.630 ANDREW LO: Well, from a risk-reward perspective, 00:12:46.630 --> 00:12:48.250 let's take a look. 00:12:48.250 --> 00:12:51.520 IBM has a higher expected rate of return, 00:12:51.520 --> 00:12:54.290 and it's got a higher level of risk. 00:12:54.290 --> 00:13:01.120 So you really can't say that you would never prefer GM over IBM, 00:13:01.120 --> 00:13:05.050 because GM has lower risk and lower expected return. 00:13:05.050 --> 00:13:09.209 If on the other hand, GM were over here, then 00:13:09.209 --> 00:13:10.000 you would be right. 00:13:10.000 --> 00:13:14.230 Because any point to the direct Northwest 00:13:14.230 --> 00:13:18.310 of a particular point on this curve is strictly preferred. 00:13:18.310 --> 00:13:22.450 And GM and IBM don't have that relationship. 00:13:22.450 --> 00:13:25.750 In other words, the way you can identify 00:13:25.750 --> 00:13:31.900 securities that are dominated in both dimensions is-- 00:13:31.900 --> 00:13:35.890 So this is your risk dimension, this is your expected return 00:13:35.890 --> 00:13:36.580 dimension. 00:13:36.580 --> 00:13:41.130 Pick a point in this space, and ask the question, 00:13:41.130 --> 00:13:43.240 what are the other portfolios that are strictly 00:13:43.240 --> 00:13:44.620 preferred to that point. 00:13:44.620 --> 00:13:46.420 Well the answer is pretty simple. 00:13:46.420 --> 00:13:49.720 Any portfolio that has higher expected rate of return 00:13:49.720 --> 00:13:52.690 for the same level of risk, so the vertical line. 00:13:52.690 --> 00:13:56.620 Any portfolio that has less risk for the same level 00:13:56.620 --> 00:14:01.030 of expected returns, So the western direction. 00:14:01.030 --> 00:14:10.050 And anything in this segment, that orthant, or quadrant, 00:14:10.050 --> 00:14:11.910 is strictly preferred. 00:14:11.910 --> 00:14:15.390 So in the case of IBM, if you draw 00:14:15.390 --> 00:14:19.530 the vertical and the horizontal and ask the question, 00:14:19.530 --> 00:14:21.330 does GM lie in that area? 00:14:21.330 --> 00:14:21.930 No. 00:14:21.930 --> 00:14:24.390 If you do GM, and you draw the vertical and then 00:14:24.390 --> 00:14:26.280 the horizontal and asked does IBM 00:14:26.280 --> 00:14:29.100 lie in that strictly preferred quadrant? 00:14:29.100 --> 00:14:30.820 The answer is no. 00:14:30.820 --> 00:14:33.780 So the answer to your question about IBM versus GM, 00:14:33.780 --> 00:14:36.300 no, there isn't any strict relationship 00:14:36.300 --> 00:14:38.970 that would say one would always dominate the other. 00:14:38.970 --> 00:14:43.080 But if GM were here, then IBM is clearly 00:14:43.080 --> 00:14:45.850 contained in that preferred quadrant. 00:14:45.850 --> 00:14:49.210 So then the answer to your question would be yes. 00:14:49.210 --> 00:14:50.670 Yeah, Justin? 00:14:50.670 --> 00:14:53.070 AUDIENCE: Theoretically then, wouldn't everyone just 00:14:53.070 --> 00:14:56.270 buy IBM, sell GM, then wouldn't there 00:14:56.270 --> 00:14:58.830 be some sort of equilibrium where then GM-- 00:14:58.830 --> 00:15:00.840 ANDREW LO: So the answer is, it depends 00:15:00.840 --> 00:15:02.830 on other things going on. 00:15:02.830 --> 00:15:04.595 Everyone would not do that. 00:15:04.595 --> 00:15:05.970 Everyone would do something else, 00:15:05.970 --> 00:15:07.360 and I'm about to tell you. 00:15:07.360 --> 00:15:09.180 So I'm about to give you the tools 00:15:09.180 --> 00:15:11.820 to make that exact conclusion, and the reason 00:15:11.820 --> 00:15:14.670 is that when I show you what people will do, 00:15:14.670 --> 00:15:16.410 that's going to far dominate what 00:15:16.410 --> 00:15:17.960 you think people want to do. 00:15:17.960 --> 00:15:19.390 Just with pairs. 00:15:19.390 --> 00:15:21.100 So instead of doing it with pairs, 00:15:21.100 --> 00:15:22.980 let's do it with all the securities, 00:15:22.980 --> 00:15:24.570 as Zeke wanted to do. 00:15:24.570 --> 00:15:26.320 We're going to do that in just a minute. 00:15:26.320 --> 00:15:28.195 But I want to make sure everybody understands 00:15:28.195 --> 00:15:29.910 this basic framework first, because we're 00:15:29.910 --> 00:15:33.540 going to now start making this a little bit more complex. 00:15:33.540 --> 00:15:37.290 Where we left off at the very last moment of Wednesday's 00:15:37.290 --> 00:15:41.219 lecture was I showed you this diagram 00:15:41.219 --> 00:15:43.260 with the tangency portfolio, but we hadn't really 00:15:43.260 --> 00:15:44.790 gotten to talking about it. 00:15:44.790 --> 00:15:48.510 Remember the case where we had only one risky asset and one 00:15:48.510 --> 00:15:50.710 riskless asset, treasury bills? 00:15:50.710 --> 00:15:54.420 And in that case, when you are combining a portfolio 00:15:54.420 --> 00:15:57.450 with one risky asset and one riskless, 00:15:57.450 --> 00:15:58.680 you've got a straight line. 00:15:58.680 --> 00:16:01.300 It turns out that that is much more general. 00:16:01.300 --> 00:16:05.460 You get a straight line anytime you combine a riskless asset 00:16:05.460 --> 00:16:07.880 with any number of risky assets. 00:16:07.880 --> 00:16:09.820 So let me give you an example. 00:16:09.820 --> 00:16:12.720 Suppose we picked an arbitrary portfolio which 00:16:12.720 --> 00:16:14.990 is this red dot, p. 00:16:14.990 --> 00:16:17.180 And I wanted you to tell me what is 00:16:17.180 --> 00:16:19.400 the risk-reward possibilities that you 00:16:19.400 --> 00:16:25.204 could achieve by mixing p with treasury bills. 00:16:25.204 --> 00:16:26.870 Well, you get that straight line, right? 00:16:26.870 --> 00:16:29.450 We derived that last time. 00:16:29.450 --> 00:16:33.680 So any point along the straight line is what you could achieve, 00:16:33.680 --> 00:16:34.820 right? 00:16:34.820 --> 00:16:37.010 Anybody tell me where the portfolio 00:16:37.010 --> 00:16:42.890 would be that invests 100% of your assets in T-Bills? 00:16:42.890 --> 00:16:44.300 Where is that on this graph? 00:16:44.300 --> 00:16:45.650 AUDIENCE: [INAUDIBLE] 00:16:45.650 --> 00:16:47.191 ANDREW LO: Right this dot right here. 00:16:47.191 --> 00:16:49.600 How about 100% in portfolio p? 00:16:49.600 --> 00:16:50.930 Right, the red dot over there. 00:16:50.930 --> 00:16:57.110 How about 25% in T-Bills, 75% in p? 00:16:57.110 --> 00:16:59.189 Where would that lie? 00:16:59.189 --> 00:17:00.730 AUDIENCE: [INAUDIBLE] along the line. 00:17:00.730 --> 00:17:02.771 ANDREW LO: It would be along the line, but where? 00:17:02.771 --> 00:17:04.770 Here? 00:17:04.770 --> 00:17:07.386 25% T-Bills, 75%-- 00:17:07.386 --> 00:17:08.260 AUDIENCE: [INAUDIBLE] 00:17:08.260 --> 00:17:09.609 ANDREW LO: Right, exactly. 00:17:09.609 --> 00:17:13.240 It would be 3/4 of the way up towards this dot, 00:17:13.240 --> 00:17:18.579 because it's 75% of the risky, 25% of the riskless, 00:17:18.579 --> 00:17:22.270 so you're going to get closer to the risky asset. 00:17:22.270 --> 00:17:23.470 OK, great. 00:17:23.470 --> 00:17:28.569 So we've now demonstrated that what 00:17:28.569 --> 00:17:32.140 I can achieve as an investor, just mixing 00:17:32.140 --> 00:17:35.110 portfolio p with the risk-free rate, 00:17:35.110 --> 00:17:38.620 is anywhere along that line. 00:17:38.620 --> 00:17:43.730 Now this analysis applies to any portfolio p. 00:17:43.730 --> 00:17:46.550 So for example, suppose I wanted to ask you, 00:17:46.550 --> 00:17:48.200 what risk-reward trade-offs could I 00:17:48.200 --> 00:17:54.527 generate by mixing the risk-free rate with General Motors? 00:17:54.527 --> 00:17:55.610 What would that look like? 00:18:02.810 --> 00:18:04.487 Yeah, Ken? 00:18:04.487 --> 00:18:07.455 AUDIENCE: The line from T-Bills through GM. 00:18:07.455 --> 00:18:08.830 ANDREW LO: Exactly, that's right. 00:18:08.830 --> 00:18:10.788 If I wanted to mix T-Bills with General Motors, 00:18:10.788 --> 00:18:13.270 I get that straight line right through that dot. 00:18:13.270 --> 00:18:16.930 If I wanted to mix T-Bills with IBM, I'd go through that dot, 00:18:16.930 --> 00:18:17.530 with IBM. 00:18:17.530 --> 00:18:19.240 If I wanted to mix T-Bills with Motorola, 00:18:19.240 --> 00:18:20.710 I'd go through Motorola. 00:18:20.710 --> 00:18:25.210 And if I wanted to mix T-Bills with any portfolio 00:18:25.210 --> 00:18:28.870 on that frontier, on that upper branch, 00:18:28.870 --> 00:18:31.900 it would just be a line between T-Bills 00:18:31.900 --> 00:18:34.330 and that point on the upper branch. 00:18:34.330 --> 00:18:36.080 Right? 00:18:36.080 --> 00:18:37.900 So question. 00:18:37.900 --> 00:18:40.920 If I were to give you the choice of mixing T-Bills 00:18:40.920 --> 00:18:50.242 with only one portfolio, just one, which would it be? 00:18:50.242 --> 00:18:51.200 Which would you prefer? 00:18:54.068 --> 00:18:55.525 [INAUDIBLE]? 00:18:55.525 --> 00:18:57.900 AUDIENCE: The one where the line is tangent to the curve. 00:18:57.900 --> 00:18:59.816 ANDREW LO: The one where the line is tangent-- 00:18:59.816 --> 00:19:02.910 so you're talking about right around here, right? 00:19:02.910 --> 00:19:04.380 Somewhere here. 00:19:04.380 --> 00:19:08.190 That's where the line is just tangent to that curve. 00:19:08.190 --> 00:19:10.050 Now why is that? 00:19:10.050 --> 00:19:11.388 How'd you come up with that? 00:19:11.388 --> 00:19:12.131 Yeah? 00:19:12.131 --> 00:19:14.256 AUDIENCE: If you took anything below that then it'd 00:19:14.256 --> 00:19:17.124 be, I would say, preferable to stay back. 00:19:17.124 --> 00:19:18.879 [INAUDIBLE] 00:19:18.879 --> 00:19:19.670 ANDREW LO: Exactly. 00:19:19.670 --> 00:19:21.440 If you picked any other portfolio 00:19:21.440 --> 00:19:24.950 besides the tangency portfolio, let's pick one and see. 00:19:24.950 --> 00:19:28.070 If you picked, let's say this one right here. 00:19:28.070 --> 00:19:31.910 If you drew a line between this point and that portfolio, 00:19:31.910 --> 00:19:33.380 it's going to turn out that there 00:19:33.380 --> 00:19:36.860 are other points over here that are strictly 00:19:36.860 --> 00:19:42.850 in the Northwest of that line, that you could do better. 00:19:42.850 --> 00:19:46.450 There exists only one portfolio that you 00:19:46.450 --> 00:19:52.930 can mix with T-Bills, such that you can never, ever do better 00:19:52.930 --> 00:19:56.050 in terms of generating risk-reward trade-offs 00:19:56.050 --> 00:20:00.670 for everybody that likes expected return, 00:20:00.670 --> 00:20:01.870 and doesn't like risk. 00:20:01.870 --> 00:20:04.600 And it turns out that that portfolio happens 00:20:04.600 --> 00:20:08.140 to be the tangency portfolio. 00:20:08.140 --> 00:20:11.350 That's the portfolio that all of you in this room 00:20:11.350 --> 00:20:12.646 would love to have. 00:20:12.646 --> 00:20:15.270 I don't know anything about you, I don't know your backgrounds, 00:20:15.270 --> 00:20:16.180 I don't know your risk aversions, 00:20:16.180 --> 00:20:17.320 but I don't have to know. 00:20:17.320 --> 00:20:20.150 As long as I know that you like expected return 00:20:20.150 --> 00:20:22.900 and you don't like risk, those are the only assumptions 00:20:22.900 --> 00:20:23.590 that I need. 00:20:23.590 --> 00:20:26.025 Then I know, all of you in this room, 00:20:26.025 --> 00:20:27.400 are going to want that portfolio. 00:20:27.400 --> 00:20:30.550 You may not be at that portfolio. 00:20:30.550 --> 00:20:32.650 For example, some of you who don't like risk, 00:20:32.650 --> 00:20:34.860 you're going to be down here. 00:20:34.860 --> 00:20:37.500 Those of you who are budding hedge fund managers, 00:20:37.500 --> 00:20:40.390 you're going to be up here. 00:20:40.390 --> 00:20:42.710 But the point is, you're going to be on this line. 00:20:42.710 --> 00:20:45.170 You're not going to be on this line down here. 00:20:45.170 --> 00:20:45.670 Why? 00:20:45.670 --> 00:20:49.510 Because why be on that line when you could get higher return 00:20:49.510 --> 00:20:51.700 for a given level of risk, or a lower risk 00:20:51.700 --> 00:20:53.400 for a given level of return. 00:20:53.400 --> 00:20:55.570 You're giving up something for no good reason. 00:20:59.660 --> 00:21:03.400 So this is a remarkable insight of modern portfolio theory. 00:21:03.400 --> 00:21:06.610 This basically tells us that regardless 00:21:06.610 --> 00:21:09.370 of our differences in preferences, 00:21:09.370 --> 00:21:11.590 as long as we satisfy the hypothesis 00:21:11.590 --> 00:21:14.770 that we like expected return and we don't like risk, that 00:21:14.770 --> 00:21:16.450 means that everybody in this room 00:21:16.450 --> 00:21:19.480 will agree that the only line that they would ever want 00:21:19.480 --> 00:21:23.020 to be on is that tangency line. 00:21:23.020 --> 00:21:23.610 Questions? 00:21:23.610 --> 00:21:24.812 Ingrid? 