1 00:00:00,090 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,820 Commons license. 3 00:00:03,820 --> 00:00:06,030 Your support will help MIT OpenCourseWare 4 00:00:06,030 --> 00:00:10,120 continue to offer high quality educational resources for free. 5 00:00:10,120 --> 00:00:12,690 To make a donation, or to view additional materials 6 00:00:12,690 --> 00:00:16,620 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,620 --> 00:00:17,830 at ocw.mit.edu. 8 00:00:21,522 --> 00:00:23,980 ANDREW LO: Well, let me pick up where we left off last time 9 00:00:23,980 --> 00:00:28,030 and give you just a very quick overview of where we're at now, 10 00:00:28,030 --> 00:00:32,350 because we're on the brink of a very important set of results 11 00:00:32,350 --> 00:00:36,290 that I think will change your perspective permanently 12 00:00:36,290 --> 00:00:38,410 on risk and expected return. 13 00:00:38,410 --> 00:00:41,590 Last time, remember, we looked at this trade-off 14 00:00:41,590 --> 00:00:45,430 between expected return and volatility. 15 00:00:45,430 --> 00:00:47,440 And we made the argument that when 16 00:00:47,440 --> 00:00:50,290 you combined a bunch of different securities 17 00:00:50,290 --> 00:00:53,200 that are not all perfectly correlated, 18 00:00:53,200 --> 00:00:56,890 what you get is this bullet-shaped curve in terms 19 00:00:56,890 --> 00:01:01,900 of the possible trade-offs between that expected return 20 00:01:01,900 --> 00:01:05,590 and riskiness of various different portfolios. 21 00:01:05,590 --> 00:01:10,570 So every single dot on this bullet-shaped curve 22 00:01:10,570 --> 00:01:15,160 corresponds to a specific portfolio, or weighting, 23 00:01:15,160 --> 00:01:18,100 or vector of portfolio weights, omega. 24 00:01:20,680 --> 00:01:24,520 So now what I want to ask you to do for the next lecture or two 25 00:01:24,520 --> 00:01:28,780 is to exhibit a little bit of a split personality 26 00:01:28,780 --> 00:01:30,910 kind of a perspective. 27 00:01:30,910 --> 00:01:33,860 I'm going to ask you to look at the geometry of risk 28 00:01:33,860 --> 00:01:36,400 and expected return, but at the same time, 29 00:01:36,400 --> 00:01:39,010 in the back of your brain, I want 30 00:01:39,010 --> 00:01:42,100 you to keep in mind the analytics 31 00:01:42,100 --> 00:01:44,650 of that set of geometries. 32 00:01:44,650 --> 00:01:46,960 In other words, I want you to keep in mind how we 33 00:01:46,960 --> 00:01:50,080 got this bullet-shaped curve. 34 00:01:50,080 --> 00:01:51,850 The way we got it was from taking 35 00:01:51,850 --> 00:01:55,720 different weighted averages of the securities 36 00:01:55,720 --> 00:01:59,320 that we have access to as investments. 37 00:01:59,320 --> 00:02:01,810 So every one of these points on the bullet 38 00:02:01,810 --> 00:02:03,970 corresponds to a specific weighting. 39 00:02:03,970 --> 00:02:05,680 As you change those weightings, you 40 00:02:05,680 --> 00:02:08,740 change the risk and return characteristics 41 00:02:08,740 --> 00:02:10,160 of your portfolio. 42 00:02:10,160 --> 00:02:13,150 So the example that I gave after showing you this 43 00:02:13,150 --> 00:02:17,440 curve where I argued that the upper branch of this bullet 44 00:02:17,440 --> 00:02:21,160 is where any rational person would want to be. 45 00:02:21,160 --> 00:02:25,030 And by rational, I've defined that as somebody 46 00:02:25,030 --> 00:02:30,100 who prefers more expected return to less, and somebody 47 00:02:30,100 --> 00:02:35,930 who prefers less risk to more, other things equal. 48 00:02:35,930 --> 00:02:38,380 So if you've got those kind of preferences, 49 00:02:38,380 --> 00:02:40,600 then you want to be in the Northeast. 50 00:02:40,600 --> 00:02:44,290 You want to be as north, sorry, Northwest as possible. 51 00:02:44,290 --> 00:02:48,160 And you would never want to be down in this lower branch when 52 00:02:48,160 --> 00:02:50,500 you could be in the upper branch because you'd 53 00:02:50,500 --> 00:02:55,570 have a higher expected return for the same level of risk. 54 00:02:55,570 --> 00:02:59,590 So after we developed this basic idea, 55 00:02:59,590 --> 00:03:01,750 I gave you this numerical example 56 00:03:01,750 --> 00:03:04,150 where you've got three stocks in your universe. 57 00:03:04,150 --> 00:03:07,240 General Motors, IBM, and Motorola. 58 00:03:07,240 --> 00:03:10,330 And these are the parameters that we've estimated 59 00:03:10,330 --> 00:03:11,500 using historical data. 60 00:03:11,500 --> 00:03:13,150 Now there's going to be a question, 61 00:03:13,150 --> 00:03:16,300 and we've already raised that question, of how stable 62 00:03:16,300 --> 00:03:17,200 are these parameters. 63 00:03:17,200 --> 00:03:19,533 Are they really parameters, or do they change over time. 64 00:03:19,533 --> 00:03:22,610 And I told you, in reality of course, they change over time. 65 00:03:22,610 --> 00:03:26,170 But for now, let's play the game and assume 66 00:03:26,170 --> 00:03:27,790 that they are constant over time, 67 00:03:27,790 --> 00:03:30,320 and see what we can do with those parameters. 68 00:03:30,320 --> 00:03:33,820 So with the means, the standard deviations, 69 00:03:33,820 --> 00:03:38,090 and most importantly, the covariance matrix-- 70 00:03:38,090 --> 00:03:41,290 So this is the matrix of variances and covariances-- 71 00:03:41,290 --> 00:03:45,010 With these data as inputs, we can now 72 00:03:45,010 --> 00:03:48,190 construct that bullet-shaped curve. 73 00:03:48,190 --> 00:03:51,310 The way we do it is of course, to recognize 74 00:03:51,310 --> 00:03:53,830 that the expected return of the portfolio 75 00:03:53,830 --> 00:03:56,830 is just a weighted average of the expected returns 76 00:03:56,830 --> 00:04:00,820 of the component securities, where the weights are 77 00:04:00,820 --> 00:04:02,180 our choice variables. 78 00:04:02,180 --> 00:04:04,060 That's what we are getting to pick, 79 00:04:04,060 --> 00:04:08,560 is how we allocate the 100% of our wealth 80 00:04:08,560 --> 00:04:11,860 to these three different securities. 81 00:04:11,860 --> 00:04:14,410 And the variance, of course, is going 82 00:04:14,410 --> 00:04:17,680 to be given by a somewhat more complicated expression where 83 00:04:17,680 --> 00:04:21,519 you have the individual security variances entering here 84 00:04:21,519 --> 00:04:23,230 from the diagonals. 85 00:04:23,230 --> 00:04:27,320 But you also have the off diagonal terms 86 00:04:27,320 --> 00:04:30,640 entering in that same equation for that variance 87 00:04:30,640 --> 00:04:32,320 of the portfolio. 88 00:04:32,320 --> 00:04:35,210 And when we put these two equations together, 89 00:04:35,210 --> 00:04:37,819 the mean and the variance, and we take the square root 90 00:04:37,819 --> 00:04:39,610 the variance to get the standard deviation, 91 00:04:39,610 --> 00:04:44,060 and we plot it on a graph, we get this. 92 00:04:44,060 --> 00:04:46,790 This is the curve, the bullet-shaped curve, 93 00:04:46,790 --> 00:04:49,430 that we generate just from three securities, 94 00:04:49,430 --> 00:04:51,970 and from their covariances. 95 00:04:51,970 --> 00:04:54,650 And where we left off last time is 96 00:04:54,650 --> 00:04:56,630 that I pointed out a couple of things that was 97 00:04:56,630 --> 00:04:58,370 interesting about this curve. 98 00:04:58,370 --> 00:05:03,710 One is that unlike the two asset example, where when you start 99 00:05:03,710 --> 00:05:05,990 with two assets, the curve, the bullet 100 00:05:05,990 --> 00:05:08,600 goes through the two assets. 101 00:05:08,600 --> 00:05:12,380 In this case, with three or more assets, 102 00:05:12,380 --> 00:05:16,430 it's going to turn out that the bullet is actually 103 00:05:16,430 --> 00:05:20,630 going to include these assets as special cases, 104 00:05:20,630 --> 00:05:22,950 but they won't be on the curve. 105 00:05:22,950 --> 00:05:25,430 In other words, what this curve suggests 106 00:05:25,430 --> 00:05:28,460 is that any rational person is going to want 107 00:05:28,460 --> 00:05:31,490 to be on this upper branch. 108 00:05:31,490 --> 00:05:35,120 What that means is that it never makes sense 109 00:05:35,120 --> 00:05:40,430 to put all your money in one single security. 110 00:05:40,430 --> 00:05:41,630 You see that? 111 00:05:41,630 --> 00:05:45,620 In other words, if we agree that any rational investor is going 112 00:05:45,620 --> 00:05:49,730 to want to be on that efficient frontier, that upper branch, 113 00:05:49,730 --> 00:05:52,340 why would you ever want to be off of that branch? 114 00:05:52,340 --> 00:05:54,740 You'd like to be Northwest of that, but you can't. 115 00:05:54,740 --> 00:05:56,900 You'd never want to be below that branch, 116 00:05:56,900 --> 00:06:00,050 or to the right of that branch because you could do better 117 00:06:00,050 --> 00:06:02,310 by being on that branch. 118 00:06:02,310 --> 00:06:05,900 So what this suggests is that we never 119 00:06:05,900 --> 00:06:08,510 are going to want to hold 100% of IBM, 120 00:06:08,510 --> 00:06:12,470 or 100% of General Motors, or 100% of Motorola. 121 00:06:12,470 --> 00:06:14,360 If we did, we'd be on those dots, 122 00:06:14,360 --> 00:06:18,080 and those dots would lie on that efficient frontier. 123 00:06:18,080 --> 00:06:21,200 But in fact, they don't. 124 00:06:21,200 --> 00:06:23,660 So right away, we have now departed 125 00:06:23,660 --> 00:06:26,930 from Warren Buffett's world of, I want to pick a few stocks 126 00:06:26,930 --> 00:06:28,845 and watch them very, very carefully. 127 00:06:28,845 --> 00:06:29,420 Yeah, Brian? 128 00:06:29,420 --> 00:06:31,711 AUDIENCE: Would you expand that to say that you'd never 129 00:06:31,711 --> 00:06:35,690 want to invest in less than three stocks at a given time? 130 00:06:35,690 --> 00:06:37,500 ANDREW LO: That's not necessarily true. 131 00:06:37,500 --> 00:06:41,120 There are points on this line where-- 132 00:06:41,120 --> 00:06:43,890 and they may be pathological, so in other words, 133 00:06:43,890 --> 00:06:44,960 they may be very rare-- 134 00:06:44,960 --> 00:06:46,670 but there may be points on the line 135 00:06:46,670 --> 00:06:50,070 where you are holding two stocks, but not the third. 