00:21:24.812 --> 00:21:27.758 AUDIENCE: Is there a particular level of risk 00:21:27.758 --> 00:21:30.220 that makes you accepting to the tangency fund? 00:21:30.220 --> 00:21:31.450 ANDREW LO: Yes. 00:21:31.450 --> 00:21:35.020 In fact, this tangency portfolio is one very 00:21:35.020 --> 00:21:38.060 particular and special portfolio. 00:21:38.060 --> 00:21:41.980 So in other words, it's a particular weighting of IBM, 00:21:41.980 --> 00:21:46.270 General Motors, and Motorola, that gives you 00:21:46.270 --> 00:21:48.078 this particular portfolio. 00:21:48.078 --> 00:21:49.034 AUDIENCE: Which one? 00:21:49.034 --> 00:21:53.340 [INAUDIBLE] something intuitive? 00:21:53.340 --> 00:21:56.740 ANDREW LO: It's something you can solve analytically. 00:21:56.740 --> 00:22:00.840 It has a solution, and if we were using matrix algebra, 00:22:00.840 --> 00:22:02.250 I can actually solve it for you. 00:22:02.250 --> 00:22:04.830 But it's a little bit complicated, 00:22:04.830 --> 00:22:08.580 so I'm not requiring that people know how to do that. 00:22:08.580 --> 00:22:10.700 Only that you know that it exists. 00:22:10.700 --> 00:22:11.818 Yeah? 00:22:11.818 --> 00:22:13.694 AUDIENCE: You go above the red dot, 00:22:13.694 --> 00:22:15.570 and into leverage [INAUDIBLE]? 00:22:15.570 --> 00:22:16.571 ANDREW LO: Exactly 00:22:16.571 --> 00:22:19.160 AUDIENCE: So it means that I have costs for my debt. 00:22:19.160 --> 00:22:20.124 ANDREW LO: Yes 00:22:20.124 --> 00:22:22.394 AUDIENCE: So maybe some debt, it would 00:22:22.394 --> 00:22:27.160 be more efficient to buy a portfolio with other weights-- 00:22:27.160 --> 00:22:28.400 ANDREW LO: Yes. 00:22:28.400 --> 00:22:31.290 If you assume that there are borrowing and lending 00:22:31.290 --> 00:22:36.780 differences, then obviously these analyses don't apply. 00:22:36.780 --> 00:22:44.504 So in particular, if you're here, you're actually lending. 00:22:44.504 --> 00:22:46.920 If you're here, you're fully invested in the stock market. 00:22:46.920 --> 00:22:49.020 If you're here, you're borrowing. 00:22:49.020 --> 00:22:51.840 If you're borrowing and lending rates are different, 00:22:51.840 --> 00:22:55.830 then it turns out that the curve that you want to be on actually 00:22:55.830 --> 00:22:57.150 has a kink in it. 00:22:57.150 --> 00:23:00.644 And that means that there is a potential for being 00:23:00.644 --> 00:23:03.060 on this curve, and then there's another tangency line that 00:23:03.060 --> 00:23:04.740 goes out at a different slope. 00:23:04.740 --> 00:23:05.490 That's possible. 00:23:05.490 --> 00:23:07.590 But that's more complicated than what we want 00:23:07.590 --> 00:23:09.040 to talk about at this point. 00:23:09.040 --> 00:23:12.900 So here I am assuming borrowing and lending rates are the same. 00:23:12.900 --> 00:23:15.000 Zeke, and then Rami. 00:23:15.000 --> 00:23:17.220 AUDIENCE: If I had a choice of, if I 00:23:17.220 --> 00:23:20.730 had control over the volatility of the market, 00:23:20.730 --> 00:23:25.470 then if the yield goes down, of the deals, 00:23:25.470 --> 00:23:28.764 then I would want to have a more volatile market so 00:23:28.764 --> 00:23:32.364 that I can intersect the curve at the higher return point. 00:23:32.364 --> 00:23:33.280 Tangent to the curve-- 00:23:33.280 --> 00:23:34.238 ANDREW LO: OK, hold on. 00:23:34.238 --> 00:23:36.450 You're changing the assumptions here. 00:23:36.450 --> 00:23:38.820 Why are you controlling the volatility of the market? 00:23:38.820 --> 00:23:41.460 The volatility of the market is a data point 00:23:41.460 --> 00:23:43.290 that you're basically using as an input. 00:23:43.290 --> 00:23:43.550 OK? 00:23:43.550 --> 00:23:44.591 AUDIENCE: I know, I know. 00:23:44.591 --> 00:23:48.030 I'm just trying to figure out what I am-- because you're 00:23:48.030 --> 00:23:50.550 basically connecting, I see this as a connection 00:23:50.550 --> 00:23:52.770 between the yield curve and the market. 00:23:52.770 --> 00:23:56.305 Because it's not only retrospective in the sense 00:23:56.305 --> 00:23:57.866 that if the yield goes down, there's 00:23:57.866 --> 00:23:59.420 cash flowing from the market-- 00:23:59.420 --> 00:24:01.336 ANDREW LO: Let's not worry about the dynamics. 00:24:01.336 --> 00:24:03.540 This is not meant to be a dynamic story. 00:24:03.540 --> 00:24:06.390 I didn't say anything about this happening over time, 00:24:06.390 --> 00:24:08.550 and there's lots of different changes going on. 00:24:08.550 --> 00:24:12.270 This is a static snapshot, today versus next period. 00:24:12.270 --> 00:24:14.760 These returns and covariances and all that 00:24:14.760 --> 00:24:17.680 apply to the returns from this period to the next, 00:24:17.680 --> 00:24:19.710 whether it's monthly or annual, that's 00:24:19.710 --> 00:24:21.800 a static snapshot as of today. 00:24:21.800 --> 00:24:26.290 So we're not talking about any term structure effects yet. 00:24:26.290 --> 00:24:28.754 Yeah, Rami? 00:24:28.754 --> 00:24:30.722 AUDIENCE: This is assuming three stocks. 00:24:30.722 --> 00:24:32.362 So if you had four or five, are you 00:24:32.362 --> 00:24:34.131 going to actually move the bullet left? 00:24:34.131 --> 00:24:35.380 And then you're gonna change-- 00:24:35.380 --> 00:24:37.240 ANDREW LO: Yes, absolutely. 00:24:37.240 --> 00:24:40.750 If you start adding more stocks to this cocktail, what's 00:24:40.750 --> 00:24:44.290 going to happen is the bullet is going to shift to the left, 00:24:44.290 --> 00:24:46.210 and it's going to shift up. 00:24:46.210 --> 00:24:49.630 And so the tangency point will change. 00:24:49.630 --> 00:24:53.380 But the curve, that straight line, the tangent line, 00:24:53.380 --> 00:24:57.700 what you're going to see is that tangent line 00:24:57.700 --> 00:24:59.520 is going to go like that. 00:24:59.520 --> 00:25:01.740 The slope, that's right. 00:25:01.740 --> 00:25:06.545 You're going to get more expected return per unit risk. 00:25:06.545 --> 00:25:07.920 And that is something we're going 00:25:07.920 --> 00:25:12.870 to take as a measure of how good this particular trade-off is. 00:25:12.870 --> 00:25:16.120 We're going to look at that slope of this line. 00:25:16.120 --> 00:25:17.970 And the slope of this line will give us 00:25:17.970 --> 00:25:23.340 a measure of the expected rate of return per unit risk. 00:25:23.340 --> 00:25:25.874 That's exactly what it's going to do for us. 00:25:25.874 --> 00:25:27.762 AUDIENCE: So if you're in the upper right 00:25:27.762 --> 00:25:30.440 beyond the tangency [INAUDIBLE], you know that line? 00:25:30.440 --> 00:25:32.940 Then you're borrowing [INAUDIBLE] portfolio? 00:25:32.940 --> 00:25:34.420 ANDREW LO: That's correct. 00:25:34.420 --> 00:25:37.088 AUDIENCE: If you were to extend the line leftwards, down 00:25:37.088 --> 00:25:39.171 to the left, would you be then shorting the market 00:25:39.171 --> 00:25:40.047 to invest in T-Bills? 00:25:40.047 --> 00:25:40.671 ANDREW LO: Yes. 00:25:40.671 --> 00:25:42.710 And if that happens, you know what you would do? 00:25:42.710 --> 00:25:44.970 It would not go this way, because of course 00:25:44.970 --> 00:25:46.860 standard deviation can't be negative. 00:25:46.860 --> 00:25:47.950 It would go like this. 00:25:47.950 --> 00:25:49.680 It would go this way. 00:25:49.680 --> 00:25:51.809 Because standard deviation is always non-negative. 00:25:51.809 --> 00:25:53.350 It's the square root of the variance, 00:25:53.350 --> 00:25:54.670 which is always positive. 00:25:54.670 --> 00:25:58.290 So if you decided to short the tangency portfolio 00:25:58.290 --> 00:26:03.540 and put it in T-Bills, well, you'd be a knucklehead. 00:26:03.540 --> 00:26:06.360 But you would be on this line right here. 00:26:06.360 --> 00:26:09.270 You would have higher and higher risk, because you're 00:26:09.270 --> 00:26:11.790 taking a short position on equities, 00:26:11.790 --> 00:26:13.470 and you'd have a lower and lower return 00:26:13.470 --> 00:26:15.303 because you're shorting the high yield asset 00:26:15.303 --> 00:26:16.680 and buying the low yield asset. 00:26:22.010 --> 00:26:25.690 Any other questions about the geometry of this point. 00:26:25.690 --> 00:26:28.770 It's very important, this is a major insight. 00:26:28.770 --> 00:26:29.664 Yeah? 00:26:29.664 --> 00:26:31.330 AUDIENCE: Earlier we discussed about how 00:26:31.330 --> 00:26:34.676 we bring market [INAUDIBLE] portfolios and expected return 00:26:34.676 --> 00:26:37.384 changes as a function of n and standard deviation 00:26:37.384 --> 00:26:39.950 from certain changes [INAUDIBLE] 00:26:39.950 --> 00:26:40.778 ANDREW LO: Yes. 00:26:40.778 --> 00:26:43.560 AUDIENCE: Is that the reason why the shift is more towards left? 00:26:43.560 --> 00:26:46.410 Because as we add more and more portfolios, 00:26:46.410 --> 00:26:47.940 the n dominates the [INAUDIBLE]? 00:26:47.940 --> 00:26:49.523 ANDREW LO: That's right, that's right. 00:26:49.523 --> 00:26:52.560 As we add more securities, you get more and more impact 00:26:52.560 --> 00:26:54.550 of diversification. 00:26:54.550 --> 00:26:58.950 So that increases your expected rate of return per unit risk 00:26:58.950 --> 00:27:01.110 because you can't make somebody worse off 00:27:01.110 --> 00:27:03.000 by giving them choices. 00:27:03.000 --> 00:27:05.610 They can always put a 0 for the new stocks 00:27:05.610 --> 00:27:07.690 that you give them if they don't like it. 00:27:07.690 --> 00:27:11.070 So the only thing you can do is to make you better off, meaning 00:27:11.070 --> 00:27:12.930 the only thing you can do is to give you 00:27:12.930 --> 00:27:15.750 a higher level of expected return per unit risk, 00:27:15.750 --> 00:27:19.560 or a lower level of risk per unit of expected return. 00:27:19.560 --> 00:27:21.870 So by adding more securities, you're 00:27:21.870 --> 00:27:26.290 basically increasing the slope of this line. 00:27:26.290 --> 00:27:28.840 So let's talk about the slope of the line. 00:27:28.840 --> 00:27:33.450 The slope of that line is equal to the expected return 00:27:33.450 --> 00:27:38.010 of that tangency portfolio, minus the T-Bill rate, divided 00:27:38.010 --> 00:27:41.460 by the volatility of that tangency portfolio. 00:27:41.460 --> 00:27:43.890 If you just calculate rise over run, 00:27:43.890 --> 00:27:45.990 that's what you get as the slope. 00:27:45.990 --> 00:27:47.370 There's a name for this. 00:27:47.370 --> 00:27:49.290 The name for this is called the Sharpe ratio. 00:27:49.290 --> 00:27:50.970 You may have heard of this, particularly 00:27:50.970 --> 00:27:53.620 those of you who have interest in hedge fund investments. 00:27:53.620 --> 00:27:56.700 Hedge fund managers will often quote their Sharpe ratio very 00:27:56.700 --> 00:27:57.930 proudly. 00:27:57.930 --> 00:28:00.330 The Sharpe ratio is simply a measure 00:28:00.330 --> 00:28:01.950 of that risk-reward trade-off. 00:28:01.950 --> 00:28:05.950 The higher the Sharpe ratio, the better you're doing. 00:28:05.950 --> 00:28:10.070 If you're a mean variance optimizer, meaning you 00:28:10.070 --> 00:28:14.440 prefer more expected return and less risk. 00:28:17.510 --> 00:28:20.720 So the idea behind the tangency portfolio 00:28:20.720 --> 00:28:22.610 is that it is the one that will give you 00:28:22.610 --> 00:28:24.320 the highest Sharpe ratio. 00:28:24.320 --> 00:28:25.760 Let's look at it again. 00:28:25.760 --> 00:28:28.820 If you pick a portfolio like here, take a look. 00:28:28.820 --> 00:28:29.730 Look at the slope. 00:28:29.730 --> 00:28:32.060 The slope is going to be lower. 00:28:32.060 --> 00:28:35.750 Take a point over here, in the inefficient branch 00:28:35.750 --> 00:28:36.740 of the bullet. 00:28:36.740 --> 00:28:38.300 Then the slope is going to be even 00:28:38.300 --> 00:28:40.250 lower than the upper branch. 00:28:40.250 --> 00:28:47.330 The biggest slope occurs when you invest between T-Bills 00:28:47.330 --> 00:28:50.370 and that tangency portfolio. 00:28:50.370 --> 00:28:54.282 That's what you're optimizing. 00:28:54.282 --> 00:28:55.746 Yeah? 00:28:55.746 --> 00:28:58.674 AUDIENCE: I have a hard time understanding 00:28:58.674 --> 00:29:03.570 why the bullet would go left when I add additional stocks. 00:29:03.570 --> 00:29:07.300 I understand it analytically, standard deviation goes down. 00:29:07.300 --> 00:29:11.499 But then on the other hand, wouldn't it 00:29:11.499 --> 00:29:16.030 happen that the likelihood of correlation 00:29:16.030 --> 00:29:19.509 between these additional stocks would 00:29:19.509 --> 00:29:22.765 decrease and therefore as we-- 00:29:22.765 --> 00:29:24.640 ANDREW LO: How would the correlation increase 00:29:24.640 --> 00:29:25.870 by adding another stock? 