136 00:06:50,070 --> 00:06:51,850 So you've got to be careful about that. 137 00:06:51,850 --> 00:06:53,720 But those are exceptions. 138 00:06:53,720 --> 00:06:57,170 As a generic statement, you're absolutely right. 139 00:06:57,170 --> 00:07:00,140 The typical portfolio is going to have some of all three 140 00:07:00,140 --> 00:07:00,800 of them. 141 00:07:00,800 --> 00:07:03,080 And if you had four stocks, the typical portfolio 142 00:07:03,080 --> 00:07:04,982 would have some of all four. 143 00:07:04,982 --> 00:07:05,732 Yeah, [INAUDIBLE]. 144 00:07:05,732 --> 00:07:07,190 AUDIENCE: You answered my question, 145 00:07:07,190 --> 00:07:09,370 which is if you take one more stock, 146 00:07:09,370 --> 00:07:13,825 you'll always have your package [INAUDIBLE] n stocks include 147 00:07:13,825 --> 00:07:17,785 all the [INAUDIBLE], all the n stocks so, at the limit 148 00:07:17,785 --> 00:07:20,270 you should have an infinite number of stocks [INAUDIBLE] 149 00:07:20,270 --> 00:07:22,639 ANDREW LO: Well, let me put it another way that may 150 00:07:22,639 --> 00:07:23,930 be a little bit more intuitive. 151 00:07:23,930 --> 00:07:27,830 What this diagram suggests-- you guys are already groping 152 00:07:27,830 --> 00:07:28,350 towards-- 153 00:07:28,350 --> 00:07:32,870 is the insight that the more, the merrier. 154 00:07:32,870 --> 00:07:37,880 As you add more stocks, you cannot make this investor worse 155 00:07:37,880 --> 00:07:39,830 off. 156 00:07:39,830 --> 00:07:42,380 So in other words, I've now shown you an example with three 157 00:07:42,380 --> 00:07:43,520 stocks, we used to do two. 158 00:07:46,820 --> 00:07:49,720 Is it possible that by giving you an extra stock 159 00:07:49,720 --> 00:07:53,365 to invest in, I've made you worse off? 160 00:07:53,365 --> 00:07:53,865 Yeah? 161 00:07:53,865 --> 00:07:54,420 AUDIENCE: No 162 00:07:54,420 --> 00:07:54,690 ANDREW LO: Why 163 00:07:54,690 --> 00:07:57,023 AUDIENCE: Because you can just not invest in that stock. 164 00:07:57,023 --> 00:07:57,930 ANDREW LO: Exactly. 165 00:07:57,930 --> 00:08:01,170 I can never make you worse off in a world 166 00:08:01,170 --> 00:08:03,420 where you're free to choose, that is. 167 00:08:03,420 --> 00:08:06,900 Because you always have the option of getting 168 00:08:06,900 --> 00:08:09,630 rid of the stock that you don't like. 169 00:08:09,630 --> 00:08:11,490 You can always put 0 on it. 170 00:08:11,490 --> 00:08:14,350 So to your point, [INAUDIBLE], as I add more stocks, 171 00:08:14,350 --> 00:08:16,620 first of all my risk-reward trade-off 172 00:08:16,620 --> 00:08:18,250 curve will get better. 173 00:08:18,250 --> 00:08:20,032 What does it mean to get better? 174 00:08:20,032 --> 00:08:21,990 What does it mean for the risk-reward trade-off 175 00:08:21,990 --> 00:08:22,680 to be better? 176 00:08:24,990 --> 00:08:25,490 Yes? 177 00:08:25,490 --> 00:08:29,205 AUDIENCE: It means you get a higher return 178 00:08:29,205 --> 00:08:30,330 for the same level of risk. 179 00:08:30,330 --> 00:08:30,870 ANDREW LO: That's right. 180 00:08:30,870 --> 00:08:32,661 A higher return for the same level of risk, 181 00:08:32,661 --> 00:08:34,770 or a lower risk for the same level of return. 182 00:08:34,770 --> 00:08:39,570 In other words, your upper branch actually 183 00:08:39,570 --> 00:08:41,970 moves to the Northwest. 184 00:08:41,970 --> 00:08:43,559 That's what it means to get better. 185 00:08:43,559 --> 00:08:47,610 As I add more stocks, this will move to the Northwest. 186 00:08:47,610 --> 00:08:50,640 And therefore, you have available 187 00:08:50,640 --> 00:08:54,090 all of the opportunities to the south and to the east, 188 00:08:54,090 --> 00:08:56,400 but you would never take those because you're 189 00:08:56,400 --> 00:08:58,110 rational in the sense that you always 190 00:08:58,110 --> 00:09:03,304 prefer less risk to more, and more return to less. 191 00:09:03,304 --> 00:09:03,804 Yeah? 192 00:09:03,804 --> 00:09:06,790 AUDIENCE: If we put all of the stocks on the index, 193 00:09:06,790 --> 00:09:07,880 on [INAUDIBLE]. 194 00:09:07,880 --> 00:09:11,618 And if we looked at all the possible combinations that-- 195 00:09:11,618 --> 00:09:13,326 we can look at them all at the same time, 196 00:09:13,326 --> 00:09:15,284 but then all the subsets that you can think of, 197 00:09:15,284 --> 00:09:19,290 then you must come up with some most efficient frontier 198 00:09:19,290 --> 00:09:20,959 in that market. 199 00:09:20,959 --> 00:09:23,000 ANDREW LO: Hold onto that thought for 10 minutes. 200 00:09:23,000 --> 00:09:24,375 We're going to come back to that. 201 00:09:24,375 --> 00:09:26,930 Let's do three first, and then we could do all of them. 202 00:09:26,930 --> 00:09:27,431 Yeah, Chris? 203 00:09:27,431 --> 00:09:29,763 AUDIENCE: Trying to better understand Buffett's strategy 204 00:09:29,763 --> 00:09:30,740 relative to this one. 205 00:09:30,740 --> 00:09:33,030 Is the correlation of these stocks due primarily 206 00:09:33,030 --> 00:09:35,370 to just psychological factors of the market, 207 00:09:35,370 --> 00:09:39,430 or is it due to intrinsic correlation? 208 00:09:39,430 --> 00:09:42,290 And then the follow on is when Buffet says invest in one stock 209 00:09:42,290 --> 00:09:44,623 and just watch it carefully, isn't that sort of assuming 210 00:09:44,623 --> 00:09:46,507 that the market will determine at some point 211 00:09:46,507 --> 00:09:49,069 that the stock is undervalued, and what was he-- 212 00:09:49,069 --> 00:09:50,860 ANDREW LO: So those are two good questions. 213 00:09:50,860 --> 00:09:52,360 Let me take each of them separately. 214 00:09:52,360 --> 00:09:54,270 Let's first talk about the correlation. 215 00:09:54,270 --> 00:09:55,440 Why is there correlation? 216 00:09:55,440 --> 00:09:57,270 We haven't really talked much about it, 217 00:09:57,270 --> 00:09:59,910 but it turns out that there are many different arguments 218 00:09:59,910 --> 00:10:02,370 for why there is correlation. 219 00:10:02,370 --> 00:10:06,630 Probably the most compelling is that a rising tide lifts 220 00:10:06,630 --> 00:10:08,610 all boats, and vice versa. 221 00:10:08,610 --> 00:10:10,800 In other words, when business conditions are good, 222 00:10:10,800 --> 00:10:12,630 then that helps all companies. 223 00:10:12,630 --> 00:10:15,210 Just like when business conditions are bad, 224 00:10:15,210 --> 00:10:16,600 it hurts all companies. 225 00:10:16,600 --> 00:10:19,110 So there's some macroeconomic type 226 00:10:19,110 --> 00:10:23,250 of commonality among businesses that create correlation. 227 00:10:23,250 --> 00:10:24,345 That's one reason. 228 00:10:24,345 --> 00:10:26,470 But the second reason is something you pointed out, 229 00:10:26,470 --> 00:10:29,095 which is quite apt, particularly over the last few weeks, which 230 00:10:29,095 --> 00:10:30,510 is the psychological factor. 231 00:10:30,510 --> 00:10:33,690 When the entire economy is under stress, 232 00:10:33,690 --> 00:10:35,924 and people are scared to death about what's 233 00:10:35,924 --> 00:10:37,590 going to happen to the market, what they 234 00:10:37,590 --> 00:10:41,670 will do is withdraw money in mass from equities 235 00:10:41,670 --> 00:10:45,570 and put them into safer assets like cash, or treasury 236 00:10:45,570 --> 00:10:48,540 bills, or money market funds, or whatever 237 00:10:48,540 --> 00:10:51,240 they can do to get to safety. 238 00:10:51,240 --> 00:10:53,760 So I would say that the answer is both. 239 00:10:53,760 --> 00:10:58,170 There are good economic reasons where correlations should exist 240 00:10:58,170 --> 00:11:00,060 among different companies, but there are also 241 00:11:00,060 --> 00:11:03,030 psychological or behavior reasons that exacerbate 242 00:11:03,030 --> 00:11:04,530 those kinds of commonalities. 243 00:11:04,530 --> 00:11:09,000 Now your second question about Buffett versus this approach. 244 00:11:09,000 --> 00:11:10,620 There's one fundamental difference 245 00:11:10,620 --> 00:11:13,710 between what Buffett would say about a company 246 00:11:13,710 --> 00:11:18,150 that he decides to buy versus how we're approaching it. 247 00:11:18,150 --> 00:11:20,040 The fundamental difference is that Buffett 248 00:11:20,040 --> 00:11:25,670 would say that he's been able to identify a severe mispricing. 249 00:11:25,670 --> 00:11:28,800 In other words, he would argue that markets are not 250 00:11:28,800 --> 00:11:29,880 in equilibrium. 251 00:11:29,880 --> 00:11:32,580 He would argue that Goldman Sachs is dramatically 252 00:11:32,580 --> 00:11:35,580 undervalued where it is today. 253 00:11:35,580 --> 00:11:38,790 And seven years from now, he may be right. 254 00:11:38,790 --> 00:11:40,860 And that's the kind of time frame he has in mind, 255 00:11:40,860 --> 00:11:43,470 if not longer. 256 00:11:43,470 --> 00:11:46,890 So far, I've made no such argument at all 257 00:11:46,890 --> 00:11:49,110 about deriving these analyses. 258 00:11:49,110 --> 00:11:51,990 I've not made any argument about whether prices are good or bad. 259 00:11:51,990 --> 00:11:55,500 In fact, I'm arguing, in a way, that these prices 260 00:11:55,500 --> 00:11:57,090 I'm taking as given. 261 00:11:57,090 --> 00:11:58,560 And the question is, what can I do 262 00:11:58,560 --> 00:12:00,570 to construct a good portfolio irrespective 263 00:12:00,570 --> 00:12:04,270 of whether markets are crazy or markets are rational. 264 00:12:04,270 --> 00:12:06,490 In a few minutes, I'm going to argue 265 00:12:06,490 --> 00:12:12,700 that when markets are rational and in equilibrium, then 266 00:12:12,700 --> 00:12:15,070 there is something that we can say 267 00:12:15,070 --> 00:12:17,710 about the relationship between risk and reward 268 00:12:17,710 --> 00:12:21,790 that's extraordinarily sharp and meaningful from the perspective 269 00:12:21,790 --> 00:12:23,530 of financial decision making. 270 00:12:23,530 --> 00:12:25,056 And then at the end of the course, 271 00:12:25,056 --> 00:12:26,680 I'm going to try to explain to you what 272 00:12:26,680 --> 00:12:30,037 the limitations of that set of assumptions are. 273 00:12:30,037 --> 00:12:31,767 Dennis? 274 00:12:31,767 --> 00:12:33,516 AUDIENCE: Just as we wouldn't put anything 275 00:12:33,516 --> 00:12:37,492 in bond's half of this frontier, does this graph 276 00:12:37,492 --> 00:12:39,977 imply that we strictly prefer IBM over GM? 277 00:12:39,977 --> 00:12:42,970 That we pretty much never weigh anything for GM? 278 00:12:42,970 --> 00:12:46,630 ANDREW LO: Well, from a risk-reward perspective, 279 00:12:46,630 --> 00:12:48,250 let's take a look. 280 00:12:48,250 --> 00:12:51,520 IBM has a higher expected rate of return, 281 00:12:51,520 --> 00:12:54,290 and it's got a higher level of risk. 282 00:12:54,290 --> 00:13:01,120 So you really can't say that you would never prefer GM over IBM, 283 00:13:01,120 --> 00:13:05,050 because GM has lower risk and lower expected return. 284 00:13:05,050 --> 00:13:09,209 If on the other hand, GM were over here, then 285 00:13:09,209 --> 00:13:10,000 you would be right. 286 00:13:10,000 --> 00:13:14,230 Because any point to the direct Northwest 287 00:13:14,230 --> 00:13:18,310 of a particular point on this curve is strictly preferred. 288 00:13:18,310 --> 00:13:22,450 And GM and IBM don't have that relationship. 289 00:13:22,450 --> 00:13:25,750 In other words, the way you can identify 290 00:13:25,750 --> 00:13:31,900 securities that are dominated in both dimensions is-- 291 00:13:31,900 --> 00:13:35,890 So this is your risk dimension, this is your expected return 292 00:13:35,890 --> 00:13:36,580 dimension. 293 00:13:36,580 --> 00:13:41,130 Pick a point in this space, and ask the question, 294 00:13:41,130 --> 00:13:43,240 what are the other portfolios that are strictly 295 00:13:43,240 --> 00:13:44,620 preferred to that point. 296 00:13:44,620 --> 00:13:46,420 Well the answer is pretty simple. 297 00:13:46,420 --> 00:13:49,720 Any portfolio that has higher expected rate of return 298 00:13:49,720 --> 00:13:52,690 for the same level of risk, so the vertical line. 299 00:13:52,690 --> 00:13:56,620 Any portfolio that has less risk for the same level 300 00:13:56,620 --> 00:14:01,030 of expected returns, So the western direction. 301 00:14:01,030 --> 00:14:10,050 And anything in this segment, that orthant, or quadrant, 302 00:14:10,050 --> 00:14:11,910 is strictly preferred. 303 00:14:11,910 --> 00:14:15,390 So in the case of IBM, if you draw 304 00:14:15,390 --> 00:14:19,530 the vertical and the horizontal and ask the question, 305 00:14:19,530 --> 00:14:21,330 does GM lie in that area? 306 00:14:21,330 --> 00:14:21,930 No. 307 00:14:21,930 --> 00:14:24,390 If you do GM, and you draw the vertical and then 308 00:14:24,390 --> 00:14:26,280 the horizontal and asked does IBM 309 00:14:26,280 --> 00:14:29,100 lie in that strictly preferred quadrant? 310 00:14:29,100 --> 00:14:30,820 The answer is no. 311 00:14:30,820 --> 00:14:33,780 So the answer to your question about IBM versus GM, 312 00:14:33,780 --> 00:14:36,300 no, there isn't any strict relationship 313 00:14:36,300 --> 00:14:38,970 that would say one would always dominate the other. 314 00:14:38,970 --> 00:14:43,080 But if GM were here, then IBM is clearly 315 00:14:43,080 --> 00:14:45,850 contained in that preferred quadrant. 316 00:14:45,850 --> 00:14:49,210 So then the answer to your question would be yes. 317 00:14:49,210 --> 00:14:50,670 Yeah, Justin? 318 00:14:50,670 --> 00:14:53,070 AUDIENCE: Theoretically then, wouldn't everyone just 319 00:14:53,070 --> 00:14:56,270 buy IBM, sell GM, then wouldn't there 320 00:14:56,270 --> 00:14:58,830 be some sort of equilibrium where then GM-- 321 00:14:58,830 --> 00:15:00,840 ANDREW LO: So the answer is, it depends 322 00:15:00,840 --> 00:15:02,830 on other things going on. 323 00:15:02,830 --> 00:15:04,595 Everyone would not do that. 324 00:15:04,595 --> 00:15:05,970 Everyone would do something else, 325 00:15:05,970 --> 00:15:07,360 and I'm about to tell you. 326 00:15:07,360 --> 00:15:09,180 So I'm about to give you the tools 327 00:15:09,180 --> 00:15:11,820 to make that exact conclusion, and the reason 328 00:15:11,820 --> 00:15:14,670 is that when I show you what people will do, 329 00:15:14,670 --> 00:15:16,410 that's going to far dominate what 330 00:15:16,410 --> 00:15:17,960 you think people want to do. 331 00:15:17,960 --> 00:15:19,390 Just with pairs. 332 00:15:19,390 --> 00:15:21,100 So instead of doing it with pairs, 333 00:15:21,100 --> 00:15:22,980 let's do it with all the securities, 334 00:15:22,980 --> 00:15:24,570 as Zeke wanted to do. 335 00:15:24,570 --> 00:15:26,320 We're going to do that in just a minute. 336 00:15:26,320 --> 00:15:28,195 But I want to make sure everybody understands 337 00:15:28,195 --> 00:15:29,910 this basic framework first, because we're 338 00:15:29,910 --> 00:15:33,540 going to now start making this a little bit more complex. 339 00:15:33,540 --> 00:15:37,290 Where we left off at the very last moment of Wednesday's 340 00:15:37,290 --> 00:15:41,219 lecture was I showed you this diagram 341 00:15:41,219 --> 00:15:43,260 with the tangency portfolio, but we hadn't really 342 00:15:43,260 --> 00:15:44,790 gotten to talking about it. 343 00:15:44,790 --> 00:15:48,510 Remember the case where we had only one risky asset and one 344 00:15:48,510 --> 00:15:50,710 riskless asset, treasury bills? 345 00:15:50,710 --> 00:15:54,420 And in that case, when you are combining a portfolio 346 00:15:54,420 --> 00:15:57,450 with one risky asset and one riskless, 347 00:15:57,450 --> 00:15:58,680 you've got a straight line. 348 00:15:58,680 --> 00:16:01,300 It turns out that that is much more general. 349 00:16:01,300 --> 00:16:05,460 You get a straight line anytime you combine a riskless asset 350 00:16:05,460 --> 00:16:07,880 with any number of risky assets. 351 00:16:07,880 --> 00:16:09,820 So let me give you an example. 352 00:16:09,820 --> 00:16:12,720 Suppose we picked an arbitrary portfolio which 353 00:16:12,720 --> 00:16:14,990 is this red dot, p. 354 00:16:14,990 --> 00:16:17,180 And I wanted you to tell me what is 355 00:16:17,180 --> 00:16:19,400 the risk-reward possibilities that you 356 00:16:19,400 --> 00:16:25,204 could achieve by mixing p with treasury bills. 357 00:16:25,204 --> 00:16:26,870 Well, you get that straight line, right? 358 00:16:26,870 --> 00:16:29,450 We derived that last time. 359 00:16:29,450 --> 00:16:33,680 So any point along the straight line is what you could achieve, 360 00:16:33,680 --> 00:16:34,820 right? 361 00:16:34,820 --> 00:16:37,010 Anybody tell me where the portfolio 362 00:16:37,010 --> 00:16:42,890 would be that invests 100% of your assets in T-Bills? 363 00:16:42,890 --> 00:16:44,300 Where is that on this graph? 364 00:16:44,300 --> 00:16:45,650 AUDIENCE: [INAUDIBLE] 365 00:16:45,650 --> 00:16:47,191 ANDREW LO: Right this dot right here. 366 00:16:47,191 --> 00:16:49,600 How about 100% in portfolio p? 367 00:16:49,600 --> 00:16:50,930 Right, the red dot over there. 368 00:16:50,930 --> 00:16:57,110 How about 25% in T-Bills, 75% in p? 369 00:16:57,110 --> 00:16:59,189 Where would that lie? 370 00:16:59,189 --> 00:17:00,730 AUDIENCE: [INAUDIBLE] along the line. 371 00:17:00,730 --> 00:17:02,771 ANDREW LO: It would be along the line, but where? 372 00:17:02,771 --> 00:17:04,770 Here? 373 00:17:04,770 --> 00:17:07,386 25% T-Bills, 75%-- 374 00:17:07,386 --> 00:17:08,260 AUDIENCE: [INAUDIBLE] 375 00:17:08,260 --> 00:17:09,609 ANDREW LO: Right, exactly. 376 00:17:09,609 --> 00:17:13,240 It would be 3/4 of the way up towards this dot, 377 00:17:13,240 --> 00:17:18,579 because it's 75% of the risky, 25% of the riskless, 378 00:17:18,579 --> 00:17:22,270 so you're going to get closer to the risky asset. 379 00:17:22,270 --> 00:17:23,470 OK, great. 380 00:17:23,470 --> 00:17:28,569 So we've now demonstrated that what 381 00:17:28,569 --> 00:17:32,140 I can achieve as an investor, just mixing 382 00:17:32,140 --> 00:17:35,110 portfolio p with the risk-free rate, 383 00:17:35,110 --> 00:17:38,620 is anywhere along that line. 384 00:17:38,620 --> 00:17:43,730 Now this analysis applies to any portfolio p. 385 00:17:43,730 --> 00:17:46,550 So for example, suppose I wanted to ask you, 386 00:17:46,550 --> 00:17:48,200 what risk-reward trade-offs could I 387 00:17:48,200 --> 00:17:54,527 generate by mixing the risk-free rate with General Motors? 388 00:17:54,527 --> 00:17:55,610 What would that look like? 389 00:18:02,810 --> 00:18:04,487 Yeah, Ken? 390 00:18:04,487 --> 00:18:07,455 AUDIENCE: The line from T-Bills through GM. 391 00:18:07,455 --> 00:18:08,830 ANDREW LO: Exactly, that's right. 392 00:18:08,830 --> 00:18:10,788 If I wanted to mix T-Bills with General Motors, 393 00:18:10,788 --> 00:18:13,270 I get that straight line right through that dot. 394 00:18:13,270 --> 00:18:16,930 If I wanted to mix T-Bills with IBM, I'd go through that dot, 395 00:18:16,930 --> 00:18:17,530 with IBM. 396 00:18:17,530 --> 00:18:19,240 If I wanted to mix T-Bills with Motorola, 397 00:18:19,240 --> 00:18:20,710 I'd go through Motorola. 398 00:18:20,710 --> 00:18:25,210 And if I wanted to mix T-Bills with any portfolio 399 00:18:25,210 --> 00:18:28,870 on that frontier, on that upper branch, 400 00:18:28,870 --> 00:18:31,900 it would just be a line between T-Bills 401 00:18:31,900 --> 00:18:34,330 and that point on the upper branch. 402 00:18:34,330 --> 00:18:36,080 Right? 403 00:18:36,080 --> 00:18:37,900 So question. 404 00:18:37,900 --> 00:18:40,920 If I were to give you the choice of mixing T-Bills 405 00:18:40,920 --> 00:18:50,242 with only one portfolio, just one, which would it be? 406 00:18:50,242 --> 00:18:51,200 Which would you prefer? 407 00:18:54,068 --> 00:18:55,525 [INAUDIBLE]? 408 00:18:55,525 --> 00:18:57,900 AUDIENCE: The one where the line is tangent to the curve. 