00:29:25.870 --> 00:29:30.160 AUDIENCE: I mean the likelihood that I have 20 stocks, 00:29:30.160 --> 00:29:35.312 the overall correlation is higher rather than [INAUDIBLE] 00:29:35.312 --> 00:29:36.520 ANDREW LO: How could that be? 00:29:36.520 --> 00:29:39.310 You've got 20 stocks, and they've 00:29:39.310 --> 00:29:42.470 got a correlation among those 20 stocks. 00:29:42.470 --> 00:29:46.730 Now, I want you to think about adding a 21st stock. 00:29:46.730 --> 00:29:48.980 When you add that 21st stock, you 00:29:48.980 --> 00:29:51.380 don't affect the existing correlations, right? 00:29:51.380 --> 00:29:53.540 I mean, it is whatever it is. 00:29:53.540 --> 00:29:54.950 Those are parameters. 00:29:54.950 --> 00:29:57.790 At least for now, we're going to call them parameters. 00:29:57.790 --> 00:30:02.330 When I add my 21st stock, I'm giving the investor 00:30:02.330 --> 00:30:03.840 an extra degree of freedom. 00:30:03.840 --> 00:30:07.460 Now instead of investing among 20 securities, 00:30:07.460 --> 00:30:09.140 I'm going to let you invest among 21. 00:30:09.140 --> 00:30:11.772 You don't have to invest in the 21st. 00:30:11.772 --> 00:30:13.230 Or another way of thinking about it 00:30:13.230 --> 00:30:16.590 is that when you only had 20 stocks, 00:30:16.590 --> 00:30:19.710 you really had 21 portfolio weights. 00:30:19.710 --> 00:30:21.870 But the 21st weight, I've arbitrarily 00:30:21.870 --> 00:30:24.320 constrained to be 0. 00:30:24.320 --> 00:30:26.630 Now, I'm going to loosen the constraint 00:30:26.630 --> 00:30:28.070 and I'm going to say, OK, now you 00:30:28.070 --> 00:30:31.390 can invest in the 21st stock. 00:30:31.390 --> 00:30:33.340 You won't affect the existing correlations, 00:30:33.340 --> 00:30:34.900 but the new stock that you add in 00:30:34.900 --> 00:30:39.320 can benefit in providing additional diversification 00:30:39.320 --> 00:30:39.820 benefits. 00:30:39.820 --> 00:30:41.507 AUDIENCE: But as a general rule, we've 00:30:41.507 --> 00:30:45.467 always been trying to have negative correlation so 00:30:45.467 --> 00:30:46.515 that the bullet was left. 00:30:46.515 --> 00:30:47.890 ANDREW LO: Oh, well actually, you 00:30:47.890 --> 00:30:49.210 don't need a negative correlation 00:30:49.210 --> 00:30:50.751 to make this go to the left, you just 00:30:50.751 --> 00:30:53.540 need to have something less than 1. 00:30:53.540 --> 00:30:54.200 Remember? 00:30:54.200 --> 00:30:55.440 From the last lecture? 00:30:55.440 --> 00:30:59.000 This is a case where you had perfect correlation. 00:30:59.000 --> 00:31:01.850 Anything less than perfect correlation, 00:31:01.850 --> 00:31:04.850 brings you to the left. 00:31:04.850 --> 00:31:08.180 So as long as my 21st stock is not 00:31:08.180 --> 00:31:10.670 perfectly correlated with the existing 00:31:10.670 --> 00:31:13.400 stocks in that portfolio of 20, I'm 00:31:13.400 --> 00:31:14.894 going to move things to the left. 00:31:14.894 --> 00:31:17.190 AUDIENCE: In general, [INAUDIBLE] any stocks? 00:31:17.190 --> 00:31:18.690 ANDREW LO: In general, that is true. 00:31:18.690 --> 00:31:22.440 AUDIENCE: I mean I would try to have the negative correlation. 00:31:22.440 --> 00:31:24.690 ANDREW LO: You would, but what this suggests 00:31:24.690 --> 00:31:26.820 is that negative correlation is a very rare thing. 00:31:26.820 --> 00:31:27.690 AUDIENCE: It's difficult. 00:31:27.690 --> 00:31:30.064 ANDREW LO: It's very difficult, it's extremely difficult. 00:31:30.064 --> 00:31:32.760 Now, from the analytical perspective, 00:31:32.760 --> 00:31:34.260 we can conclude it's very difficult. 00:31:34.260 --> 00:31:37.440 Let me ask you from an economic perspective, 00:31:37.440 --> 00:31:40.680 why is it difficult to find a stock that's negatively 00:31:40.680 --> 00:31:42.540 correlated with all other stocks? 00:31:42.540 --> 00:31:46.350 Anybody give me a business rationale for that? 00:31:46.350 --> 00:31:47.502 Yeah, Ingrid? 00:31:47.502 --> 00:31:49.150 AUDIENCE: There's what you said before 00:31:49.150 --> 00:31:52.004 that when the economy goes down, everything goes down 00:31:52.004 --> 00:31:52.980 and we get vice versa. 00:31:52.980 --> 00:31:57.082 Just that we mention [INAUDIBLE] different countries, 00:31:57.082 --> 00:32:00.435 in different economic regions, in different industry, 00:32:00.435 --> 00:32:01.740 and they should not be-- 00:32:01.740 --> 00:32:03.990 ANDREW LO: Let's actually spend a little bit more time 00:32:03.990 --> 00:32:05.020 thinking about this. 00:32:05.020 --> 00:32:07.770 I want you guys to tell me right now, give me 00:32:07.770 --> 00:32:12.440 a stock that you would put your money in right now, today. 00:32:12.440 --> 00:32:17.640 S&P has gone down by 45% since the high several months ago. 00:32:17.640 --> 00:32:19.500 The stock market's doing terribly, 00:32:19.500 --> 00:32:21.610 and it doesn't look like it's getting any better. 00:32:21.610 --> 00:32:23.860 So you tell me, what stock would you put your money in 00:32:23.860 --> 00:32:25.090 right now, today? 00:32:25.090 --> 00:32:25.707 Yeah, Terry. 00:32:25.707 --> 00:32:26.790 AUDIENCE: Campbell's Soup. 00:32:26.790 --> 00:32:28.470 ANDREW LO: Campbell's Soup. 00:32:28.470 --> 00:32:29.380 Why is that? 00:32:29.380 --> 00:32:30.546 AUDIENCE: It's a food stock. 00:32:30.546 --> 00:32:35.848 It's a foodstuff people need, will purchase, inexpensive, 00:32:35.848 --> 00:32:37.300 pretty much just for-- 00:32:37.300 --> 00:32:38.760 ANDREW LO: OK. 00:32:38.760 --> 00:32:43.440 But on the other hand, if people are poorer all around, 00:32:43.440 --> 00:32:46.650 might not they start consuming even less of canned soup 00:32:46.650 --> 00:32:49.680 and try to make their own soup from little packages of ketchup 00:32:49.680 --> 00:32:51.390 and hot water? 00:32:51.390 --> 00:32:54.580 I saw that on an I Love Lucy episode years ago. 00:32:54.580 --> 00:32:56.980 It's pretty cool. 00:32:56.980 --> 00:32:58.286 So are you sure? 00:32:58.286 --> 00:32:59.660 Are you sure that Campbell's Soup 00:32:59.660 --> 00:33:02.250 is going to go up over the next few months, 00:33:02.250 --> 00:33:03.720 in response to the current crisis? 00:33:03.720 --> 00:33:05.332 AUDIENCE: It'll stay pretty stable. 00:33:05.332 --> 00:33:06.540 ANDREW LO: It'll stay stable. 00:33:06.540 --> 00:33:09.210 Ah, but that's not negative correlation. 00:33:09.210 --> 00:33:10.812 That's 0 correlation. 00:33:10.812 --> 00:33:12.270 I want something that's going to go 00:33:12.270 --> 00:33:15.300 the opposite direction of where the economy is heading. 00:33:15.300 --> 00:33:18.690 Tell me where that is? 00:33:18.690 --> 00:33:19.770 Yeah? 00:33:19.770 --> 00:33:21.907 AUDIENCE: [INAUDIBLE] short financials. 00:33:21.907 --> 00:33:22.740 ANDREW LO: OK, fine. 00:33:22.740 --> 00:33:24.240 So you're going to short the market. 00:33:24.240 --> 00:33:26.850 That's a cheap answer. 00:33:26.850 --> 00:33:29.610 Sorry, you don't get any credit for that. 00:33:29.610 --> 00:33:31.200 I want to answer the question that 00:33:31.200 --> 00:33:33.690 was raised by David which is, show me 00:33:33.690 --> 00:33:37.260 a stock that can get me even more to that left. 00:33:37.260 --> 00:33:39.990 I want a negatively correlated stock. 00:33:39.990 --> 00:33:41.090 Yeah, [INAUDIBLE]. 00:33:41.090 --> 00:33:42.230 AUDIENCE: Wal-Mart. 00:33:42.230 --> 00:33:43.890 ANDREW LO: Wal-Mart? 00:33:43.890 --> 00:33:45.255 AUDIENCE: It's been going up. 00:33:45.255 --> 00:33:47.130 ANDREW LO: Well, that's not the same thing 00:33:47.130 --> 00:33:48.644 as saying that it is going to go up 00:33:48.644 --> 00:33:50.310 over the next several months in response 00:33:50.310 --> 00:33:51.310 to this economic crisis. 00:33:51.310 --> 00:33:52.726 You don't think that there's going 00:33:52.726 --> 00:33:54.300 to be a decline in consumer spending 00:33:54.300 --> 00:33:57.570 that will affect retail as well? 00:33:57.570 --> 00:33:59.650 AUDIENCE: So far, everybody's going to Wal-Mart. 00:33:59.650 --> 00:34:01.900 If they don't go to Wal-Mart, where would they go? 00:34:01.900 --> 00:34:02.700 ANDREW LO: Well, that's what I'm asking you. 00:34:02.700 --> 00:34:03.750 Where are you going to go? 00:34:03.750 --> 00:34:05.458 So you're telling me now that you believe 00:34:05.458 --> 00:34:06.848 that Wal-Mart is the answer? 00:34:06.848 --> 00:34:08.639 You think it will be negatively correlated? 00:34:08.639 --> 00:34:10.382 Historically, just to let you know, 00:34:10.382 --> 00:34:12.090 retail has not been negatively correlated 00:34:12.090 --> 00:34:14.835 with the business cycle. 00:34:14.835 --> 00:34:15.460 Yeah, Zeke? 00:34:15.460 --> 00:34:17.011 AUDIENCE: What about Freddie Mac? 00:34:17.011 --> 00:34:17.969 ANDREW LO: Freddie Mac? 00:34:17.969 --> 00:34:18.594 AUDIENCE: Yeah. 00:34:18.594 --> 00:34:22.165 [LAUGHTER] 00:34:22.165 --> 00:34:23.790 ANDREW LO: If you like that investment, 00:34:23.790 --> 00:34:24.870 I have something else for you-- 00:34:24.870 --> 00:34:25.712 [INTERPOSING VOICES] 00:34:25.712 --> 00:34:26.554 --afterwards. 00:34:26.554 --> 00:34:28.482 AUDIENCE: We could do it this time. 00:34:28.482 --> 00:34:29.940 ANDREW LO: I don't know if you want 00:34:29.940 --> 00:34:32.159 to argue that Freddie Mac is negatively correlated 00:34:32.159 --> 00:34:33.076 with market downturns. 00:34:33.076 --> 00:34:35.409 I mean, the reason that Freddie Mac got into the trouble 00:34:35.409 --> 00:34:37.504 that it did was because of the economic downturn. 00:34:37.504 --> 00:34:38.625 All right, one more. 00:34:38.625 --> 00:34:39.380 [INAUDIBLE]? 00:34:39.380 --> 00:34:40.779 AUDIENCE: Philip Morris. 00:34:40.779 --> 00:34:41.820 ANDREW LO: Philip Morris. 00:34:41.820 --> 00:34:44.250 That's an interesting one. 00:34:44.250 --> 00:34:46.050 Obviously, people are very nervous now. 00:34:46.050 --> 00:34:49.230 When you're nervous, you're going to be smoking. 00:34:51.989 --> 00:34:53.940 On the other hand, again, one could 00:34:53.940 --> 00:34:56.580 argue that it's not negatively correlated. 00:34:56.580 --> 00:35:02.100 It might be either slightly positively correlated, 00:35:02.100 --> 00:35:04.200 but even there, people have argued 00:35:04.200 --> 00:35:07.350 that cigarettes are a consumption good that can get 00:35:07.350 --> 00:35:10.724 hit with a downturn in markets. 00:35:10.724 --> 00:35:12.390 The bottom line is that it's really hard 00:35:12.390 --> 00:35:14.832 to come up with negative correlated stocks. 00:35:14.832 --> 00:35:17.040 Let me tell you, if you found one that was negatively 00:35:17.040 --> 00:35:20.490 correlated, if you found one that was really, really, 00:35:20.490 --> 00:35:23.112 negatively correlated, what would all of you do? 00:35:23.112 --> 00:35:23.820 AUDIENCE: Buy it. 00:35:23.820 --> 00:35:26.310 ANDREW LO: Exactly, you'd buy it. 00:35:26.310 --> 00:35:30.210 The effect of that would be to increase the price 00:35:30.210 --> 00:35:33.570 and depress the expected return. 00:35:33.570 --> 00:35:35.160 Remember what the expected return is. 00:35:35.160 --> 00:35:39.270 It's the expected future price, divided by the current price. 00:35:39.270 --> 00:35:42.150 If now all of you go out and buy Wal-Mart, 00:35:42.150 --> 00:35:44.900 or whatever stock you think is negatively correlated, 00:35:44.900 --> 00:35:48.990 that would have the impact of increasing the current price 00:35:48.990 --> 00:35:52.164 and therefore decreasing the expected return. 00:35:52.164 --> 00:35:53.580 Now if you have a stock that's got 00:35:53.580 --> 00:35:56.821 a negative covariance and a negative return, 00:35:56.821 --> 00:35:57.570 that doesn't help. 00:35:57.570 --> 00:35:59.340 Because in fact, that was a suggestion 00:35:59.340 --> 00:36:00.660 that was put forward here. 00:36:00.660 --> 00:36:03.960 Let's just take the S&P and short it, 00:36:03.960 --> 00:36:06.280 and then you get a negatively correlated stock. 00:36:06.280 --> 00:36:09.000 The problem is that it's also got a negative expected return 00:36:09.000 --> 00:36:11.306 and then you're not helping things. 00:36:11.306 --> 00:36:12.930 The key is to find negative correlation 00:36:12.930 --> 00:36:14.100 with a positive return. 00:36:14.100 --> 00:36:16.410 If you can find that, then you've 00:36:16.410 --> 00:36:19.110 really found something worthwhile. 00:36:19.110 --> 00:36:22.830 But my guess is it won't last, for exactly this reason. 00:36:22.830 --> 00:36:24.420 Other questions? 00:36:24.420 --> 00:36:27.650 OK, so now, let's go back and ask the question, what 00:36:27.650 --> 00:36:30.920 does this mean if we agree that all of us 00:36:30.920 --> 00:36:33.920 want to be on that tangency portfolio. 00:36:33.920 --> 00:36:35.940 What does that tell us? 00:36:35.940 --> 00:36:41.250 Well, that allows us to then make an argument 00:36:41.250 --> 00:36:46.170 that managers that are trying to provide value 00:36:46.170 --> 00:36:51.560 added services for us, they need to be doing something above 00:36:51.560 --> 00:36:55.700 and beyond what we can do ourselves. 