409 00:18:57,900 --> 00:18:59,816 ANDREW LO: The one where the line is tangent-- 410 00:18:59,816 --> 00:19:02,910 so you're talking about right around here, right? 411 00:19:02,910 --> 00:19:04,380 Somewhere here. 412 00:19:04,380 --> 00:19:08,190 That's where the line is just tangent to that curve. 413 00:19:08,190 --> 00:19:10,050 Now why is that? 414 00:19:10,050 --> 00:19:11,388 How'd you come up with that? 415 00:19:11,388 --> 00:19:12,131 Yeah? 416 00:19:12,131 --> 00:19:14,256 AUDIENCE: If you took anything below that then it'd 417 00:19:14,256 --> 00:19:17,124 be, I would say, preferable to stay back. 418 00:19:17,124 --> 00:19:18,879 [INAUDIBLE] 419 00:19:18,879 --> 00:19:19,670 ANDREW LO: Exactly. 420 00:19:19,670 --> 00:19:21,440 If you picked any other portfolio 421 00:19:21,440 --> 00:19:24,950 besides the tangency portfolio, let's pick one and see. 422 00:19:24,950 --> 00:19:28,070 If you picked, let's say this one right here. 423 00:19:28,070 --> 00:19:31,910 If you drew a line between this point and that portfolio, 424 00:19:31,910 --> 00:19:33,380 it's going to turn out that there 425 00:19:33,380 --> 00:19:36,860 are other points over here that are strictly 426 00:19:36,860 --> 00:19:42,850 in the Northwest of that line, that you could do better. 427 00:19:42,850 --> 00:19:46,450 There exists only one portfolio that you 428 00:19:46,450 --> 00:19:52,930 can mix with T-Bills, such that you can never, ever do better 429 00:19:52,930 --> 00:19:56,050 in terms of generating risk-reward trade-offs 430 00:19:56,050 --> 00:20:00,670 for everybody that likes expected return, 431 00:20:00,670 --> 00:20:01,870 and doesn't like risk. 432 00:20:01,870 --> 00:20:04,600 And it turns out that that portfolio happens 433 00:20:04,600 --> 00:20:08,140 to be the tangency portfolio. 434 00:20:08,140 --> 00:20:11,350 That's the portfolio that all of you in this room 435 00:20:11,350 --> 00:20:12,646 would love to have. 436 00:20:12,646 --> 00:20:15,270 I don't know anything about you, I don't know your backgrounds, 437 00:20:15,270 --> 00:20:16,180 I don't know your risk aversions, 438 00:20:16,180 --> 00:20:17,320 but I don't have to know. 439 00:20:17,320 --> 00:20:20,150 As long as I know that you like expected return 440 00:20:20,150 --> 00:20:22,900 and you don't like risk, those are the only assumptions 441 00:20:22,900 --> 00:20:23,590 that I need. 442 00:20:23,590 --> 00:20:26,025 Then I know, all of you in this room, 443 00:20:26,025 --> 00:20:27,400 are going to want that portfolio. 444 00:20:27,400 --> 00:20:30,550 You may not be at that portfolio. 445 00:20:30,550 --> 00:20:32,650 For example, some of you who don't like risk, 446 00:20:32,650 --> 00:20:34,860 you're going to be down here. 447 00:20:34,860 --> 00:20:37,500 Those of you who are budding hedge fund managers, 448 00:20:37,500 --> 00:20:40,390 you're going to be up here. 449 00:20:40,390 --> 00:20:42,710 But the point is, you're going to be on this line. 450 00:20:42,710 --> 00:20:45,170 You're not going to be on this line down here. 451 00:20:45,170 --> 00:20:45,670 Why? 452 00:20:45,670 --> 00:20:49,510 Because why be on that line when you could get higher return 453 00:20:49,510 --> 00:20:51,700 for a given level of risk, or a lower risk 454 00:20:51,700 --> 00:20:53,400 for a given level of return. 455 00:20:53,400 --> 00:20:55,570 You're giving up something for no good reason. 456 00:20:59,660 --> 00:21:03,400 So this is a remarkable insight of modern portfolio theory. 457 00:21:03,400 --> 00:21:06,610 This basically tells us that regardless 458 00:21:06,610 --> 00:21:09,370 of our differences in preferences, 459 00:21:09,370 --> 00:21:11,590 as long as we satisfy the hypothesis 460 00:21:11,590 --> 00:21:14,770 that we like expected return and we don't like risk, that 461 00:21:14,770 --> 00:21:16,450 means that everybody in this room 462 00:21:16,450 --> 00:21:19,480 will agree that the only line that they would ever want 463 00:21:19,480 --> 00:21:23,020 to be on is that tangency line. 464 00:21:23,020 --> 00:21:23,610 Questions? 465 00:21:23,610 --> 00:21:24,812 Ingrid? 466 00:21:24,812 --> 00:21:27,758 AUDIENCE: Is there a particular level of risk 467 00:21:27,758 --> 00:21:30,220 that makes you accepting to the tangency fund? 468 00:21:30,220 --> 00:21:31,450 ANDREW LO: Yes. 469 00:21:31,450 --> 00:21:35,020 In fact, this tangency portfolio is one very 470 00:21:35,020 --> 00:21:38,060 particular and special portfolio. 471 00:21:38,060 --> 00:21:41,980 So in other words, it's a particular weighting of IBM, 472 00:21:41,980 --> 00:21:46,270 General Motors, and Motorola, that gives you 473 00:21:46,270 --> 00:21:48,078 this particular portfolio. 474 00:21:48,078 --> 00:21:49,034 AUDIENCE: Which one? 475 00:21:49,034 --> 00:21:53,340 [INAUDIBLE] something intuitive? 476 00:21:53,340 --> 00:21:56,740 ANDREW LO: It's something you can solve analytically. 477 00:21:56,740 --> 00:22:00,840 It has a solution, and if we were using matrix algebra, 478 00:22:00,840 --> 00:22:02,250 I can actually solve it for you. 479 00:22:02,250 --> 00:22:04,830 But it's a little bit complicated, 480 00:22:04,830 --> 00:22:08,580 so I'm not requiring that people know how to do that. 481 00:22:08,580 --> 00:22:10,700 Only that you know that it exists. 482 00:22:10,700 --> 00:22:11,818 Yeah? 483 00:22:11,818 --> 00:22:13,694 AUDIENCE: You go above the red dot, 484 00:22:13,694 --> 00:22:15,570 and into leverage [INAUDIBLE]? 485 00:22:15,570 --> 00:22:16,571 ANDREW LO: Exactly 486 00:22:16,571 --> 00:22:19,160 AUDIENCE: So it means that I have costs for my debt. 487 00:22:19,160 --> 00:22:20,124 ANDREW LO: Yes 488 00:22:20,124 --> 00:22:22,394 AUDIENCE: So maybe some debt, it would 489 00:22:22,394 --> 00:22:27,160 be more efficient to buy a portfolio with other weights-- 490 00:22:27,160 --> 00:22:28,400 ANDREW LO: Yes. 491 00:22:28,400 --> 00:22:31,290 If you assume that there are borrowing and lending 492 00:22:31,290 --> 00:22:36,780 differences, then obviously these analyses don't apply. 493 00:22:36,780 --> 00:22:44,504 So in particular, if you're here, you're actually lending. 494 00:22:44,504 --> 00:22:46,920 If you're here, you're fully invested in the stock market. 495 00:22:46,920 --> 00:22:49,020 If you're here, you're borrowing. 496 00:22:49,020 --> 00:22:51,840 If you're borrowing and lending rates are different, 497 00:22:51,840 --> 00:22:55,830 then it turns out that the curve that you want to be on actually 498 00:22:55,830 --> 00:22:57,150 has a kink in it. 499 00:22:57,150 --> 00:23:00,644 And that means that there is a potential for being 500 00:23:00,644 --> 00:23:03,060 on this curve, and then there's another tangency line that 501 00:23:03,060 --> 00:23:04,740 goes out at a different slope. 502 00:23:04,740 --> 00:23:05,490 That's possible. 503 00:23:05,490 --> 00:23:07,590 But that's more complicated than what we want 504 00:23:07,590 --> 00:23:09,040 to talk about at this point. 505 00:23:09,040 --> 00:23:12,900 So here I am assuming borrowing and lending rates are the same. 506 00:23:12,900 --> 00:23:15,000 Zeke, and then Rami. 507 00:23:15,000 --> 00:23:17,220 AUDIENCE: If I had a choice of, if I 508 00:23:17,220 --> 00:23:20,730 had control over the volatility of the market, 509 00:23:20,730 --> 00:23:25,470 then if the yield goes down, of the deals, 510 00:23:25,470 --> 00:23:28,764 then I would want to have a more volatile market so 511 00:23:28,764 --> 00:23:32,364 that I can intersect the curve at the higher return point. 512 00:23:32,364 --> 00:23:33,280 Tangent to the curve-- 513 00:23:33,280 --> 00:23:34,238 ANDREW LO: OK, hold on. 514 00:23:34,238 --> 00:23:36,450 You're changing the assumptions here. 515 00:23:36,450 --> 00:23:38,820 Why are you controlling the volatility of the market? 516 00:23:38,820 --> 00:23:41,460 The volatility of the market is a data point 517 00:23:41,460 --> 00:23:43,290 that you're basically using as an input. 518 00:23:43,290 --> 00:23:43,550 OK? 519 00:23:43,550 --> 00:23:44,591 AUDIENCE: I know, I know. 520 00:23:44,591 --> 00:23:48,030 I'm just trying to figure out what I am-- because you're 521 00:23:48,030 --> 00:23:50,550 basically connecting, I see this as a connection 522 00:23:50,550 --> 00:23:52,770 between the yield curve and the market. 523 00:23:52,770 --> 00:23:56,305 Because it's not only retrospective in the sense 524 00:23:56,305 --> 00:23:57,866 that if the yield goes down, there's 525 00:23:57,866 --> 00:23:59,420 cash flowing from the market-- 526 00:23:59,420 --> 00:24:01,336 ANDREW LO: Let's not worry about the dynamics. 527 00:24:01,336 --> 00:24:03,540 This is not meant to be a dynamic story. 528 00:24:03,540 --> 00:24:06,390 I didn't say anything about this happening over time, 529 00:24:06,390 --> 00:24:08,550 and there's lots of different changes going on. 530 00:24:08,550 --> 00:24:12,270 This is a static snapshot, today versus next period. 531 00:24:12,270 --> 00:24:14,760 These returns and covariances and all that 532 00:24:14,760 --> 00:24:17,680 apply to the returns from this period to the next, 533 00:24:17,680 --> 00:24:19,710 whether it's monthly or annual, that's 534 00:24:19,710 --> 00:24:21,800 a static snapshot as of today. 535 00:24:21,800 --> 00:24:26,290 So we're not talking about any term structure effects yet. 536 00:24:26,290 --> 00:24:28,754 Yeah, Rami? 537 00:24:28,754 --> 00:24:30,722 AUDIENCE: This is assuming three stocks. 538 00:24:30,722 --> 00:24:32,362 So if you had four or five, are you 539 00:24:32,362 --> 00:24:34,131 going to actually move the bullet left? 540 00:24:34,131 --> 00:24:35,380 And then you're gonna change-- 541 00:24:35,380 --> 00:24:37,240 ANDREW LO: Yes, absolutely. 542 00:24:37,240 --> 00:24:40,750 If you start adding more stocks to this cocktail, what's 543 00:24:40,750 --> 00:24:44,290 going to happen is the bullet is going to shift to the left, 544 00:24:44,290 --> 00:24:46,210 and it's going to shift up. 545 00:24:46,210 --> 00:24:49,630 And so the tangency point will change. 546 00:24:49,630 --> 00:24:53,380 But the curve, that straight line, the tangent line, 547 00:24:53,380 --> 00:24:57,700 what you're going to see is that tangent line 548 00:24:57,700 --> 00:24:59,520 is going to go like that. 549 00:24:59,520 --> 00:25:01,740 The slope, that's right. 550 00:25:01,740 --> 00:25:06,545 You're going to get more expected return per unit risk. 551 00:25:06,545 --> 00:25:07,920 And that is something we're going 552 00:25:07,920 --> 00:25:12,870 to take as a measure of how good this particular trade-off is. 553 00:25:12,870 --> 00:25:16,120 We're going to look at that slope of this line. 554 00:25:16,120 --> 00:25:17,970 And the slope of this line will give us 555 00:25:17,970 --> 00:25:23,340 a measure of the expected rate of return per unit risk. 556 00:25:23,340 --> 00:25:25,874 That's exactly what it's going to do for us. 557 00:25:25,874 --> 00:25:27,762 AUDIENCE: So if you're in the upper right 558 00:25:27,762 --> 00:25:30,440 beyond the tangency [INAUDIBLE], you know that line? 559 00:25:30,440 --> 00:25:32,940 Then you're borrowing [INAUDIBLE] portfolio? 560 00:25:32,940 --> 00:25:34,420 ANDREW LO: That's correct. 561 00:25:34,420 --> 00:25:37,088 AUDIENCE: If you were to extend the line leftwards, down 562 00:25:37,088 --> 00:25:39,171 to the left, would you be then shorting the market 563 00:25:39,171 --> 00:25:40,047 to invest in T-Bills? 564 00:25:40,047 --> 00:25:40,671 ANDREW LO: Yes. 565 00:25:40,671 --> 00:25:42,710 And if that happens, you know what you would do? 566 00:25:42,710 --> 00:25:44,970 It would not go this way, because of course 567 00:25:44,970 --> 00:25:46,860 standard deviation can't be negative. 568 00:25:46,860 --> 00:25:47,950 It would go like this. 569 00:25:47,950 --> 00:25:49,680 It would go this way. 570 00:25:49,680 --> 00:25:51,809 Because standard deviation is always non-negative. 571 00:25:51,809 --> 00:25:53,350 It's the square root of the variance, 572 00:25:53,350 --> 00:25:54,670 which is always positive. 573 00:25:54,670 --> 00:25:58,290 So if you decided to short the tangency portfolio 574 00:25:58,290 --> 00:26:03,540 and put it in T-Bills, well, you'd be a knucklehead. 575 00:26:03,540 --> 00:26:06,360 But you would be on this line right here. 576 00:26:06,360 --> 00:26:09,270 You would have higher and higher risk, because you're 577 00:26:09,270 --> 00:26:11,790 taking a short position on equities, 578 00:26:11,790 --> 00:26:13,470 and you'd have a lower and lower return 579 00:26:13,470 --> 00:26:15,303 because you're shorting the high yield asset 580 00:26:15,303 --> 00:26:16,680 and buying the low yield asset. 581 00:26:22,010 --> 00:26:25,690 Any other questions about the geometry of this point. 582 00:26:25,690 --> 00:26:28,770 It's very important, this is a major insight. 583 00:26:28,770 --> 00:26:29,664 Yeah? 584 00:26:29,664 --> 00:26:31,330 AUDIENCE: Earlier we discussed about how 585 00:26:31,330 --> 00:26:34,676 we bring market [INAUDIBLE] portfolios and expected return 586 00:26:34,676 --> 00:26:37,384 changes as a function of n and standard deviation 587 00:26:37,384 --> 00:26:39,950 from certain changes [INAUDIBLE] 588 00:26:39,950 --> 00:26:40,778 ANDREW LO: Yes. 589 00:26:40,778 --> 00:26:43,560 AUDIENCE: Is that the reason why the shift is more towards left? 590 00:26:43,560 --> 00:26:46,410 Because as we add more and more portfolios, 591 00:26:46,410 --> 00:26:47,940 the n dominates the [INAUDIBLE]? 592 00:26:47,940 --> 00:26:49,523 ANDREW LO: That's right, that's right. 593 00:26:49,523 --> 00:26:52,560 As we add more securities, you get more and more impact 594 00:26:52,560 --> 00:26:54,550 of diversification. 595 00:26:54,550 --> 00:26:58,950 So that increases your expected rate of return per unit risk 596 00:26:58,950 --> 00:27:01,110 because you can't make somebody worse off 597 00:27:01,110 --> 00:27:03,000 by giving them choices. 598 00:27:03,000 --> 00:27:05,610 They can always put a 0 for the new stocks 599 00:27:05,610 --> 00:27:07,690 that you give them if they don't like it. 600 00:27:07,690 --> 00:27:11,070 So the only thing you can do is to make you better off, meaning 601 00:27:11,070 --> 00:27:12,930 the only thing you can do is to give you 602 00:27:12,930 --> 00:27:15,750 a higher level of expected return per unit risk, 603 00:27:15,750 --> 00:27:19,560 or a lower level of risk per unit of expected return. 604 00:27:19,560 --> 00:27:21,870 So by adding more securities, you're 605 00:27:21,870 --> 00:27:26,290 basically increasing the slope of this line. 606 00:27:26,290 --> 00:27:28,840 So let's talk about the slope of the line. 607 00:27:28,840 --> 00:27:33,450 The slope of that line is equal to the expected return 608 00:27:33,450 --> 00:27:38,010 of that tangency portfolio, minus the T-Bill rate, divided 609 00:27:38,010 --> 00:27:41,460 by the volatility of that tangency portfolio. 610 00:27:41,460 --> 00:27:43,890 If you just calculate rise over run, 611 00:27:43,890 --> 00:27:45,990 that's what you get as the slope. 612 00:27:45,990 --> 00:27:47,370 There's a name for this. 613 00:27:47,370 --> 00:27:49,290 The name for this is called the Sharpe ratio. 614 00:27:49,290 --> 00:27:50,970 You may have heard of this, particularly 615 00:27:50,970 --> 00:27:53,620 those of you who have interest in hedge fund investments. 616 00:27:53,620 --> 00:27:56,700 Hedge fund managers will often quote their Sharpe ratio very 617 00:27:56,700 --> 00:27:57,930 proudly. 618 00:27:57,930 --> 00:28:00,330 The Sharpe ratio is simply a measure 619 00:28:00,330 --> 00:28:01,950 of that risk-reward trade-off. 620 00:28:01,950 --> 00:28:05,950 The higher the Sharpe ratio, the better you're doing. 621 00:28:05,950 --> 00:28:10,070 If you're a mean variance optimizer, meaning you 622 00:28:10,070 --> 00:28:14,440 prefer more expected return and less risk. 623 00:28:17,510 --> 00:28:20,720 So the idea behind the tangency portfolio 624 00:28:20,720 --> 00:28:22,610 is that it is the one that will give you 625 00:28:22,610 --> 00:28:24,320 the highest Sharpe ratio. 626 00:28:24,320 --> 00:28:25,760 Let's look at it again. 627 00:28:25,760 --> 00:28:28,820 If you pick a portfolio like here, take a look. 628 00:28:28,820 --> 00:28:29,730 Look at the slope. 629 00:28:29,730 --> 00:28:32,060 The slope is going to be lower. 630 00:28:32,060 --> 00:28:35,750 Take a point over here, in the inefficient branch 631 00:28:35,750 --> 00:28:36,740 of the bullet. 632 00:28:36,740 --> 00:28:38,300 Then the slope is going to be even 633 00:28:38,300 --> 00:28:40,250 lower than the upper branch. 634 00:28:40,250 --> 00:28:47,330 The biggest slope occurs when you invest between T-Bills 635 00:28:47,330 --> 00:28:50,370 and that tangency portfolio. 636 00:28:50,370 --> 00:28:54,282 That's what you're optimizing. 637 00:28:54,282 --> 00:28:55,746 Yeah? 638 00:28:55,746 --> 00:28:58,674 AUDIENCE: I have a hard time understanding 639 00:28:58,674 --> 00:29:03,570 why the bullet would go left when I add additional stocks. 640 00:29:03,570 --> 00:29:07,300 I understand it analytically, standard deviation goes down. 641 00:29:07,300 --> 00:29:11,499 But then on the other hand, wouldn't it 642 00:29:11,499 --> 00:29:16,030 happen that the likelihood of correlation 643 00:29:16,030 --> 00:29:19,509 between these additional stocks would 644 00:29:19,509 --> 00:29:22,765 decrease and therefore as we-- 645 00:29:22,765 --> 00:29:24,640 ANDREW LO: How would the correlation increase 646 00:29:24,640 --> 00:29:25,870 by adding another stock? 647 00:29:25,870 --> 00:29:30,160 AUDIENCE: I mean the likelihood that I have 20 stocks, 648 00:29:30,160 --> 00:29:35,312 the overall correlation is higher rather than [INAUDIBLE] 649 00:29:35,312 --> 00:29:36,520 ANDREW LO: How could that be? 650 00:29:36,520 --> 00:29:39,310 You've got 20 stocks, and they've 651 00:29:39,310 --> 00:29:42,470 got a correlation among those 20 stocks. 652 00:29:42,470 --> 00:29:46,730 Now, I want you to think about adding a 21st stock. 653 00:29:46,730 --> 00:29:48,980 When you add that 21st stock, you 654 00:29:48,980 --> 00:29:51,380 don't affect the existing correlations, right? 655 00:29:51,380 --> 00:29:53,540 I mean, it is whatever it is. 656 00:29:53,540 --> 00:29:54,950 Those are parameters. 657 00:29:54,950 --> 00:29:57,790 At least for now, we're going to call them parameters. 658 00:29:57,790 --> 00:30:02,330 When I add my 21st stock, I'm giving the investor 659 00:30:02,330 --> 00:30:03,840 an extra degree of freedom. 660 00:30:03,840 --> 00:30:07,460 Now instead of investing among 20 securities, 661 00:30:07,460 --> 00:30:09,140 I'm going to let you invest among 21. 662 00:30:09,140 --> 00:30:11,772 You don't have to invest in the 21st. 663 00:30:11,772 --> 00:30:13,230 Or another way of thinking about it 664 00:30:13,230 --> 00:30:16,590 is that when you only had 20 stocks, 665 00:30:16,590 --> 00:30:19,710 you really had 21 portfolio weights. 666 00:30:19,710 --> 00:30:21,870 But the 21st weight, I've arbitrarily 667 00:30:21,870 --> 00:30:24,320 constrained to be 0. 668 00:30:24,320 --> 00:30:26,630 Now, I'm going to loosen the constraint 669 00:30:26,630 --> 00:30:28,070 and I'm going to say, OK, now you 670 00:30:28,070 --> 00:30:31,390 can invest in the 21st stock. 671 00:30:31,390 --> 00:30:33,340 You won't affect the existing correlations, 672 00:30:33,340 --> 00:30:34,900 but the new stock that you add in 673 00:30:34,900 --> 00:30:39,320 can benefit in providing additional diversification 674 00:30:39,320 --> 00:30:39,820 benefits. 675 00:30:39,820 --> 00:30:41,507 AUDIENCE: But as a general rule, we've 676 00:30:41,507 --> 00:30:45,467 always been trying to have negative correlation so 677 00:30:45,467 --> 00:30:46,515 that the bullet was left. 678 00:30:46,515 --> 00:30:47,890 ANDREW LO: Oh, well actually, you 679 00:30:47,890 --> 00:30:49,210 don't need a negative correlation 680 00:30:49,210 --> 00:30:50,751 to make this go to the left, you just 681 00:30:50,751 --> 00:30:53,540 need to have something less than 1. 682 00:30:53,540 --> 00:30:54,200 Remember? 683 00:30:54,200 --> 00:30:55,440 From the last lecture? 684 00:30:55,440 --> 00:30:59,000 This is a case where you had perfect correlation. 