00:36:55.700 --> 00:36:58.280 Now, here's where Warren Buffett meets modern finance theory, 00:36:58.280 --> 00:37:01.090 in a way. 00:37:01.090 --> 00:37:02.740 If I want to see whether or not Warren 00:37:02.740 --> 00:37:05.290 Buffett or any other managers are adding value, 00:37:05.290 --> 00:37:08.990 one simple criterion that I can put forward is this. 00:37:08.990 --> 00:37:10.960 This Is what I can do on my own. 00:37:10.960 --> 00:37:16.240 I can get that line pretty much by just using my basic finance 00:37:16.240 --> 00:37:19.670 skills that I've learned here at MIT. 00:37:19.670 --> 00:37:23.570 If you're going to manage my money and charge me 2 and 20, 00:37:23.570 --> 00:37:26.420 show me what you can do above and beyond this. 00:37:26.420 --> 00:37:29.720 I want you to tell me where you can get me on this graph. 00:37:29.720 --> 00:37:32.870 Can you get me up here? 00:37:32.870 --> 00:37:34.540 Can you get me over here? 00:37:34.540 --> 00:37:37.690 Can you get me anywhere either to the left 00:37:37.690 --> 00:37:40.850 or above that curve? 00:37:40.850 --> 00:37:43.650 We can use that as a measure of performance, 00:37:43.650 --> 00:37:45.680 and there's a name for that. 00:37:45.680 --> 00:37:46.910 It's called Alpha. 00:37:46.910 --> 00:37:48.980 Typically, when people talk about Alpha, 00:37:48.980 --> 00:37:52.022 they're talking about deviations from a line like this. 00:37:52.022 --> 00:37:53.730 We're going to get to that more formally, 00:37:53.730 --> 00:37:56.230 you don't have to write it down or make note of it just yet. 00:37:56.230 --> 00:37:57.544 It's on the next slide. 00:37:57.544 --> 00:37:59.960 But we're going to show you how to measure that explicitly 00:37:59.960 --> 00:38:04.460 so now, not only is this a good idea for you as a baseline 00:38:04.460 --> 00:38:07.310 to manage your own portfolio, but you can then 00:38:07.310 --> 00:38:09.650 use it as a metric to gauge whether other people are 00:38:09.650 --> 00:38:11.220 adding value to you. 00:38:11.220 --> 00:38:14.780 So Warren Buffett would say, no problem. 00:38:14.780 --> 00:38:16.310 I think I've got Alpha. 00:38:16.310 --> 00:38:18.110 So I'm not going to bother with this, 00:38:18.110 --> 00:38:20.710 I think I can get you up here. 00:38:20.710 --> 00:38:23.320 That is, if you want to invest with me. 00:38:23.320 --> 00:38:26.380 And in fact, if you looked at Warren Buffett's performance 00:38:26.380 --> 00:38:30.310 over the last 25 years that he's been doing it, or 30 years, 00:38:30.310 --> 00:38:36.250 his Sharpe ratio is a lot better than the tangency portfolios. 00:38:36.250 --> 00:38:40.420 So he actually has added value if you use this as a criterion. 00:38:40.420 --> 00:38:43.780 But the problem is, you have to identify the Warren 00:38:43.780 --> 00:38:46.171 Buffetts before they become Warren Buffetts. 00:38:46.171 --> 00:38:47.920 Because after they become Warren Buffetts, 00:38:47.920 --> 00:38:49.030 it's not clear that they're adding 00:38:49.030 --> 00:38:50.350 the same amount of value. 00:38:50.350 --> 00:38:52.490 It's already, the cat's out of the bag. 00:38:52.490 --> 00:38:54.028 Yeah, [INAUDIBLE]? 00:38:54.028 --> 00:38:55.819 AUDIENCE: Question about the Sharpe ration. 00:38:55.819 --> 00:38:57.996 So is this a stag ratio, or is it dynamic? 00:38:57.996 --> 00:39:02.956 Because in my mind, as you gain more stock options, 00:39:02.956 --> 00:39:04.227 it's going to become sharper. 00:39:04.227 --> 00:39:04.810 ANDREW LO: Yes 00:39:04.810 --> 00:39:07.175 AUDIENCE: Opportunity cost for switching to T-Bills 00:39:07.175 --> 00:39:08.712 is going to be greater, so you're 00:39:08.712 --> 00:39:10.587 going to shift preferences away from T-Bills. 00:39:10.587 --> 00:39:13.230 Then isn't that point going to increase and flatten out? 00:39:13.230 --> 00:39:17.350 ANDREW LO: Yes, so the dynamics of this are very complex. 00:39:17.350 --> 00:39:19.060 This is, right now, a static theory. 00:39:19.060 --> 00:39:21.700 Static meaning today versus next period. 00:39:21.700 --> 00:39:23.534 We're not looking at the dynamics over time. 00:39:23.534 --> 00:39:25.783 In order to do that, there's lots of different effects 00:39:25.783 --> 00:39:27.540 that are much, much more complicated. 00:39:27.540 --> 00:39:30.010 For that, you've really got to take 15 433 00:39:30.010 --> 00:39:32.680 and even 433 won't cover those kinds of questions 00:39:32.680 --> 00:39:36.850 in complete detail because they rely on some very complex kinds 00:39:36.850 --> 00:39:37.444 of analysis. 00:39:37.444 --> 00:39:39.110 But I'm going to get to that at the end. 00:39:39.110 --> 00:39:41.500 So if I don't, please bring it up again. 00:39:41.500 --> 00:39:43.390 I want to make a comment about that, 00:39:43.390 --> 00:39:46.630 and how you can take this relatively simple static theory 00:39:46.630 --> 00:39:49.240 and make it dynamic in an informal way 00:39:49.240 --> 00:39:51.790 even though the analytics become very hard when 00:39:51.790 --> 00:39:54.640 you try to do it formally. 00:39:54.640 --> 00:39:58.840 So the key points of this lecture are, oh, sorry, 00:39:58.840 --> 00:40:00.076 question? 00:40:00.076 --> 00:40:02.850 AUDIENCE: [INAUDIBLE] every point on the line, 00:40:02.850 --> 00:40:04.230 it's indifferent? 00:40:04.230 --> 00:40:04.865 Or-- 00:40:04.865 --> 00:40:05.740 ANDREW LO: No, no no. 00:40:05.740 --> 00:40:07.870 Not at all, not at all indifferent. 00:40:07.870 --> 00:40:10.900 Any point on this line is a different risk-reward 00:40:10.900 --> 00:40:12.040 combination. 00:40:12.040 --> 00:40:14.162 So in other words, it depends on your preferences. 00:40:14.162 --> 00:40:16.120 AUDIENCE: You take a function for the industry? 00:40:16.120 --> 00:40:17.020 ANDREW LO: Yes, yes. 00:40:17.020 --> 00:40:19.450 Now, we haven't talked about utility functions yet, 00:40:19.450 --> 00:40:21.380 but we're going to in a little while. 00:40:21.380 --> 00:40:24.790 Let me preview that, since you asked. 00:40:24.790 --> 00:40:27.370 You all remember what indifference curves are? 00:40:27.370 --> 00:40:29.380 From basic economics? 00:40:29.380 --> 00:40:32.110 An indifference curve, when I first came across that, 00:40:32.110 --> 00:40:36.070 I was rather offended because I don't view myself 00:40:36.070 --> 00:40:37.990 as an indifferent individual. 00:40:37.990 --> 00:40:39.820 I have lots of passions. 00:40:39.820 --> 00:40:43.666 And so, why should we be indifferent about two choices? 00:40:43.666 --> 00:40:45.040 In fact, that's an economic term. 00:40:45.040 --> 00:40:49.000 It simply means that you are just as well off between two 00:40:49.000 --> 00:40:51.430 combinations and therefore, these two 00:40:51.430 --> 00:40:55.600 combinations you're indifferent to, you're indifferent between. 00:40:55.600 --> 00:41:01.690 So if I had to ask you to draw on this graph, an indifference 00:41:01.690 --> 00:41:06.501 curve of risk-reward trade-offs for you, 00:41:06.501 --> 00:41:08.500 the typical individual, what would it look like? 00:41:08.500 --> 00:41:11.650 Can anybody give me a sense of what different kinds 00:41:11.650 --> 00:41:15.880 of risk-reward trade-offs you would be indifferent among? 00:41:15.880 --> 00:41:18.790 And to make the question a little bit simpler, 00:41:18.790 --> 00:41:20.890 let's start off with a particular point. 00:41:20.890 --> 00:41:22.780 So let's suppose that this point right here 00:41:22.780 --> 00:41:25.450 is the point that I want you to draw the indifferent 00:41:25.450 --> 00:41:27.610 curve from. 00:41:27.610 --> 00:41:31.570 Which is a standard deviation on a monthly basis of about 6%, 00:41:31.570 --> 00:41:38.180 and an expected return of about, say 1.4% or something. 00:41:38.180 --> 00:41:45.780 So you've got a monthly return of 1.4% and a risk of about 6%. 00:41:45.780 --> 00:41:50.760 Give me another point that you would be indifferent between, 00:41:50.760 --> 00:41:51.560 versus that one? 00:41:56.140 --> 00:41:56.640 Anybody? 00:41:56.640 --> 00:41:57.480 Any volunteers? 00:42:00.917 --> 00:42:01.899 Yeah? 00:42:01.899 --> 00:42:06.640 AUDIENCE: There could be a point above the tangent line, 00:42:06.640 --> 00:42:07.607 but to your right. 00:42:07.607 --> 00:42:08.940 Somewhere where you're pointing. 00:42:08.940 --> 00:42:11.580 ANDREW LO: OK, so do you have a particular number in mind? 00:42:11.580 --> 00:42:13.760 In other words, let me ask you this. 00:42:13.760 --> 00:42:18.524 If I cranked up your volatility from 6% to 8%, 00:42:18.524 --> 00:42:19.940 how much extra return would I have 00:42:19.940 --> 00:42:21.898 to give you in order for you to be just as well 00:42:21.898 --> 00:42:26.299 off as you were at 6% and 1.4%? 00:42:26.299 --> 00:42:28.340 AUDIENCE: Something slightly higher than the 1.8% 00:42:28.340 --> 00:42:29.720 [INAUDIBLE]. 00:42:29.720 --> 00:42:32.430 ANDREW LO: OK, higher than 1.8% OK. 00:42:32.430 --> 00:42:35.353 AUDIENCE: Or what about the corresponding value. 00:42:35.353 --> 00:42:35.853 1.41%. 00:42:35.853 --> 00:42:38.790 1.41% 00:42:38.790 --> 00:42:40.940 ANDREW LO: OK, this is 1.41%. 00:42:40.940 --> 00:42:43.670 And if I said, now I want to be at 8%, 00:42:43.670 --> 00:42:45.171 how much risk do I have to give you, 00:42:45.171 --> 00:42:47.586 how much expected return do I have to give you to make you 00:42:47.586 --> 00:42:49.180 just as well off as this point? 00:42:52.180 --> 00:42:54.390 AUDIENCE: Higher then 1.2%, right? 00:42:54.390 --> 00:42:56.097 Or 1.4%, higher than 1.4%. 00:42:56.097 --> 00:42:57.680 ANDREW LO: Right, but how much higher? 00:42:57.680 --> 00:42:58.513 That's the question. 00:42:58.513 --> 00:42:59.701 It's a personal question. 00:42:59.701 --> 00:43:00.200 Rami? 00:43:00.200 --> 00:43:05.190 AUDIENCE: 1.3 times your initial expected return. 00:43:05.190 --> 00:43:06.746 So 33% on top. 00:43:06.746 --> 00:43:08.120 ANDREW LO: You would have to have 00:43:08.120 --> 00:43:11.150 an increase in 33% of your expected return, 00:43:11.150 --> 00:43:15.134 even though I'm only giving you a 25% increase in the risk. 00:43:15.134 --> 00:43:16.118 AUDIENCE: Well, no. 00:43:16.118 --> 00:43:19.316 You'd have to at least do 25% of the-- sorry. 00:43:19.316 --> 00:43:20.494 Put 6%-8% is-- 00:43:20.494 --> 00:43:22.160 ANDREW LO: That's a third, you're right. 00:43:22.160 --> 00:43:23.990 So I'm increasing the risk by 1/3, 00:43:23.990 --> 00:43:26.510 you want me to increase the expected return by 1/3. 00:43:26.510 --> 00:43:28.580 So your trade-off is linear, is that right? 00:43:28.580 --> 00:43:31.040 You're looking at it linearly? 00:43:31.040 --> 00:43:32.404 Anybody else? 00:43:32.404 --> 00:43:34.070 You know, you may want to translate this 00:43:34.070 --> 00:43:35.630 into annual numbers. 00:43:35.630 --> 00:43:38.480 Because I'm sensing that you may not have a good 00:43:38.480 --> 00:43:40.520 feel for what your own preferences are. 00:43:40.520 --> 00:43:42.260 And by the way, this is a challenge. 00:43:42.260 --> 00:43:46.370 Not everybody understands what their own personal preferences 00:43:46.370 --> 00:43:47.970 are for these numbers. 00:43:47.970 --> 00:43:51.230 This is not a natural act of human nature, 00:43:51.230 --> 00:43:55.670 that we automatically have preferences on these numbers. 00:43:55.670 --> 00:43:59.770 But the bottom line is that if I make you take more risk, 00:43:59.770 --> 00:44:02.020 I'm going to have to compensate you and give you 00:44:02.020 --> 00:44:02.960 more expected return. 00:44:02.960 --> 00:44:05.980 There's got to be a reason why you want to take that risk. 00:44:05.980 --> 00:44:08.486 For some people, it's linear. 00:44:08.486 --> 00:44:10.360 For other people, it's much more than linear. 00:44:10.360 --> 00:44:11.943 They don't want to take any more risk. 00:44:11.943 --> 00:44:14.140 In fact, right now most investors 00:44:14.140 --> 00:44:16.320 don't even want to answer that question. 00:44:16.320 --> 00:44:18.070 Because they don't want to take more risk. 00:44:18.070 --> 00:44:19.160 And you say, well what if you did? 00:44:19.160 --> 00:44:20.210 Well I don't want to. 00:44:20.210 --> 00:44:21.330 Well, but just what if? 00:44:21.330 --> 00:44:22.330 I don't want to what if. 00:44:22.330 --> 00:44:24.460 I just don't want to take the risk. 00:44:24.460 --> 00:44:26.290 So they can't even answer that question. 00:44:26.290 --> 00:44:29.450 But if they could, my guess is that it would be way up here. 00:44:29.450 --> 00:44:31.960 So you'd have to give them a lot of expected return 00:44:31.960 --> 00:44:34.360 to make people take more risk today. 00:44:34.360 --> 00:44:37.480 Alternatively, if you want to give people less risk, 00:44:37.480 --> 00:44:39.040 my guess is that you can actually 00:44:39.040 --> 00:44:42.610 subtract a lot of return in order to take away 00:44:42.610 --> 00:44:43.640 a little bit of risk. 00:44:43.640 --> 00:44:44.620 How do I know that? 00:44:44.620 --> 00:44:46.828 Take a look at the yield on the three month treasury. 00:44:50.420 --> 00:44:53.370 So an indifference curve. 00:44:53.370 --> 00:44:54.730 It's going to look like this. 00:44:54.730 --> 00:44:58.530 It's going to look like it'll be increasing, 00:44:58.530 --> 00:45:01.680 but it'll actually be bowed this way. 00:45:01.680 --> 00:45:04.940 And the theory behind why it's going to be convex, 00:45:04.940 --> 00:45:07.240 holds water as opposed to spills water, 00:45:07.240 --> 00:45:08.880 the reason it's going to be convex 00:45:08.880 --> 00:45:12.000 is because there is a decreasing, or diminishing, 00:45:12.000 --> 00:45:15.600 marginal utility between risk and expected rate of return. 00:45:15.600 --> 00:45:18.420 Like anything else, economists have this notion 00:45:18.420 --> 00:45:22.320 of diminishing marginal utility between any two commodities. 00:45:22.320 --> 00:45:29.040 If you've got ice cream sundaes and basketballs, 00:45:29.040 --> 00:45:30.750 there's only so many basketballs that you 00:45:30.750 --> 00:45:34.380 can enjoy before the next incremental basketball provides 00:45:34.380 --> 00:45:36.240 relatively little pleasure for you. 00:45:36.240 --> 00:45:37.890 The same thing with ice cream sundaes. 00:45:37.890 --> 00:45:39.870 You can only consume so many ice cream sundaes 00:45:39.870 --> 00:45:42.990 before the next incremental sundae provides 00:45:42.990 --> 00:45:45.840 somewhat less benefit to you. 00:45:45.840 --> 00:45:48.480 That kind of diminishing marginal utility 00:45:48.480 --> 00:45:50.700 gives you this kind of a bowed curve. 00:45:50.700 --> 00:45:53.250 So where you are on this straight line 00:45:53.250 --> 00:45:56.070 depends upon how bowed your curve is. 00:45:56.070 --> 00:45:59.430 Somebody that's really risk-averse has a curve that 00:45:59.430 --> 00:45:59.930 looks-- 00:45:59.930 --> 00:46:01.971 let me draw this because it's a little bit easier 00:46:01.971 --> 00:46:04.230 to see rather than trying to follow my laser pointer. 00:46:07.890 --> 00:46:10.650 So here's the trade-off, this is the line. 00:46:10.650 --> 00:46:12.570 Somebody that's extremely risk-averse 00:46:12.570 --> 00:46:14.355 is going to have curves like this. 00:46:17.500 --> 00:46:19.310 Those are indifference curves. 00:46:19.310 --> 00:46:23.640 And as you go to the Northwest, you're happier and happier. 00:46:23.640 --> 00:46:28.140 So the optimal point is where this indifference curve 00:46:28.140 --> 00:46:31.970 hits this particular line. 00:46:31.970 --> 00:46:34.970 On the other hand, if you're very risk-seeking, 00:46:34.970 --> 00:46:36.980 if you don't need a lot of compensation 00:46:36.980 --> 00:46:39.205 of expected return per unit risk, 00:46:39.205 --> 00:46:41.580 then your difference curve's not going to look like that. 00:46:41.580 --> 00:46:45.420 It's going to look like this. 00:46:45.420 --> 00:46:50.970 In which case, you're tangency point will be farther 00:46:50.970 --> 00:46:53.620 to the Northeast. 00:46:53.620 --> 00:46:55.600 You'll be taking more risk and getting 00:46:55.600 --> 00:46:58.630 more expected rates of return. 00:46:58.630 --> 00:47:01.180 But the bottom line for this graph and this lecture 00:47:01.180 --> 00:47:05.395 is that everybody, no matter what your risk preferences are, 00:47:05.395 --> 00:47:07.270 everybody's going to want to be on that line, 00:47:07.270 --> 00:47:09.570 that tangency line. 00:47:09.570 --> 00:47:12.020 And it turns out that that insight 00:47:12.020 --> 00:47:16.070 is going to translate into a remarkable, remarkable 00:47:16.070 --> 00:47:21.140 conclusion about risk-reward trade-offs. 00:47:21.140 --> 00:47:23.590 So the key points for this lecture 00:47:23.590 --> 00:47:27.110 are diversification reduces risk. 00:47:27.110 --> 00:47:29.600 In diversified portfolios, covariances 00:47:29.600 --> 00:47:31.940 are the most important characteristics 00:47:31.940 --> 00:47:32.840 of that portfolio. 00:47:32.840 --> 00:47:37.150 It's not the variances, but the covariances. 00:47:37.150 --> 00:47:39.250 Investors should try to hold portfolios 00:47:39.250 --> 00:47:41.920 on the efficient frontier, that upper branch. 00:47:41.920 --> 00:47:45.160 And with the riskless asset, everybody 00:47:45.160 --> 00:47:48.790 is going to want to be on the tangency line. 00:47:48.790 --> 00:47:51.715 Those are the major conclusions from this analysis. 00:47:51.715 --> 00:47:53.590 And you can work all of this out analytically 00:47:53.590 --> 00:47:56.320 using the mathematics of optimization theory, 00:47:56.320 --> 00:47:59.080 but in fact, all of this can be done graphically 00:47:59.080 --> 00:48:00.935 as we have geometrically. 00:48:03.790 --> 00:48:05.071 Question, [INAUDIBLE]? 00:48:05.071 --> 00:48:07.487 AUDIENCE: Sorry, this is sort of a simple-minded question, 00:48:07.487 --> 00:48:12.928 but I'm having trouble thinking of the expected return. 00:48:12.928 --> 00:48:20.188 I know it's absolute, but so much of the portfolio 00:48:20.188 --> 00:48:24.060 metrics, I guess, are relative to the benchmark of the market 00:48:24.060 --> 00:48:25.512 rather than [INAUDIBLE]. 00:48:25.512 --> 00:48:26.480 So I don't know. 00:48:26.480 --> 00:48:28.360 ANDREW LO: OK, so we're going to get to that. 00:48:28.360 --> 00:48:30.485 We're going to talk about benchmarks because you're 00:48:30.485 --> 00:48:34.370 right, that most investments today are all benchmarked 00:48:34.370 --> 00:48:36.470 against something, right? 00:48:36.470 --> 00:48:39.830 And you're probably wondering how that got to be. 00:48:39.830 --> 00:48:45.810 That whole direction of analysis and performance attribution, 00:48:45.810 --> 00:48:47.805 that came out of this. 00:48:47.805 --> 00:48:49.180 In other words, it was because of 00:48:49.180 --> 00:48:51.040 this particular academic framework 00:48:51.040 --> 00:48:53.560 that was developed by Harry Markowitz, and Bill 00:48:53.560 --> 00:48:56.380 Sharpe, and others, that indexation 00:48:56.380 --> 00:48:58.494 and benchmarking came to be. 00:48:58.494 --> 00:48:59.660 So I'm going to get to that. 00:48:59.660 --> 00:49:02.050 Let me put that off for another lecture or so. 00:49:02.050 --> 00:49:05.260 After we derive the implications of everybody wanting 00:49:05.260 --> 00:49:07.210 to hold the tangency portfolio, it's 00:49:07.210 --> 00:49:10.510 going to turn out that that tangency portfolio happens 00:49:10.510 --> 00:49:11.740 to be the benchmark. 00:49:11.740 --> 00:49:13.020 So we'll get to that. 00:49:13.020 --> 00:49:14.639 Yeah? 00:49:14.639 --> 00:49:16.014 AUDIENCE: Is there any assumption 00:49:16.014 --> 00:49:19.402 behind all this analysis that would say, 00:49:19.402 --> 00:49:21.338 if you have these preferences, then you 00:49:21.338 --> 00:49:24.242 should choose this portfolio. 00:49:24.242 --> 00:49:28.320 If everyone did that, does that change [INAUDIBLE] 00:49:28.320 --> 00:49:30.980 ANDREW LO: So you would think that it would, but in fact, 00:49:30.980 --> 00:49:33.610 I'm going to show you that there is exactly one case where it 00:49:33.610 --> 00:49:34.167 doesn't. 00:49:34.167 --> 00:49:35.750 And that's the case of the equilibrium 00:49:35.750 --> 00:49:37.240 that I'm about describe. 00:49:37.240 --> 00:49:39.010 So let me turn to that right now. 00:49:39.010 --> 00:49:40.120 There any other questions? 00:49:40.120 --> 00:49:42.369 AUDIENCE: We've been using risk and standard deviation 00:49:42.369 --> 00:49:44.260 kind of interchangeably, whereas I 00:49:44.260 --> 00:49:47.438 think of risk as the risk of not making anything. 00:49:47.438 --> 00:49:49.188 Is there a way to mathematically translate 00:49:49.188 --> 00:49:50.987 from a standard deviation in your portfolio 00:49:50.987 --> 00:49:52.570 to the risk of not making [INAUDIBLE]? 00:49:52.570 --> 00:49:54.580 ANDREW LO: Well, there is. 00:49:54.580 --> 00:49:56.410 Although, I would have to say that if you 00:49:56.410 --> 00:50:01.010 have a preference about the downside, 00:50:01.010 --> 00:50:03.240 so not making anything as you point out. 00:50:03.240 --> 00:50:05.240 Then that changes this analysis. 00:50:05.240 --> 00:50:09.370 So this analysis really requires that you use standard deviation 00:50:09.370 --> 00:50:13.270 as the sum total of your perception 00:50:13.270 --> 00:50:15.310 of the risk of a portfolio. 00:50:15.310 --> 00:50:17.890 If you have other kinds of sensitivities, 00:50:17.890 --> 00:50:20.170 then you need to bring them into the analysis, 00:50:20.170 --> 00:50:22.668 and that will change these outcomes. 00:50:22.668 --> 00:50:25.720 AUDIENCE: The way we do this, if we 00:50:25.720 --> 00:50:30.998 measure the risk of a company, if they're historically-- 00:50:30.998 --> 00:50:31.962 Nevermind. 00:50:31.962 --> 00:50:34.372 I guess if they were historically, 00:50:34.372 --> 00:50:37.060 they varied higher, if that wasn't strictly normal, 00:50:37.060 --> 00:50:39.820 and they end up being higher in market [INAUDIBLE] lower, 00:50:39.820 --> 00:50:42.647 they would still have a larger deviation 00:50:42.647 --> 00:50:46.383 so you're correlating that with companies that are also 00:50:46.383 --> 00:50:47.784 [INAUDIBLE]. 00:50:47.784 --> 00:50:48.730 Does that make sens? 00:50:48.730 --> 00:50:50.438 ANDREW LO: That's true, but again, you've 00:50:50.438 --> 00:50:52.684 made an assumption there that I'm not making. 00:50:52.684 --> 00:50:54.850 Which is you're assuming companies are outperforming 00:50:54.850 --> 00:50:56.230 or underperforming. 00:50:56.230 --> 00:50:58.300 I'm assuming that the data are given, 00:50:58.300 --> 00:51:00.790 and I'm not making a bet on whether any companies are 00:51:00.790 --> 00:51:02.830 likely to succeed or fail. 00:51:02.830 --> 00:51:06.040 I'm merely looking at companies as investment opportunities 00:51:06.040 --> 00:51:08.410 that provide certain expected returns, 00:51:08.410 --> 00:51:10.682 volatilities, and covariances. 00:51:10.682 --> 00:51:12.640 You want to go down the path of Warren Buffett, 00:51:12.640 --> 00:51:14.290 and I'm resisting that because I don't have 00:51:14.290 --> 00:51:15.581 the skills of a Warren Buffett. 00:51:15.581 --> 00:51:17.200 So I don't know what's a good value 00:51:17.200 --> 00:51:18.820 and what's not a good value. 00:51:18.820 --> 00:51:21.180 And the case in point is a discussion we just had today. 00:51:21.180 --> 00:51:22.805 You tell me what is a good value today? 00:51:22.805 --> 00:51:25.300 Do you really believe that Campbell's Soup or Wal-Mart 00:51:25.300 --> 00:51:27.370 should be the companies you invest in today? 00:51:27.370 --> 00:51:28.171 I don't know. 00:51:28.171 --> 00:51:29.920 I mean, another argument is entertainment. 00:51:29.920 --> 00:51:31.300 Why don't you invest in movie theaters? 00:51:31.300 --> 00:51:33.730 Lots of people now are going to see the James Bond movie, 00:51:33.730 --> 00:51:35.362 and they want to escape from reality. 00:51:35.362 --> 00:51:36.820 Wouldn't that be a growth industry, 00:51:36.820 --> 00:51:37.780 given market conditions? 00:51:37.780 --> 00:51:39.363 Well, that's true, but how many people 00:51:39.363 --> 00:51:40.594 have $12 to spend on a movie? 00:51:40.594 --> 00:51:42.760 Plus, you've got to get the popcorn and the bonbons, 00:51:42.760 --> 00:51:43.343 and all those. 00:51:43.343 --> 00:51:46.570 And by the time you're done, it's like a $60 evening. 00:51:46.570 --> 00:51:47.770 I mean, I don't know. 00:51:47.770 --> 00:51:49.720 So the point is that unless you are 00:51:49.720 --> 00:51:52.630 willing to make predictions, this 00:51:52.630 --> 00:51:55.960 is the only alternative that provides a disciplined 00:51:55.960 --> 00:52:01.630 approach to investing in so-called good portfolios. 00:52:01.630 --> 00:52:05.690 So it's a different approach. 00:52:05.690 --> 00:52:10.220 So now, let me turn to the next lectures. 00:52:10.220 --> 00:52:13.040 Lectures 15 through 17, where we're now going 00:52:13.040 --> 00:52:15.480 to talk about equilibrium. 00:52:15.480 --> 00:52:18.800 We've already identified that all of us in this room, 00:52:18.800 --> 00:52:22.370 assuming we have mean variance preferences, that's 00:52:22.370 --> 00:52:24.800 an important assumption, I grant you, 00:52:24.800 --> 00:52:26.420 but it's not an unreasonable one. 00:52:26.420 --> 00:52:29.895 It's just, it is an important assumption. 