685 00:30:59,000 --> 00:31:01,850 Anything less than perfect correlation, 686 00:31:01,850 --> 00:31:04,850 brings you to the left. 687 00:31:04,850 --> 00:31:08,180 So as long as my 21st stock is not 688 00:31:08,180 --> 00:31:10,670 perfectly correlated with the existing 689 00:31:10,670 --> 00:31:13,400 stocks in that portfolio of 20, I'm 690 00:31:13,400 --> 00:31:14,894 going to move things to the left. 691 00:31:14,894 --> 00:31:17,190 AUDIENCE: In general, [INAUDIBLE] any stocks? 692 00:31:17,190 --> 00:31:18,690 ANDREW LO: In general, that is true. 693 00:31:18,690 --> 00:31:22,440 AUDIENCE: I mean I would try to have the negative correlation. 694 00:31:22,440 --> 00:31:24,690 ANDREW LO: You would, but what this suggests 695 00:31:24,690 --> 00:31:26,820 is that negative correlation is a very rare thing. 696 00:31:26,820 --> 00:31:27,690 AUDIENCE: It's difficult. 697 00:31:27,690 --> 00:31:30,064 ANDREW LO: It's very difficult, it's extremely difficult. 698 00:31:30,064 --> 00:31:32,760 Now, from the analytical perspective, 699 00:31:32,760 --> 00:31:34,260 we can conclude it's very difficult. 700 00:31:34,260 --> 00:31:37,440 Let me ask you from an economic perspective, 701 00:31:37,440 --> 00:31:40,680 why is it difficult to find a stock that's negatively 702 00:31:40,680 --> 00:31:42,540 correlated with all other stocks? 703 00:31:42,540 --> 00:31:46,350 Anybody give me a business rationale for that? 704 00:31:46,350 --> 00:31:47,502 Yeah, Ingrid? 705 00:31:47,502 --> 00:31:49,150 AUDIENCE: There's what you said before 706 00:31:49,150 --> 00:31:52,004 that when the economy goes down, everything goes down 707 00:31:52,004 --> 00:31:52,980 and we get vice versa. 708 00:31:52,980 --> 00:31:57,082 Just that we mention [INAUDIBLE] different countries, 709 00:31:57,082 --> 00:32:00,435 in different economic regions, in different industry, 710 00:32:00,435 --> 00:32:01,740 and they should not be-- 711 00:32:01,740 --> 00:32:03,990 ANDREW LO: Let's actually spend a little bit more time 712 00:32:03,990 --> 00:32:05,020 thinking about this. 713 00:32:05,020 --> 00:32:07,770 I want you guys to tell me right now, give me 714 00:32:07,770 --> 00:32:12,440 a stock that you would put your money in right now, today. 715 00:32:12,440 --> 00:32:17,640 S&P has gone down by 45% since the high several months ago. 716 00:32:17,640 --> 00:32:19,500 The stock market's doing terribly, 717 00:32:19,500 --> 00:32:21,610 and it doesn't look like it's getting any better. 718 00:32:21,610 --> 00:32:23,860 So you tell me, what stock would you put your money in 719 00:32:23,860 --> 00:32:25,090 right now, today? 720 00:32:25,090 --> 00:32:25,707 Yeah, Terry. 721 00:32:25,707 --> 00:32:26,790 AUDIENCE: Campbell's Soup. 722 00:32:26,790 --> 00:32:28,470 ANDREW LO: Campbell's Soup. 723 00:32:28,470 --> 00:32:29,380 Why is that? 724 00:32:29,380 --> 00:32:30,546 AUDIENCE: It's a food stock. 725 00:32:30,546 --> 00:32:35,848 It's a foodstuff people need, will purchase, inexpensive, 726 00:32:35,848 --> 00:32:37,300 pretty much just for-- 727 00:32:37,300 --> 00:32:38,760 ANDREW LO: OK. 728 00:32:38,760 --> 00:32:43,440 But on the other hand, if people are poorer all around, 729 00:32:43,440 --> 00:32:46,650 might not they start consuming even less of canned soup 730 00:32:46,650 --> 00:32:49,680 and try to make their own soup from little packages of ketchup 731 00:32:49,680 --> 00:32:51,390 and hot water? 732 00:32:51,390 --> 00:32:54,580 I saw that on an I Love Lucy episode years ago. 733 00:32:54,580 --> 00:32:56,980 It's pretty cool. 734 00:32:56,980 --> 00:32:58,286 So are you sure? 735 00:32:58,286 --> 00:32:59,660 Are you sure that Campbell's Soup 736 00:32:59,660 --> 00:33:02,250 is going to go up over the next few months, 737 00:33:02,250 --> 00:33:03,720 in response to the current crisis? 738 00:33:03,720 --> 00:33:05,332 AUDIENCE: It'll stay pretty stable. 739 00:33:05,332 --> 00:33:06,540 ANDREW LO: It'll stay stable. 740 00:33:06,540 --> 00:33:09,210 Ah, but that's not negative correlation. 741 00:33:09,210 --> 00:33:10,812 That's 0 correlation. 742 00:33:10,812 --> 00:33:12,270 I want something that's going to go 743 00:33:12,270 --> 00:33:15,300 the opposite direction of where the economy is heading. 744 00:33:15,300 --> 00:33:18,690 Tell me where that is? 745 00:33:18,690 --> 00:33:19,770 Yeah? 746 00:33:19,770 --> 00:33:21,907 AUDIENCE: [INAUDIBLE] short financials. 747 00:33:21,907 --> 00:33:22,740 ANDREW LO: OK, fine. 748 00:33:22,740 --> 00:33:24,240 So you're going to short the market. 749 00:33:24,240 --> 00:33:26,850 That's a cheap answer. 750 00:33:26,850 --> 00:33:29,610 Sorry, you don't get any credit for that. 751 00:33:29,610 --> 00:33:31,200 I want to answer the question that 752 00:33:31,200 --> 00:33:33,690 was raised by David which is, show me 753 00:33:33,690 --> 00:33:37,260 a stock that can get me even more to that left. 754 00:33:37,260 --> 00:33:39,990 I want a negatively correlated stock. 755 00:33:39,990 --> 00:33:41,090 Yeah, [INAUDIBLE]. 756 00:33:41,090 --> 00:33:42,230 AUDIENCE: Wal-Mart. 757 00:33:42,230 --> 00:33:43,890 ANDREW LO: Wal-Mart? 758 00:33:43,890 --> 00:33:45,255 AUDIENCE: It's been going up. 759 00:33:45,255 --> 00:33:47,130 ANDREW LO: Well, that's not the same thing 760 00:33:47,130 --> 00:33:48,644 as saying that it is going to go up 761 00:33:48,644 --> 00:33:50,310 over the next several months in response 762 00:33:50,310 --> 00:33:51,310 to this economic crisis. 763 00:33:51,310 --> 00:33:52,726 You don't think that there's going 764 00:33:52,726 --> 00:33:54,300 to be a decline in consumer spending 765 00:33:54,300 --> 00:33:57,570 that will affect retail as well? 766 00:33:57,570 --> 00:33:59,650 AUDIENCE: So far, everybody's going to Wal-Mart. 767 00:33:59,650 --> 00:34:01,900 If they don't go to Wal-Mart, where would they go? 768 00:34:01,900 --> 00:34:02,700 ANDREW LO: Well, that's what I'm asking you. 769 00:34:02,700 --> 00:34:03,750 Where are you going to go? 770 00:34:03,750 --> 00:34:05,458 So you're telling me now that you believe 771 00:34:05,458 --> 00:34:06,848 that Wal-Mart is the answer? 772 00:34:06,848 --> 00:34:08,639 You think it will be negatively correlated? 773 00:34:08,639 --> 00:34:10,382 Historically, just to let you know, 774 00:34:10,382 --> 00:34:12,090 retail has not been negatively correlated 775 00:34:12,090 --> 00:34:14,835 with the business cycle. 776 00:34:14,835 --> 00:34:15,460 Yeah, Zeke? 777 00:34:15,460 --> 00:34:17,011 AUDIENCE: What about Freddie Mac? 778 00:34:17,011 --> 00:34:17,969 ANDREW LO: Freddie Mac? 779 00:34:17,969 --> 00:34:18,594 AUDIENCE: Yeah. 780 00:34:18,594 --> 00:34:22,165 [LAUGHTER] 781 00:34:22,165 --> 00:34:23,790 ANDREW LO: If you like that investment, 782 00:34:23,790 --> 00:34:24,870 I have something else for you-- 783 00:34:24,870 --> 00:34:25,712 [INTERPOSING VOICES] 784 00:34:25,712 --> 00:34:26,554 --afterwards. 785 00:34:26,554 --> 00:34:28,482 AUDIENCE: We could do it this time. 786 00:34:28,482 --> 00:34:29,940 ANDREW LO: I don't know if you want 787 00:34:29,940 --> 00:34:32,159 to argue that Freddie Mac is negatively correlated 788 00:34:32,159 --> 00:34:33,076 with market downturns. 789 00:34:33,076 --> 00:34:35,409 I mean, the reason that Freddie Mac got into the trouble 790 00:34:35,409 --> 00:34:37,504 that it did was because of the economic downturn. 791 00:34:37,504 --> 00:34:38,625 All right, one more. 792 00:34:38,625 --> 00:34:39,380 [INAUDIBLE]? 793 00:34:39,380 --> 00:34:40,779 AUDIENCE: Philip Morris. 794 00:34:40,779 --> 00:34:41,820 ANDREW LO: Philip Morris. 795 00:34:41,820 --> 00:34:44,250 That's an interesting one. 796 00:34:44,250 --> 00:34:46,050 Obviously, people are very nervous now. 797 00:34:46,050 --> 00:34:49,230 When you're nervous, you're going to be smoking. 798 00:34:51,989 --> 00:34:53,940 On the other hand, again, one could 799 00:34:53,940 --> 00:34:56,580 argue that it's not negatively correlated. 800 00:34:56,580 --> 00:35:02,100 It might be either slightly positively correlated, 801 00:35:02,100 --> 00:35:04,200 but even there, people have argued 802 00:35:04,200 --> 00:35:07,350 that cigarettes are a consumption good that can get 803 00:35:07,350 --> 00:35:10,724 hit with a downturn in markets. 804 00:35:10,724 --> 00:35:12,390 The bottom line is that it's really hard 805 00:35:12,390 --> 00:35:14,832 to come up with negative correlated stocks. 806 00:35:14,832 --> 00:35:17,040 Let me tell you, if you found one that was negatively 807 00:35:17,040 --> 00:35:20,490 correlated, if you found one that was really, really, 808 00:35:20,490 --> 00:35:23,112 negatively correlated, what would all of you do? 809 00:35:23,112 --> 00:35:23,820 AUDIENCE: Buy it. 810 00:35:23,820 --> 00:35:26,310 ANDREW LO: Exactly, you'd buy it. 811 00:35:26,310 --> 00:35:30,210 The effect of that would be to increase the price 812 00:35:30,210 --> 00:35:33,570 and depress the expected return. 813 00:35:33,570 --> 00:35:35,160 Remember what the expected return is. 814 00:35:35,160 --> 00:35:39,270 It's the expected future price, divided by the current price. 815 00:35:39,270 --> 00:35:42,150 If now all of you go out and buy Wal-Mart, 816 00:35:42,150 --> 00:35:44,900 or whatever stock you think is negatively correlated, 817 00:35:44,900 --> 00:35:48,990 that would have the impact of increasing the current price 818 00:35:48,990 --> 00:35:52,164 and therefore decreasing the expected return. 819 00:35:52,164 --> 00:35:53,580 Now if you have a stock that's got 820 00:35:53,580 --> 00:35:56,821 a negative covariance and a negative return, 821 00:35:56,821 --> 00:35:57,570 that doesn't help. 822 00:35:57,570 --> 00:35:59,340 Because in fact, that was a suggestion 823 00:35:59,340 --> 00:36:00,660 that was put forward here. 824 00:36:00,660 --> 00:36:03,960 Let's just take the S&P and short it, 825 00:36:03,960 --> 00:36:06,280 and then you get a negatively correlated stock. 