00:52:29.895 --> 00:52:31.270 We've all agreed that we're going 00:52:31.270 --> 00:52:37.900 to take on portfolios that lie on that line, and therefore, 00:52:37.900 --> 00:52:41.470 the portfolio that is the tangency portfolio, 00:52:41.470 --> 00:52:43.930 I'm going to give it a special name. 00:52:43.930 --> 00:52:50.990 I'm going to call it M, portfolio M. What we now 00:52:50.990 --> 00:52:56.780 know is that, given a choice between holding n 00:52:56.780 --> 00:53:04.940 securities and T-Bills, versus holding 00:53:04.940 --> 00:53:11.030 T-Bills and a single portfolio, all of you 00:53:11.030 --> 00:53:13.710 would be indifferent between those two choices, 00:53:13.710 --> 00:53:21.470 if that single portfolio were M. The tangency portfolio. 00:53:21.470 --> 00:53:24.270 Do we agree on that? 00:53:24.270 --> 00:53:28.230 So therefore, I could, in principle, 00:53:28.230 --> 00:53:35.110 construct a mutual fund called M. This mutual fund holds 00:53:35.110 --> 00:53:39.610 stocks in exact proportion to the weights given 00:53:39.610 --> 00:53:41.740 by that tangency portfolio. 00:53:41.740 --> 00:53:46.800 In other words, it is the tangency portfolio. 00:53:46.800 --> 00:53:51.730 So what that suggests is that all of you in this room 00:53:51.730 --> 00:53:57.010 would be absolutely indifferent between investing among the n 00:53:57.010 --> 00:53:59.470 stocks and T-Bills on the one hand, 00:53:59.470 --> 00:54:02.530 versus investing in two securities on the other. 00:54:02.530 --> 00:54:05.380 One security is T-Bills, and the other security 00:54:05.380 --> 00:54:12.560 is shares of mutual fund M. Do we agree on that? 00:54:12.560 --> 00:54:14.540 Any controversy there? 00:54:14.540 --> 00:54:17.480 I know I've made a number of assumptions to get us here, 00:54:17.480 --> 00:54:20.510 but given mean variance preferences, which is not 00:54:20.510 --> 00:54:22.460 an unreasonable assumption, and given 00:54:22.460 --> 00:54:24.830 that we've assumed these parameters are 00:54:24.830 --> 00:54:27.516 stable over time, that's where we are. 00:54:27.516 --> 00:54:28.016 Rami? 00:54:28.016 --> 00:54:29.682 AUDIENCE: Somebody might have said this, 00:54:29.682 --> 00:54:33.402 but you assume all fees are trading fees? 00:54:33.402 --> 00:54:34.610 ANDREW LO: Forget about fees. 00:54:34.610 --> 00:54:36.239 There are fees no matter what you do. 00:54:36.239 --> 00:54:38.030 So for now, I'm going to forget about fees. 00:54:38.030 --> 00:54:40.817 I'll put fees back in later, and if I do that, 00:54:40.817 --> 00:54:42.650 then it's going to look even more compelling 00:54:42.650 --> 00:54:47.870 for you to want to invest in mutual fund M, versus n stocks. 00:54:47.870 --> 00:54:50.700 I don't know how many of you have traded individual stocks, 00:54:50.700 --> 00:54:54.620 but if you ever try to manage a portfolio of 1,000 stocks, 00:54:54.620 --> 00:54:58.092 it's actually fairly time consuming, right? 00:54:58.092 --> 00:55:00.300 And by the way, there are more than 1,000 securities. 00:55:00.300 --> 00:55:03.890 I mean the S&P 500 you can think of as being M, 00:55:03.890 --> 00:55:05.360 but that's an approximation, right? 00:55:05.360 --> 00:55:08.480 There's probably 7,000 or 8,000 securities that trade today. 00:55:08.480 --> 00:55:10.910 Probably only 2,000 or 3,000 that you would really 00:55:10.910 --> 00:55:13.520 take seriously, and probably only 1,500 00:55:13.520 --> 00:55:16.530 that you really need from a diversification perspective. 00:55:16.530 --> 00:55:17.676 1,500 stocks. 00:55:17.676 --> 00:55:19.550 Would you want to trade in that, or would you 00:55:19.550 --> 00:55:21.838 want to trade in one mutual fund? 00:55:21.838 --> 00:55:22.338 Yeah? 00:55:22.338 --> 00:55:26.024 AUDIENCE: Can I ask, I mean knowing that with M you're 00:55:26.024 --> 00:55:30.340 trying to get on that tangent portfolio? 00:55:30.340 --> 00:55:32.232 And you said, for example, Warren Buffet 00:55:32.232 --> 00:55:33.232 beats it the whole time. 00:55:33.232 --> 00:55:35.549 Why don't you just buy one share of Berkshire Hathaway, 00:55:35.549 --> 00:55:37.090 and you'd have a higher Sharpe ratio? 00:55:37.090 --> 00:55:38.740 ANDREW LO: Because Warren Buffett beat it in the past, 00:55:38.740 --> 00:55:40.786 do you think he's going to beat it in the future? 00:55:40.786 --> 00:55:42.160 AUDIENCE: I would [INAUDIBLE] it. 00:55:42.160 --> 00:55:43.160 ANDREW LO: I don't know. 00:55:43.160 --> 00:55:43.960 That's right. 00:55:43.960 --> 00:55:45.181 Good question, good question. 00:55:45.181 --> 00:55:47.680 I mean, if you're thinking about Warren Buffett as a 10 year 00:55:47.680 --> 00:55:51.190 investment, I think I might short that. 00:55:51.190 --> 00:55:53.140 I mean, you know, he seems healthy, 00:55:53.140 --> 00:55:56.950 but you know those Cherry Cokes have to have an impact. 00:55:56.950 --> 00:55:58.340 I'm sorry. 00:55:58.340 --> 00:56:01.180 You know, you eat enough steaks at that Omaha restaurant, 00:56:01.180 --> 00:56:03.250 I don't know what it is, and those Cherry Cokes, 00:56:03.250 --> 00:56:06.250 I don't know. 00:56:06.250 --> 00:56:06.824 OK, so fine. 00:56:06.824 --> 00:56:08.990 Let's not do Warren Buffett, let's do somebody else. 00:56:08.990 --> 00:56:09.490 Fine. 00:56:09.490 --> 00:56:11.435 You tell me who that is? 00:56:11.435 --> 00:56:13.060 Tell me who the next Warren Buffett is? 00:56:13.060 --> 00:56:14.770 Can anybody tell me? 00:56:14.770 --> 00:56:17.260 I'll be happy to do that, I'll be happy to invest in them. 00:56:17.260 --> 00:56:18.397 Who is it? 00:56:18.397 --> 00:56:21.260 AUDIENCE: Andrew Lo. 00:56:21.260 --> 00:56:24.550 ANDREW LO: Thank you, but those who can't do teach, 00:56:24.550 --> 00:56:26.130 those who can't teach, teach gym. 00:56:26.130 --> 00:56:28.600 And at least I don't teach gym. 00:56:31.360 --> 00:56:33.504 The point is that we don't know who the next Warren 00:56:33.504 --> 00:56:35.170 Buffett is going to be, and I don't want 00:56:35.170 --> 00:56:36.130 to have to figure that out. 00:56:36.130 --> 00:56:37.546 I mean, that's a pretty tall order 00:56:37.546 --> 00:56:39.910 to tell an investor that they've got to figure out who 00:56:39.910 --> 00:56:41.410 the next investment genius is. 00:56:41.410 --> 00:56:44.140 If they knew, they wouldn't have to ask them to invest. 00:56:44.140 --> 00:56:46.210 They'd invest themselves, right? 00:56:46.210 --> 00:56:49.540 So what I'm showing you is a simple way of investing 00:56:49.540 --> 00:56:51.389 that may not be as good as Warren Buffett, 00:56:51.389 --> 00:56:52.930 but it's certainly better than trying 00:56:52.930 --> 00:56:54.160 to pick the next Warren Buffett if you 00:56:54.160 --> 00:56:55.060 don't know what you're doing. 00:56:55.060 --> 00:56:55.745 Jen? 00:56:55.745 --> 00:56:58.357 AUDIENCE: Is it easier to kind of figure 00:56:58.357 --> 00:57:01.682 out the future covariances of the different than it 00:57:01.682 --> 00:57:03.774 is to pick the next Warren Buffet? 00:57:03.774 --> 00:57:06.190 ANDREW LO: Thank you, that's another way of looking at it. 00:57:06.190 --> 00:57:10.140 If you ask the question, is it easier to try-- 00:57:10.140 --> 00:57:13.650 is the historical covariances and variances 00:57:13.650 --> 00:57:16.920 and expected returns more predictive of the future 00:57:16.920 --> 00:57:20.320 than your ability to find the next Warren Buffett, 00:57:20.320 --> 00:57:22.030 then yes, that's another good argument. 00:57:22.030 --> 00:57:23.488 That in other words, this framework 00:57:23.488 --> 00:57:29.340 relies on less ability to forecast. 00:57:29.340 --> 00:57:32.130 It doesn't completely rule it out because, as I said, 00:57:32.130 --> 00:57:34.184 these parameters, they change over time. 00:57:34.184 --> 00:57:35.850 And you have to think about that impact. 00:57:35.850 --> 00:57:38.700 So it's not totally trivial, but from 00:57:38.700 --> 00:57:42.210 the theoretical perspective, it seems 00:57:42.210 --> 00:57:45.490 like it's a very internally consistent approach. 00:57:45.490 --> 00:57:47.160 Now, let me go on for a little while 00:57:47.160 --> 00:57:50.389 longer because if it were just this, then 00:57:50.389 --> 00:57:52.180 this would be an interesting rule of thumb. 00:57:52.180 --> 00:57:55.920 But this is not a theory of financial markets just yet. 00:57:55.920 --> 00:58:00.130 I haven't really done anything truly astounding 00:58:00.130 --> 00:58:02.400 because you're still left with the question of, 00:58:02.400 --> 00:58:04.770 what's the appropriate risk-reward trade-off? 00:58:04.770 --> 00:58:07.650 What should I use for my discount rate? 00:58:07.650 --> 00:58:09.319 A lot of financial decision making 00:58:09.319 --> 00:58:11.610 is not just picking stocks and making good investments. 00:58:11.610 --> 00:58:14.640 But it's whether or not should I invest in nanotechnology 00:58:14.640 --> 00:58:18.780 as a corporate officer of a particular tech company, 00:58:18.780 --> 00:58:21.130 or should I invest in green technologies? 00:58:21.130 --> 00:58:22.707 What discount rate should I use? 00:58:22.707 --> 00:58:25.290 How should I engage in capital budgeting or project financing? 00:58:25.290 --> 00:58:28.050 All of these questions seem like they have nothing 00:58:28.050 --> 00:58:29.009 to do with investments. 00:58:29.009 --> 00:58:31.299 So I don't want to make this course into an investments 00:58:31.299 --> 00:58:31.877 course. 00:58:31.877 --> 00:58:33.960 There's a lot about corporate financial management 00:58:33.960 --> 00:58:38.160 that relies on being able to understand these markets. 00:58:38.160 --> 00:58:40.080 So let me show you where we go next, 00:58:40.080 --> 00:58:44.340 because we're very close now to the big payoff. 00:58:44.340 --> 00:58:46.320 We've already identified the tangency portfolio 00:58:46.320 --> 00:58:47.490 as being special. 00:58:47.490 --> 00:58:49.450 I'm going to call that portfolio M, 00:58:49.450 --> 00:58:52.200 and I'm going to argue that everybody in their right minds 00:58:52.200 --> 00:58:55.560 are going to be indifferent between picking among these two 00:58:55.560 --> 00:58:57.960 investment opportunities, T-Bills and M, 00:58:57.960 --> 00:59:02.340 versus the n plus 1 investment opportunities of all stocks, 00:59:02.340 --> 00:59:04.800 plus T-Bills. 00:59:04.800 --> 00:59:10.170 It turns out that portfolio M, therefore, 00:59:10.170 --> 00:59:14.290 has to be a very specific portfolio. 00:59:17.370 --> 00:59:20.510 And it turns out that that portfolio 00:59:20.510 --> 00:59:28.080 is the portfolio of all assets in the entire economy, 00:59:28.080 --> 00:59:32.990 in proportion to their market capitalizations. 00:59:32.990 --> 00:59:36.140 Now what I just said is an incredibly deep result, 00:59:36.140 --> 00:59:39.010 so I don't expect you to just get it. 00:59:39.010 --> 00:59:40.000 Let me say it again. 00:59:40.000 --> 00:59:42.170 First of all, I want you to understand it, 00:59:42.170 --> 00:59:44.628 and then I'm going to try to give you the intuition for it. 00:59:46.810 --> 00:59:51.130 If it's true that everybody, not only in this room, 00:59:51.130 --> 00:59:55.520 but in the world, if everybody in the world 00:59:55.520 --> 01:00:00.110 is indifferent between investing in those n plus 1 securities, 01:00:00.110 --> 01:00:09.350 and in two, then we can argue that those two securities 01:00:09.350 --> 01:00:11.630 play a very special role. 01:00:11.630 --> 01:00:18.100 In particular, think about what that mutual fund M has to be. 01:00:18.100 --> 01:00:22.670 Everybody in the world wants to hold M. 01:00:22.670 --> 01:00:27.344 So, let's make the leap of faith that everybody does hold M. So 01:00:27.344 --> 01:00:29.510 in other words, now we're in a world where everybody 01:00:29.510 --> 01:00:33.290 is already mean variance optimizers, 01:00:33.290 --> 01:00:39.350 and they already hold two assets in their portfolio. 01:00:39.350 --> 01:00:44.440 The treasury bill asset, and the mutual fund M. So you hold M, 01:00:44.440 --> 01:00:47.100 you hold M, you hold M, you hold M, you hold M, 01:00:47.100 --> 01:00:51.490 everybody holds M. We hold different amounts of it, 01:00:51.490 --> 01:00:53.680 so as a hedge fund manager, you're 01:00:53.680 --> 01:00:56.170 holding a large amount of M. In fact, you're 01:00:56.170 --> 01:00:59.350 holding twice as much M as your wealth allows, 01:00:59.350 --> 01:01:01.582 and you're borrowing T-Bills to do so. 01:01:01.582 --> 01:01:03.040 Somebody who's very conservative is 01:01:03.040 --> 01:01:05.140 holding a very tiny little bit of M. 01:01:05.140 --> 01:01:08.770 Mostly, that person is invested in T-Bills. 01:01:08.770 --> 01:01:11.410 But the point is that every single person's portfolio 01:01:11.410 --> 01:01:13.630 you look at, when you look at their portfolio, 01:01:13.630 --> 01:01:20.940 it's M. If that's true, if what I just said is true, 01:01:20.940 --> 01:01:24.610 what portfolio does M have to be? 01:01:24.610 --> 01:01:27.670 There's only one that it can possibly be. 01:01:27.670 --> 01:01:31.960 And that is the portfolio of all equities 01:01:31.960 --> 01:01:36.530 in the marketplace, held in proportion to their market 01:01:36.530 --> 01:01:38.410 value. 