826 00:36:06,280 --> 00:36:09,000 The problem is that it's also got a negative expected return 827 00:36:09,000 --> 00:36:11,306 and then you're not helping things. 828 00:36:11,306 --> 00:36:12,930 The key is to find negative correlation 829 00:36:12,930 --> 00:36:14,100 with a positive return. 830 00:36:14,100 --> 00:36:16,410 If you can find that, then you've 831 00:36:16,410 --> 00:36:19,110 really found something worthwhile. 832 00:36:19,110 --> 00:36:22,830 But my guess is it won't last, for exactly this reason. 833 00:36:22,830 --> 00:36:24,420 Other questions? 834 00:36:24,420 --> 00:36:27,650 OK, so now, let's go back and ask the question, what 835 00:36:27,650 --> 00:36:30,920 does this mean if we agree that all of us 836 00:36:30,920 --> 00:36:33,920 want to be on that tangency portfolio. 837 00:36:33,920 --> 00:36:35,940 What does that tell us? 838 00:36:35,940 --> 00:36:41,250 Well, that allows us to then make an argument 839 00:36:41,250 --> 00:36:46,170 that managers that are trying to provide value 840 00:36:46,170 --> 00:36:51,560 added services for us, they need to be doing something above 841 00:36:51,560 --> 00:36:55,700 and beyond what we can do ourselves. 842 00:36:55,700 --> 00:36:58,280 Now, here's where Warren Buffett meets modern finance theory, 843 00:36:58,280 --> 00:37:01,090 in a way. 844 00:37:01,090 --> 00:37:02,740 If I want to see whether or not Warren 845 00:37:02,740 --> 00:37:05,290 Buffett or any other managers are adding value, 846 00:37:05,290 --> 00:37:08,990 one simple criterion that I can put forward is this. 847 00:37:08,990 --> 00:37:10,960 This Is what I can do on my own. 848 00:37:10,960 --> 00:37:16,240 I can get that line pretty much by just using my basic finance 849 00:37:16,240 --> 00:37:19,670 skills that I've learned here at MIT. 850 00:37:19,670 --> 00:37:23,570 If you're going to manage my money and charge me 2 and 20, 851 00:37:23,570 --> 00:37:26,420 show me what you can do above and beyond this. 852 00:37:26,420 --> 00:37:29,720 I want you to tell me where you can get me on this graph. 853 00:37:29,720 --> 00:37:32,870 Can you get me up here? 854 00:37:32,870 --> 00:37:34,540 Can you get me over here? 855 00:37:34,540 --> 00:37:37,690 Can you get me anywhere either to the left 856 00:37:37,690 --> 00:37:40,850 or above that curve? 857 00:37:40,850 --> 00:37:43,650 We can use that as a measure of performance, 858 00:37:43,650 --> 00:37:45,680 and there's a name for that. 859 00:37:45,680 --> 00:37:46,910 It's called Alpha. 860 00:37:46,910 --> 00:37:48,980 Typically, when people talk about Alpha, 861 00:37:48,980 --> 00:37:52,022 they're talking about deviations from a line like this. 862 00:37:52,022 --> 00:37:53,730 We're going to get to that more formally, 863 00:37:53,730 --> 00:37:56,230 you don't have to write it down or make note of it just yet. 864 00:37:56,230 --> 00:37:57,544 It's on the next slide. 865 00:37:57,544 --> 00:37:59,960 But we're going to show you how to measure that explicitly 866 00:37:59,960 --> 00:38:04,460 so now, not only is this a good idea for you as a baseline 867 00:38:04,460 --> 00:38:07,310 to manage your own portfolio, but you can then 868 00:38:07,310 --> 00:38:09,650 use it as a metric to gauge whether other people are 869 00:38:09,650 --> 00:38:11,220 adding value to you. 870 00:38:11,220 --> 00:38:14,780 So Warren Buffett would say, no problem. 871 00:38:14,780 --> 00:38:16,310 I think I've got Alpha. 872 00:38:16,310 --> 00:38:18,110 So I'm not going to bother with this, 873 00:38:18,110 --> 00:38:20,710 I think I can get you up here. 874 00:38:20,710 --> 00:38:23,320 That is, if you want to invest with me. 875 00:38:23,320 --> 00:38:26,380 And in fact, if you looked at Warren Buffett's performance 876 00:38:26,380 --> 00:38:30,310 over the last 25 years that he's been doing it, or 30 years, 877 00:38:30,310 --> 00:38:36,250 his Sharpe ratio is a lot better than the tangency portfolios. 878 00:38:36,250 --> 00:38:40,420 So he actually has added value if you use this as a criterion. 879 00:38:40,420 --> 00:38:43,780 But the problem is, you have to identify the Warren 880 00:38:43,780 --> 00:38:46,171 Buffetts before they become Warren Buffetts. 881 00:38:46,171 --> 00:38:47,920 Because after they become Warren Buffetts, 882 00:38:47,920 --> 00:38:49,030 it's not clear that they're adding 883 00:38:49,030 --> 00:38:50,350 the same amount of value. 884 00:38:50,350 --> 00:38:52,490 It's already, the cat's out of the bag. 885 00:38:52,490 --> 00:38:54,028 Yeah, [INAUDIBLE]? 886 00:38:54,028 --> 00:38:55,819 AUDIENCE: Question about the Sharpe ration. 887 00:38:55,819 --> 00:38:57,996 So is this a stag ratio, or is it dynamic? 888 00:38:57,996 --> 00:39:02,956 Because in my mind, as you gain more stock options, 889 00:39:02,956 --> 00:39:04,227 it's going to become sharper. 890 00:39:04,227 --> 00:39:04,810 ANDREW LO: Yes 891 00:39:04,810 --> 00:39:07,175 AUDIENCE: Opportunity cost for switching to T-Bills 892 00:39:07,175 --> 00:39:08,712 is going to be greater, so you're 893 00:39:08,712 --> 00:39:10,587 going to shift preferences away from T-Bills. 894 00:39:10,587 --> 00:39:13,230 Then isn't that point going to increase and flatten out? 895 00:39:13,230 --> 00:39:17,350 ANDREW LO: Yes, so the dynamics of this are very complex. 896 00:39:17,350 --> 00:39:19,060 This is, right now, a static theory. 897 00:39:19,060 --> 00:39:21,700 Static meaning today versus next period. 898 00:39:21,700 --> 00:39:23,534 We're not looking at the dynamics over time. 899 00:39:23,534 --> 00:39:25,783 In order to do that, there's lots of different effects 900 00:39:25,783 --> 00:39:27,540 that are much, much more complicated. 901 00:39:27,540 --> 00:39:30,010 For that, you've really got to take 15 433 902 00:39:30,010 --> 00:39:32,680 and even 433 won't cover those kinds of questions 903 00:39:32,680 --> 00:39:36,850 in complete detail because they rely on some very complex kinds 904 00:39:36,850 --> 00:39:37,444 of analysis. 905 00:39:37,444 --> 00:39:39,110 But I'm going to get to that at the end. 906 00:39:39,110 --> 00:39:41,500 So if I don't, please bring it up again. 907 00:39:41,500 --> 00:39:43,390 I want to make a comment about that, 908 00:39:43,390 --> 00:39:46,630 and how you can take this relatively simple static theory 909 00:39:46,630 --> 00:39:49,240 and make it dynamic in an informal way 910 00:39:49,240 --> 00:39:51,790 even though the analytics become very hard when 911 00:39:51,790 --> 00:39:54,640 you try to do it formally. 912 00:39:54,640 --> 00:39:58,840 So the key points of this lecture are, oh, sorry, 913 00:39:58,840 --> 00:40:00,076 question? 914 00:40:00,076 --> 00:40:02,850 AUDIENCE: [INAUDIBLE] every point on the line, 915 00:40:02,850 --> 00:40:04,230 it's indifferent? 916 00:40:04,230 --> 00:40:04,865 Or-- 917 00:40:04,865 --> 00:40:05,740 ANDREW LO: No, no no. 918 00:40:05,740 --> 00:40:07,870 Not at all, not at all indifferent. 919 00:40:07,870 --> 00:40:10,900 Any point on this line is a different risk-reward 920 00:40:10,900 --> 00:40:12,040 combination. 921 00:40:12,040 --> 00:40:14,162 So in other words, it depends on your preferences. 922 00:40:14,162 --> 00:40:16,120 AUDIENCE: You take a function for the industry? 923 00:40:16,120 --> 00:40:17,020 ANDREW LO: Yes, yes. 924 00:40:17,020 --> 00:40:19,450 Now, we haven't talked about utility functions yet, 925 00:40:19,450 --> 00:40:21,380 but we're going to in a little while. 926 00:40:21,380 --> 00:40:24,790 Let me preview that, since you asked. 927 00:40:24,790 --> 00:40:27,370 You all remember what indifference curves are? 928 00:40:27,370 --> 00:40:29,380 From basic economics? 929 00:40:29,380 --> 00:40:32,110 An indifference curve, when I first came across that, 930 00:40:32,110 --> 00:40:36,070 I was rather offended because I don't view myself 931 00:40:36,070 --> 00:40:37,990 as an indifferent individual. 932 00:40:37,990 --> 00:40:39,820 I have lots of passions. 933 00:40:39,820 --> 00:40:43,666 And so, why should we be indifferent about two choices? 934 00:40:43,666 --> 00:40:45,040 In fact, that's an economic term. 935 00:40:45,040 --> 00:40:49,000 It simply means that you are just as well off between two 936 00:40:49,000 --> 00:40:51,430 combinations and therefore, these two 937 00:40:51,430 --> 00:40:55,600 combinations you're indifferent to, you're indifferent between. 938 00:40:55,600 --> 00:41:01,690 So if I had to ask you to draw on this graph, an indifference 939 00:41:01,690 --> 00:41:06,501 curve of risk-reward trade-offs for you, 940 00:41:06,501 --> 00:41:08,500 the typical individual, what would it look like? 941 00:41:08,500 --> 00:41:11,650 Can anybody give me a sense of what different kinds 942 00:41:11,650 --> 00:41:15,880 of risk-reward trade-offs you would be indifferent among? 943 00:41:15,880 --> 00:41:18,790 And to make the question a little bit simpler, 944 00:41:18,790 --> 00:41:20,890 let's start off with a particular point. 945 00:41:20,890 --> 00:41:22,780 So let's suppose that this point right here 946 00:41:22,780 --> 00:41:25,450 is the point that I want you to draw the indifferent 947 00:41:25,450 --> 00:41:27,610 curve from. 948 00:41:27,610 --> 00:41:31,570 Which is a standard deviation on a monthly basis of about 6%, 949 00:41:31,570 --> 00:41:38,180 and an expected return of about, say 1.4% or something. 950 00:41:38,180 --> 00:41:45,780 So you've got a monthly return of 1.4% and a risk of about 6%. 951 00:41:45,780 --> 00:41:50,760 Give me another point that you would be indifferent between, 952 00:41:50,760 --> 00:41:51,560 versus that one? 953 00:41:56,140 --> 00:41:56,640 Anybody? 954 00:41:56,640 --> 00:41:57,480 Any volunteers? 955 00:42:00,917 --> 00:42:01,899 Yeah? 956 00:42:01,899 --> 00:42:06,640 AUDIENCE: There could be a point above the tangent line, 957 00:42:06,640 --> 00:42:07,607 but to your right. 958 00:42:07,607 --> 00:42:08,940 Somewhere where you're pointing. 