01:01:38.410 --> 01:01:40.570 Do you see the beauty of that? 01:01:40.570 --> 01:01:42.520 Now, let me try to explain it. 01:01:42.520 --> 01:01:45.220 I hope you understand it, let me explain it. 01:01:45.220 --> 01:01:47.050 Why does that have to be? 01:01:47.050 --> 01:01:48.970 This has to do with supply equaling demand. 01:01:48.970 --> 01:01:51.460 Now, I'm going to make an argument about equilibrium. 01:01:51.460 --> 01:01:53.369 I haven't done so up until now. 01:01:53.369 --> 01:01:54.910 Up until now, I haven't said anything 01:01:54.910 --> 01:01:59.540 about supply equaling demand, but I'm about to do so. 01:01:59.540 --> 01:02:03.250 If everybody is holding this portfolio M, 01:02:03.250 --> 01:02:04.990 that's the demand side, right? 01:02:04.990 --> 01:02:10.810 Everybody is demanding M. On the supply side, 01:02:10.810 --> 01:02:16.460 I'm assuming that all stocks that are being supplied 01:02:16.460 --> 01:02:17.780 are held. 01:02:17.780 --> 01:02:23.410 If all stocks that are being supplied are held by somebody, 01:02:23.410 --> 01:02:27.490 but if everybody in the world is holding the same portfolio 01:02:27.490 --> 01:02:31.044 M, when you aggregate all of the demands. 01:02:31.044 --> 01:02:33.210 So I'm going to add up your demand, and your demand, 01:02:33.210 --> 01:02:33.940 and your demand, and you're. 01:02:33.940 --> 01:02:34.910 We're going to go through the class, 01:02:34.910 --> 01:02:36.451 and go through the world, we're going 01:02:36.451 --> 01:02:38.070 to add up everybody's demand. 01:02:38.070 --> 01:02:43.690 In every single case, your weights are identical. 01:02:43.690 --> 01:02:46.400 You're holding the same portfolio M. 01:02:46.400 --> 01:02:49.200 So when I aggregate the entire world, 01:02:49.200 --> 01:02:53.900 and I get the portfolio M, what does it have to equal? 01:02:53.900 --> 01:02:57.740 It can only equal the sum total of all assets in the world, 01:02:57.740 --> 01:02:58.370 right? 01:02:58.370 --> 01:03:01.800 Supply equals demand. 01:03:01.800 --> 01:03:06.120 And therefore, when I aggregate all of your holdings of M 01:03:06.120 --> 01:03:09.780 into one big fat M, that big fat M 01:03:09.780 --> 01:03:11.700 can only be equal to one thing, which is 01:03:11.700 --> 01:03:14.530 all the equities in the world. 01:03:14.530 --> 01:03:18.231 And the weightings are just simply their market caps, 01:03:18.231 --> 01:03:18.730 right? 01:03:18.730 --> 01:03:20.740 There's only so much of General Motors. 01:03:20.740 --> 01:03:22.640 Take the entire sum total of that, 01:03:22.640 --> 01:03:25.690 that's the global investment in General Motors. 01:03:25.690 --> 01:03:27.670 And then you do that for every single stock, 01:03:27.670 --> 01:03:33.260 and you divide that by the total market capital of all stocks, 01:03:33.260 --> 01:03:37.240 you get the market portfolio, M. 01:03:37.240 --> 01:03:43.680 So this shockingly, simple, but extraordinarily powerful 01:03:43.680 --> 01:03:46.670 result is due to Bill Sharpe. 01:03:46.670 --> 01:03:50.310 Harry Markowitz came up with portfolio optimization. 01:03:50.310 --> 01:03:52.380 He applied mean variance analysis 01:03:52.380 --> 01:03:55.094 to portfolio optimization and argued that everybody 01:03:55.094 --> 01:03:56.010 has to be on the line. 01:03:56.010 --> 01:03:59.030 Bill Sharpe looked at this and said, aha. 01:03:59.030 --> 01:04:00.619 If everybody's on that line, that 01:04:00.619 --> 01:04:02.660 means that everybody's going to be either holding 01:04:02.660 --> 01:04:06.050 M or T-Bills, or both, and therefore, 01:04:06.050 --> 01:04:09.260 the only thing that M could possibly be 01:04:09.260 --> 01:04:11.680 is the market portfolio. 01:04:11.680 --> 01:04:14.620 And now we have a proxy for the market portfolio, the Russell 01:04:14.620 --> 01:04:15.700 2000. 01:04:15.700 --> 01:04:17.320 Or the S&P 500. 01:04:17.320 --> 01:04:19.944 Both of those are very well diversified stock that have 01:04:19.944 --> 01:04:21.610 lot-- they don't have everything in it-- 01:04:21.610 --> 01:04:24.950 but they have a lot of things in it, that proxy for everything. 01:04:24.950 --> 01:04:29.770 The Russell 2000 has 2,000 stocks weighted by market cap. 01:04:29.770 --> 01:04:33.310 That's as close as you're going to get to everything that you 01:04:33.310 --> 01:04:35.712 care about. 01:04:35.712 --> 01:04:37.920 So now, you'll see we're benchmarking is coming from, 01:04:37.920 --> 01:04:40.290 but I'm going to get back to that in more detail. 01:04:40.290 --> 01:04:47.380 So this equilibrium result that says supply equals demand, 01:04:47.380 --> 01:04:52.030 identifies this portfolio M. And what it says 01:04:52.030 --> 01:04:56.590 is that if everybody does this, if everybody takes finance 01:04:56.590 --> 01:04:59.980 here and learns how to do this, it's 01:04:59.980 --> 01:05:01.810 not going to kill the idea. 01:05:01.810 --> 01:05:05.320 It's going to lead to a very well-defined portfolio 01:05:05.320 --> 01:05:09.445 M. Now, let me take it one step farther, 01:05:09.445 --> 01:05:13.830 then I want to ask you to ask questions. 01:05:13.830 --> 01:05:16.940 If I know what that portfolio M is, 01:05:16.940 --> 01:05:20.945 then I've got an equation for this line. 01:05:23.840 --> 01:05:28.040 I can write down a relationship between the expected return 01:05:28.040 --> 01:05:31.280 and risk of a portfolio on this line. 01:05:31.280 --> 01:05:33.190 And this is it. 01:05:33.190 --> 01:05:36.700 The expected rate of return of an efficient portfolio, 01:05:36.700 --> 01:05:39.640 by efficient I mean a portfolio that's on that line. 01:05:39.640 --> 01:05:43.100 Anything that's not on that line, if it's below that line, 01:05:43.100 --> 01:05:45.470 it's inefficient, right? 01:05:45.470 --> 01:05:47.870 You're not getting as much expected return per unit risk, 01:05:47.870 --> 01:05:50.780 and you're not reducing your risk as much as you can, 01:05:50.780 --> 01:05:52.880 per unit of expected return. 01:05:52.880 --> 01:05:55.430 The expected return of an efficient portfolio 01:05:55.430 --> 01:06:01.710 is equal to the risk-free rate, plus the ratio 01:06:01.710 --> 01:06:05.610 of the standard deviation of that portfolio, divided 01:06:05.610 --> 01:06:09.180 by the standard deviation of the tangency portfolio, 01:06:09.180 --> 01:06:15.120 or the market, multiplied by the excess return of the market 01:06:15.120 --> 01:06:15.780 portfolio. 01:06:20.360 --> 01:06:28.340 This result is a risk-reward trade-off between risk 01:06:28.340 --> 01:06:30.610 and expected return. 01:06:30.610 --> 01:06:34.090 You see, what it says is really something quite astounding. 01:06:34.090 --> 01:06:37.060 It's telling you that, here's the risk-free rate. 01:06:37.060 --> 01:06:40.630 That's the base return for your portfolio. 01:06:40.630 --> 01:06:42.520 And what this is telling you is that what 01:06:42.520 --> 01:06:45.610 you should expect for your portfolio 01:06:45.610 --> 01:06:50.650 is that base return, plus something extra. 01:06:50.650 --> 01:06:55.800 And the extra is the market's excess return, 01:06:55.800 --> 01:06:58.650 multiplied by a factor. 01:06:58.650 --> 01:07:00.930 And the factor is simply how risky your portfolio 01:07:00.930 --> 01:07:03.990 is relative to the market. 01:07:03.990 --> 01:07:05.290 Let's do a simple example. 01:07:05.290 --> 01:07:09.960 Suppose that your portfolio is the exact same risk 01:07:09.960 --> 01:07:11.912 as the market. 01:07:11.912 --> 01:07:13.370 Well, if that's the case, then what 01:07:13.370 --> 01:07:15.216 is your expected rate of return? 01:07:15.216 --> 01:07:16.150 AUDIENCE: The market. 01:07:16.150 --> 01:07:17.274 ANDREW LO: It's the market. 01:07:17.274 --> 01:07:21.750 So it's the risk-free rate, plus the market excess return, 01:07:21.750 --> 01:07:24.122 which, when you add it together, is just the market. 01:07:24.122 --> 01:07:25.830 Suppose you're holding a portfolio that's 01:07:25.830 --> 01:07:29.040 more risky than the market. 01:07:29.040 --> 01:07:30.900 Is your rate of return greater or less 01:07:30.900 --> 01:07:32.070 than the rate of return of the market? 01:07:32.070 --> 01:07:32.760 AUDIENCE: Greater. 01:07:32.760 --> 01:07:33.551 ANDREW LO: Greater. 01:07:33.551 --> 01:07:36.570 Suppose that your portfolio has no risk. 01:07:36.570 --> 01:07:39.476 Suppose that sigma p is 0, then what's your rate of return? 01:07:39.476 --> 01:07:40.350 AUDIENCE: [INAUDIBLE] 01:07:40.350 --> 01:07:41.310 ANDREW LO: Exactly. 01:07:41.310 --> 01:07:42.270 Makes sense, right? 01:07:42.270 --> 01:07:44.370 This is very intuitive. 01:07:44.370 --> 01:07:48.000 What this tells us, now, is that we can figure out 01:07:48.000 --> 01:07:52.170 what the fair rate of return is for an efficient portfolio. 01:07:52.170 --> 01:07:54.420 For any portfolio on this line, I 01:07:54.420 --> 01:07:56.400 can tell you what my fair rate of return is, 01:07:56.400 --> 01:07:58.620 and it's an objective measure. 01:07:58.620 --> 01:08:00.060 It's not just theory now. 01:08:00.060 --> 01:08:01.551 Now, I can go into the marketplace, 01:08:01.551 --> 01:08:03.550 I can measure the expected return of the market. 01:08:03.550 --> 01:08:05.050 You know what that is, historically? 01:08:05.050 --> 01:08:06.945 Not including the last few months. 01:08:06.945 --> 01:08:08.126 AUDIENCE: 7%. 01:08:08.126 --> 01:08:09.750 ANDREW LO: It's about 7%, historically. 01:08:09.750 --> 01:08:12.180 Over the last 100 years, 7%, the expected 01:08:12.180 --> 01:08:14.730 rate of return of the market. 01:08:14.730 --> 01:08:16.316 Sorry, the expected risk premium, 01:08:16.316 --> 01:08:17.399 the excess rate of return. 01:08:17.399 --> 01:08:18.414 About 7%. 01:08:18.414 --> 01:08:20.080 What about the volatility of the market? 01:08:20.080 --> 01:08:22.229 It's been about 15% historically. 01:08:22.229 --> 01:08:24.899 So according to this relationship, 01:08:24.899 --> 01:08:26.910 I've already figured out what this number is. 01:08:26.910 --> 01:08:28.036 It's like 7%. 01:08:28.036 --> 01:08:29.910 I've already figured out what this number is. 01:08:29.910 --> 01:08:31.410 It's like 15%. 01:08:31.410 --> 01:08:34.890 So now, you should be able to get a benchmark for what 01:08:34.890 --> 01:08:38.670 to expect when you've got a particular level of risk 01:08:38.670 --> 01:08:41.069 in an efficient portfolio. 01:08:41.069 --> 01:08:42.510 You've got all the ingredients. 01:08:42.510 --> 01:08:43.593 What about risk-free rate? 01:08:43.593 --> 01:08:45.450 Well, it depends on what risk-free rate, 01:08:45.450 --> 01:08:47.460 but let's talk about over a one year period. 01:08:47.460 --> 01:08:48.960 Right now we're looking at somewhere 01:08:48.960 --> 01:08:52.109 between, I don't know, 1%, 2%, 3%, 01:08:52.109 --> 01:08:55.290 depending on what day of the week you're looking at. 01:08:55.290 --> 01:08:58.938 So one year T-Bill rate is about 1% or so, yeah? 01:08:58.938 --> 01:09:00.688 AUDIENCE: I think it was the last class we 01:09:00.688 --> 01:09:02.064 talked about unsystematic risk. 01:09:02.064 --> 01:09:02.689 ANDREW LO: Yes. 01:09:02.689 --> 01:09:06.870 AUDIENCE: Is that defined by [INAUDIBLE] in this case? 01:09:06.870 --> 01:09:08.580 ANDREW LO: No, the unsystematic risk 01:09:08.580 --> 01:09:10.890 is risk that is not measured by sigma p, 01:09:10.890 --> 01:09:12.960 so we're going to come back to that. 01:09:12.960 --> 01:09:14.729 Let me hold off on that for now, because I 01:09:14.729 --> 01:09:16.930 want to come back to it after I finish developing this. 01:09:16.930 --> 01:09:18.626 There's going to be a connection between 01:09:18.626 --> 01:09:20.250 systematic and unsystematic risk that's 01:09:20.250 --> 01:09:22.990 going to come right out of this relationship. 01:09:22.990 --> 01:09:23.490 Yeah, Brian? 01:09:23.490 --> 01:09:28.310 AUDIENCE: So if you take the S&P 500 as M here, 01:09:28.310 --> 01:09:31.640 the market portfolio, and the capitalization is the weight, 01:09:31.640 --> 01:09:34.855 so you've got non-zero weights for all the different stocks 01:09:34.855 --> 01:09:35.430 there. 01:09:35.430 --> 01:09:38.760 Does that imply that there's no stocks in the S&P 500 01:09:38.760 --> 01:09:41.160 that are Southeast of any others? 01:09:41.160 --> 01:09:42.819 ANDREW LO: No, no, there could be. 01:09:42.819 --> 01:09:43.630 AUDIENCE: Why would you have them, 01:09:43.630 --> 01:09:45.671 because we said those are strictly non-preferred? 01:09:45.671 --> 01:09:47.910 ANDREW LO: Well, that's if you're looking 01:09:47.910 --> 01:09:49.800 at a pairwise comparison. 01:09:49.800 --> 01:09:52.200 If, now, you're trying to create an entire collection 01:09:52.200 --> 01:09:53.850 of these portfolios of securities, 01:09:53.850 --> 01:09:55.129 that's a different story. 01:09:55.129 --> 01:09:57.420 That's why I answered in response to Justin's question. 01:09:57.420 --> 01:09:59.850 Justin said, why not just trade off those two? 01:09:59.850 --> 01:10:00.380 Why not? 01:10:00.380 --> 01:10:03.090 It's because you can do far better by using all of them 01:10:03.090 --> 01:10:04.290 in this way. 01:10:04.290 --> 01:10:07.200 You see, by looking at pairwise, you can no doubt do better. 01:10:07.200 --> 01:10:10.644 But if I use all of them, I get this entire line. 01:10:10.644 --> 01:10:12.060 And you can't get that entire line 01:10:12.060 --> 01:10:13.780 just from looking at two of these stocks, 01:10:13.780 --> 01:10:14.580 you need all of them. 01:10:14.580 --> 01:10:16.250 AUDIENCE: So in this portfolio of three, 01:10:16.250 --> 01:10:18.520 if you kick GM over to the right a little bit, 01:10:18.520 --> 01:10:20.530 and made it strictly non-preferred to IBM, 01:10:20.530 --> 01:10:24.280 then you still might have a positive portfolio 01:10:24.280 --> 01:10:25.560 weight on GM? 01:10:25.560 --> 01:10:27.360 ANDREW LO: You might, but more likely, 01:10:27.360 --> 01:10:28.917 it'll be a negative portfolio weight. 01:10:28.917 --> 01:10:30.750 It'll be negative, and you'll be shorting it 01:10:30.750 --> 01:10:32.750 somewhere along the line here. 01:10:32.750 --> 01:10:36.930 However, the tangency portfolio, by assumption, 01:10:36.930 --> 01:10:41.490 if it's the market portfolio, cannot have negative weights. 01:10:41.490 --> 01:10:45.570 And so there, what will happen, is that all of the stocks 01:10:45.570 --> 01:10:47.580 will change in their relationship 01:10:47.580 --> 01:10:50.230 based upon various different kinds of equilibrium, 01:10:50.230 --> 01:10:52.440 so that you won't get into a lot of those situations 01:10:52.440 --> 01:10:55.300 where you're going to be shorting these negative stocks. 01:10:55.300 --> 01:10:55.800 Yeah? 01:10:55.800 --> 01:10:59.132 AUDIENCE: So basically, according to the Sharpe theory, 01:10:59.132 --> 01:11:04.380 every stock that the market, the capital is not 0 01:11:04.380 --> 01:11:06.500 is worth holding in some portfolio? 01:11:06.500 --> 01:11:07.500 ANDREW LO: That's right. 01:11:07.500 --> 01:11:09.030 AUDIENCE: Diversifying your portfolio. 01:11:09.030 --> 01:11:10.030 ANDREW LO: That's right. 01:11:10.030 --> 01:11:12.990 Every stock has some benefit in adding 01:11:12.990 --> 01:11:15.570 to this particular risk-reward trade-off, 01:11:15.570 --> 01:11:19.660 and the sum total benefit is summarized by this line. 01:11:19.660 --> 01:11:20.910 That's the ultimate objective. 01:11:20.910 --> 01:11:23.390 AUDIENCE: If I don't hold a specific stock in the market 01:11:23.390 --> 01:11:26.502 and I gain a diversification [INAUDIBLE]? 01:11:26.502 --> 01:11:27.210 ANDREW LO: Sorry? 01:11:27.210 --> 01:11:29.864 If you hold a specific stock? 01:11:29.864 --> 01:11:31.530 AUDIENCE: If I don't hold it, because it 01:11:31.530 --> 01:11:32.510 has market [INAUDIBLE]. 01:11:32.510 --> 01:11:34.370 ANDREW LO: Oh, if you put 0 weight. 01:11:34.370 --> 01:11:35.640 Yes. 01:11:35.640 --> 01:11:38.730 What Sharpe would argue, based upon this theory, 01:11:38.730 --> 01:11:41.190 is that you want to hold as many stocks 01:11:41.190 --> 01:11:43.770 as you can to get the most diversification. 01:11:43.770 --> 01:11:45.210 Now, that's the theory. 01:11:45.210 --> 01:11:49.410 In practice, it may well be that the benefits do not 01:11:49.410 --> 01:11:51.960 outweigh the costs, because when you hold multiple stocks, 01:11:51.960 --> 01:11:54.900 you have to manage them, and so it may cost more. 01:11:54.900 --> 01:11:57.330 So a mutual fund that has 3,000 stocks 01:11:57.330 --> 01:12:01.080 may have a higher expense ratio than a mutual fund with 500. 01:12:01.080 --> 01:12:01.800 It may not. 01:12:01.800 --> 01:12:04.500 Nowadays, actually, the technology is so good that 01:12:04.500 --> 01:12:05.580 probably it doesn't. 01:12:05.580 --> 01:12:08.250 But 15 years ago, that was not true. 01:12:08.250 --> 01:12:10.140 But apart from the transactions cost, 01:12:10.140 --> 01:12:13.420 the theory suggests more is better. 01:12:13.420 --> 01:12:16.124 Because it will always give you more opportunities, and it 01:12:16.124 --> 01:12:18.540 can never hurt you because you could always put a 0 weight 01:12:18.540 --> 01:12:19.831 on them if you don't like them. 01:12:22.330 --> 01:12:30.040 Now, it turns out that this is a trade-off between the expected 01:12:30.040 --> 01:12:35.820 return of an efficient portfolio, 01:12:35.820 --> 01:12:37.860 and the risk of that portfolio. 01:12:37.860 --> 01:12:41.460 In other words, this applies only 01:12:41.460 --> 01:12:45.100 to portfolios on that tangency line. 01:12:45.100 --> 01:12:48.040 What if you want to know what the expected rate of return 01:12:48.040 --> 01:12:51.120 is for Wal-Mart? 01:12:51.120 --> 01:12:54.510 We just said that no individual stock 01:12:54.510 --> 01:12:58.170 is going to be likely to be on that efficient frontier. 01:12:58.170 --> 01:13:00.690 And therefore, no individual stock 01:13:00.690 --> 01:13:03.016 is likely to be on this line. 01:13:03.016 --> 01:13:04.890 So this is great if what you're talking about 01:13:04.890 --> 01:13:07.230 is investing in efficient portfolios, 01:13:07.230 --> 01:13:09.500 but how does that help the corporate financial officer 01:13:09.500 --> 01:13:11.041 that's trying to figure out how to do 01:13:11.041 --> 01:13:13.650 capital budgeting for a particular pharmaceutical 01:13:13.650 --> 01:13:15.399 project? 01:13:15.399 --> 01:13:16.440 It turns out, it doesn't. 01:13:16.440 --> 01:13:17.106 It doesn't help. 01:13:17.106 --> 01:13:19.770 This doesn't answer that question. 01:13:19.770 --> 01:13:24.630 It turns out, you need to have an additional piece of theory 01:13:24.630 --> 01:13:28.590 that allows you to derive the same results, not 01:13:28.590 --> 01:13:33.370 just for the efficient portfolios here, 01:13:33.370 --> 01:13:35.590 but for any portfolio. 01:13:35.590 --> 01:13:37.780 And this is another innovation of Bill Sharpe. 01:13:37.780 --> 01:13:40.190 This is actually why Bill Sharpe won the Nobel Prize. 01:13:40.190 --> 01:13:42.250 It was not for this little picture here, 01:13:42.250 --> 01:13:45.310 but it was for this equation right here. 01:13:45.310 --> 01:13:50.110 What Bill Sharpe discovered is after computing the equilibrium 01:13:50.110 --> 01:13:53.810 relationships among various different securities, 01:13:53.810 --> 01:13:58.670 he's demonstrated that there has to be a linear relationship 01:13:58.670 --> 01:14:05.230 between any stock's expected return and the market risk 01:14:05.230 --> 01:14:06.550 premium. 01:14:06.550 --> 01:14:09.370 Just like here, where you've got the risk-free rate, 01:14:09.370 --> 01:14:11.800 plus some extra premium. 01:14:11.800 --> 01:14:14.620 So this is the premium, the second term. 01:14:14.620 --> 01:14:18.790 But what Bill Sharpe showed was that if this portfolio is not 01:14:18.790 --> 01:14:24.120 an efficient portfolio, if it's not on that line, 01:14:24.120 --> 01:14:27.810 the linear relationship still holds. 01:14:27.810 --> 01:14:31.080 But it turns out that this particular multiplier is 01:14:31.080 --> 01:14:34.210 no longer the right one to use. 01:14:34.210 --> 01:14:39.300 It turns out that the right parameter to plug in there, 01:14:39.300 --> 01:14:42.240 is something called beta. 01:14:42.240 --> 01:14:44.440 Now, you've heard all about beta, I'm sure. 01:14:44.440 --> 01:14:47.500 But now, I'm telling you exactly what beta is. 01:14:47.500 --> 01:14:51.010 Beta is the multiplier that is defined 01:14:51.010 --> 01:14:54.640 by the covariance between the return on the market 01:14:54.640 --> 01:14:57.760 and the return on the individual asset, divided 01:14:57.760 --> 01:15:00.220 by the variance of that market return. 01:15:03.860 --> 01:15:08.780 If the portfolio happens to be on that efficient frontier, 01:15:08.780 --> 01:15:13.040 then this beta reduces to this previous measure. 01:15:13.040 --> 01:15:17.090 So this is a special case of the more general relationship 01:15:17.090 --> 01:15:20.560 where beta is used as the multiplier. 01:15:20.560 --> 01:15:21.960 So let me repeat what beta is. 01:15:21.960 --> 01:15:26.250 Beta is the ratio of the covariance between the return 01:15:26.250 --> 01:15:28.860 on the particular asset or portfolio, that may or may not 01:15:28.860 --> 01:15:33.350 be efficient, it's any asset, with the return 01:15:33.350 --> 01:15:34.980 on the market portfolio. 01:15:34.980 --> 01:15:39.710 So this numerator is a measure of the covariability 01:15:39.710 --> 01:15:42.200 between the particular asset that you're 01:15:42.200 --> 01:15:45.410 trying to measure the expected return of, 01:15:45.410 --> 01:15:48.890 and that tangency portfolio, divided 01:15:48.890 --> 01:15:53.390 by the variance of that tangency portfolio. 01:15:56.010 --> 01:16:00.960 Beta, it turns out, is the right measure of risk, 01:16:00.960 --> 01:16:05.190 in the sense that it is the beta that determines what 01:16:05.190 --> 01:16:09.480 the multiplier is going to be on the market risk premium, which 01:16:09.480 --> 01:16:13.290 is to be added to your asset's expected rate of return, 01:16:13.290 --> 01:16:15.810 above and beyond the risk-free rate. 01:16:15.810 --> 01:16:19.920 That's how the cost of capital is determined for your asset. 01:16:24.030 --> 01:16:27.420 So I think you all saw how I derived this, 01:16:27.420 --> 01:16:29.070 but I didn't derive this. 01:16:29.070 --> 01:16:30.600 I'm just telling you this is really 01:16:30.600 --> 01:16:35.700 where Sharpe's ideas became extraordinarily compelling. 01:16:35.700 --> 01:16:37.800 And in order to understand how to derive 01:16:37.800 --> 01:16:41.017 that, I'm going to refer you to 433, because 01:16:41.017 --> 01:16:42.600 in that investment's course, we really 01:16:42.600 --> 01:16:46.082 delve into the underpinnings of that kind of calculation. 01:16:46.082 --> 01:16:47.540 It's a little bit more involved, it 01:16:47.540 --> 01:16:49.470 involves some matrix algebra. 01:16:49.470 --> 01:16:52.502 But it's not terribly difficult or challenging, 01:16:52.502 --> 01:16:54.960 and certainly be happy to give you references if any of you 01:16:54.960 --> 01:16:55.710 are interested. 01:16:55.710 --> 01:16:59.370 I believe it's in Brealey, Myers, and Allen. 01:16:59.370 --> 01:17:01.380 But the bottom line is that this gives you 01:17:01.380 --> 01:17:04.490 an extraordinarily important conclusion now 01:17:04.490 --> 01:17:09.930 to the several weeks that we've been working towards this goal. 01:17:09.930 --> 01:17:14.100 Which is now, finally, after eight or nine weeks, 01:17:14.100 --> 01:17:17.850 I can tell you how to come up with the appropriate discount 01:17:17.850 --> 01:17:21.120 rate for various NPV calculations. 01:17:21.120 --> 01:17:23.640 The answer is the expected rate of return, 01:17:23.640 --> 01:17:26.940 the appropriate fair rate of return, 01:17:26.940 --> 01:17:30.300 or the market equilibrium rate of return, 01:17:30.300 --> 01:17:33.960 is simply given by the beta of that security, multiplied 01:17:33.960 --> 01:17:39.140 by the expected excess return on the market portfolio. 01:17:39.140 --> 01:17:41.729 So now, this has a lot of assumptions, granted. 01:17:41.729 --> 01:17:43.520 We're going to talk about those assumptions 01:17:43.520 --> 01:17:46.430 over the next couple of lectures. 01:17:46.430 --> 01:17:50.740 But what we've done today is move the theory forward 01:17:50.740 --> 01:17:55.030 by quite a bit, because we've identified a particular method 01:17:55.030 --> 01:17:58.150 for coming up with the appropriate cost of capital 01:17:58.150 --> 01:17:59.440 as a function of the risk. 01:17:59.440 --> 01:18:03.340 Where the risk is measured, not by volatility anymore, 01:18:03.340 --> 01:18:09.710 but by the covariance between an asset and the market portfolio. 01:18:09.710 --> 01:18:11.960 And next time, I'm going to try to give you 01:18:11.960 --> 01:18:14.300 some intuition for why this should be, 01:18:14.300 --> 01:18:20.120 why this makes sense, and why, in a mean variance efficient 01:18:20.120 --> 01:18:23.840 set of portfolios, why it reduces to something that we 01:18:23.840 --> 01:18:24.650 know and love. 01:18:27.930 --> 01:18:30.670 Any questions? 01:18:30.670 --> 01:18:31.170 OK. 01:18:31.170 --> 01:18:34.010 I'll stop here, and I'll see you on Wednesday.