959 00:42:08,940 --> 00:42:11,580 ANDREW LO: OK, so do you have a particular number in mind? 960 00:42:11,580 --> 00:42:13,760 In other words, let me ask you this. 961 00:42:13,760 --> 00:42:18,524 If I cranked up your volatility from 6% to 8%, 962 00:42:18,524 --> 00:42:19,940 how much extra return would I have 963 00:42:19,940 --> 00:42:21,898 to give you in order for you to be just as well 964 00:42:21,898 --> 00:42:26,299 off as you were at 6% and 1.4%? 965 00:42:26,299 --> 00:42:28,340 AUDIENCE: Something slightly higher than the 1.8% 966 00:42:28,340 --> 00:42:29,720 [INAUDIBLE]. 967 00:42:29,720 --> 00:42:32,430 ANDREW LO: OK, higher than 1.8% OK. 968 00:42:32,430 --> 00:42:35,353 AUDIENCE: Or what about the corresponding value. 969 00:42:35,353 --> 00:42:35,853 1.41%. 970 00:42:35,853 --> 00:42:38,790 1.41% 971 00:42:38,790 --> 00:42:40,940 ANDREW LO: OK, this is 1.41%. 972 00:42:40,940 --> 00:42:43,670 And if I said, now I want to be at 8%, 973 00:42:43,670 --> 00:42:45,171 how much risk do I have to give you, 974 00:42:45,171 --> 00:42:47,586 how much expected return do I have to give you to make you 975 00:42:47,586 --> 00:42:49,180 just as well off as this point? 976 00:42:52,180 --> 00:42:54,390 AUDIENCE: Higher then 1.2%, right? 977 00:42:54,390 --> 00:42:56,097 Or 1.4%, higher than 1.4%. 978 00:42:56,097 --> 00:42:57,680 ANDREW LO: Right, but how much higher? 979 00:42:57,680 --> 00:42:58,513 That's the question. 980 00:42:58,513 --> 00:42:59,701 It's a personal question. 981 00:42:59,701 --> 00:43:00,200 Rami? 982 00:43:00,200 --> 00:43:05,190 AUDIENCE: 1.3 times your initial expected return. 983 00:43:05,190 --> 00:43:06,746 So 33% on top. 984 00:43:06,746 --> 00:43:08,120 ANDREW LO: You would have to have 985 00:43:08,120 --> 00:43:11,150 an increase in 33% of your expected return, 986 00:43:11,150 --> 00:43:15,134 even though I'm only giving you a 25% increase in the risk. 987 00:43:15,134 --> 00:43:16,118 AUDIENCE: Well, no. 988 00:43:16,118 --> 00:43:19,316 You'd have to at least do 25% of the-- sorry. 989 00:43:19,316 --> 00:43:20,494 Put 6%-8% is-- 990 00:43:20,494 --> 00:43:22,160 ANDREW LO: That's a third, you're right. 991 00:43:22,160 --> 00:43:23,990 So I'm increasing the risk by 1/3, 992 00:43:23,990 --> 00:43:26,510 you want me to increase the expected return by 1/3. 993 00:43:26,510 --> 00:43:28,580 So your trade-off is linear, is that right? 994 00:43:28,580 --> 00:43:31,040 You're looking at it linearly? 995 00:43:31,040 --> 00:43:32,404 Anybody else? 996 00:43:32,404 --> 00:43:34,070 You know, you may want to translate this 997 00:43:34,070 --> 00:43:35,630 into annual numbers. 998 00:43:35,630 --> 00:43:38,480 Because I'm sensing that you may not have a good 999 00:43:38,480 --> 00:43:40,520 feel for what your own preferences are. 1000 00:43:40,520 --> 00:43:42,260 And by the way, this is a challenge. 1001 00:43:42,260 --> 00:43:46,370 Not everybody understands what their own personal preferences 1002 00:43:46,370 --> 00:43:47,970 are for these numbers. 1003 00:43:47,970 --> 00:43:51,230 This is not a natural act of human nature, 1004 00:43:51,230 --> 00:43:55,670 that we automatically have preferences on these numbers. 1005 00:43:55,670 --> 00:43:59,770 But the bottom line is that if I make you take more risk, 1006 00:43:59,770 --> 00:44:02,020 I'm going to have to compensate you and give you 1007 00:44:02,020 --> 00:44:02,960 more expected return. 1008 00:44:02,960 --> 00:44:05,980 There's got to be a reason why you want to take that risk. 1009 00:44:05,980 --> 00:44:08,486 For some people, it's linear. 1010 00:44:08,486 --> 00:44:10,360 For other people, it's much more than linear. 1011 00:44:10,360 --> 00:44:11,943 They don't want to take any more risk. 1012 00:44:11,943 --> 00:44:14,140 In fact, right now most investors 1013 00:44:14,140 --> 00:44:16,320 don't even want to answer that question. 1014 00:44:16,320 --> 00:44:18,070 Because they don't want to take more risk. 1015 00:44:18,070 --> 00:44:19,160 And you say, well what if you did? 1016 00:44:19,160 --> 00:44:20,210 Well I don't want to. 1017 00:44:20,210 --> 00:44:21,330 Well, but just what if? 1018 00:44:21,330 --> 00:44:22,330 I don't want to what if. 1019 00:44:22,330 --> 00:44:24,460 I just don't want to take the risk. 1020 00:44:24,460 --> 00:44:26,290 So they can't even answer that question. 1021 00:44:26,290 --> 00:44:29,450 But if they could, my guess is that it would be way up here. 1022 00:44:29,450 --> 00:44:31,960 So you'd have to give them a lot of expected return 1023 00:44:31,960 --> 00:44:34,360 to make people take more risk today. 1024 00:44:34,360 --> 00:44:37,480 Alternatively, if you want to give people less risk, 1025 00:44:37,480 --> 00:44:39,040 my guess is that you can actually 1026 00:44:39,040 --> 00:44:42,610 subtract a lot of return in order to take away 1027 00:44:42,610 --> 00:44:43,640 a little bit of risk. 1028 00:44:43,640 --> 00:44:44,620 How do I know that? 1029 00:44:44,620 --> 00:44:46,828 Take a look at the yield on the three month treasury. 1030 00:44:50,420 --> 00:44:53,370 So an indifference curve. 1031 00:44:53,370 --> 00:44:54,730 It's going to look like this. 1032 00:44:54,730 --> 00:44:58,530 It's going to look like it'll be increasing, 1033 00:44:58,530 --> 00:45:01,680 but it'll actually be bowed this way. 1034 00:45:01,680 --> 00:45:04,940 And the theory behind why it's going to be convex, 1035 00:45:04,940 --> 00:45:07,240 holds water as opposed to spills water, 1036 00:45:07,240 --> 00:45:08,880 the reason it's going to be convex 1037 00:45:08,880 --> 00:45:12,000 is because there is a decreasing, or diminishing, 1038 00:45:12,000 --> 00:45:15,600 marginal utility between risk and expected rate of return. 1039 00:45:15,600 --> 00:45:18,420 Like anything else, economists have this notion 1040 00:45:18,420 --> 00:45:22,320 of diminishing marginal utility between any two commodities. 1041 00:45:22,320 --> 00:45:29,040 If you've got ice cream sundaes and basketballs, 1042 00:45:29,040 --> 00:45:30,750 there's only so many basketballs that you 1043 00:45:30,750 --> 00:45:34,380 can enjoy before the next incremental basketball provides 1044 00:45:34,380 --> 00:45:36,240 relatively little pleasure for you. 1045 00:45:36,240 --> 00:45:37,890 The same thing with ice cream sundaes. 1046 00:45:37,890 --> 00:45:39,870 You can only consume so many ice cream sundaes 1047 00:45:39,870 --> 00:45:42,990 before the next incremental sundae provides 1048 00:45:42,990 --> 00:45:45,840 somewhat less benefit to you. 1049 00:45:45,840 --> 00:45:48,480 That kind of diminishing marginal utility 1050 00:45:48,480 --> 00:45:50,700 gives you this kind of a bowed curve. 1051 00:45:50,700 --> 00:45:53,250 So where you are on this straight line 1052 00:45:53,250 --> 00:45:56,070 depends upon how bowed your curve is. 1053 00:45:56,070 --> 00:45:59,430 Somebody that's really risk-averse has a curve that 1054 00:45:59,430 --> 00:45:59,930 looks-- 1055 00:45:59,930 --> 00:46:01,971 let me draw this because it's a little bit easier 1056 00:46:01,971 --> 00:46:04,230 to see rather than trying to follow my laser pointer. 1057 00:46:07,890 --> 00:46:10,650 So here's the trade-off, this is the line. 1058 00:46:10,650 --> 00:46:12,570 Somebody that's extremely risk-averse 1059 00:46:12,570 --> 00:46:14,355 is going to have curves like this. 1060 00:46:17,500 --> 00:46:19,310 Those are indifference curves. 1061 00:46:19,310 --> 00:46:23,640 And as you go to the Northwest, you're happier and happier. 1062 00:46:23,640 --> 00:46:28,140 So the optimal point is where this indifference curve 1063 00:46:28,140 --> 00:46:31,970 hits this particular line. 1064 00:46:31,970 --> 00:46:34,970 On the other hand, if you're very risk-seeking, 1065 00:46:34,970 --> 00:46:36,980 if you don't need a lot of compensation 1066 00:46:36,980 --> 00:46:39,205 of expected return per unit risk, 1067 00:46:39,205 --> 00:46:41,580 then your difference curve's not going to look like that. 1068 00:46:41,580 --> 00:46:45,420 It's going to look like this. 1069 00:46:45,420 --> 00:46:50,970 In which case, you're tangency point will be farther 1070 00:46:50,970 --> 00:46:53,620 to the Northeast. 1071 00:46:53,620 --> 00:46:55,600 You'll be taking more risk and getting 1072 00:46:55,600 --> 00:46:58,630 more expected rates of return. 1073 00:46:58,630 --> 00:47:01,180 But the bottom line for this graph and this lecture 1074 00:47:01,180 --> 00:47:05,395 is that everybody, no matter what your risk preferences are, 1075 00:47:05,395 --> 00:47:07,270 everybody's going to want to be on that line, 1076 00:47:07,270 --> 00:47:09,570 that tangency line. 1077 00:47:09,570 --> 00:47:12,020 And it turns out that that insight 1078 00:47:12,020 --> 00:47:16,070 is going to translate into a remarkable, remarkable 1079 00:47:16,070 --> 00:47:21,140 conclusion about risk-reward trade-offs. 1080 00:47:21,140 --> 00:47:23,590 So the key points for this lecture 1081 00:47:23,590 --> 00:47:27,110 are diversification reduces risk. 1082 00:47:27,110 --> 00:47:29,600 In diversified portfolios, covariances 1083 00:47:29,600 --> 00:47:31,940 are the most important characteristics 1084 00:47:31,940 --> 00:47:32,840 of that portfolio. 1085 00:47:32,840 --> 00:47:37,150 It's not the variances, but the covariances. 1086 00:47:37,150 --> 00:47:39,250 Investors should try to hold portfolios 1087 00:47:39,250 --> 00:47:41,920 on the efficient frontier, that upper branch. 1088 00:47:41,920 --> 00:47:45,160 And with the riskless asset, everybody 1089 00:47:45,160 --> 00:47:48,790 is going to want to be on the tangency line. 1090 00:47:48,790 --> 00:47:51,715 Those are the major conclusions from this analysis. 1091 00:47:51,715 --> 00:47:53,590 And you can work all of this out analytically 1092 00:47:53,590 --> 00:47:56,320 using the mathematics of optimization theory, 1093 00:47:56,320 --> 00:47:59,080 but in fact, all of this can be done graphically 1094 00:47:59,080 --> 00:48:00,935 as we have geometrically. 1095 00:48:03,790 --> 00:48:05,071 Question, [INAUDIBLE]? 1096 00:48:05,071 --> 00:48:07,487 AUDIENCE: Sorry, this is sort of a simple-minded question, 1097 00:48:07,487 --> 00:48:12,928 but I'm having trouble thinking of the expected return. 1098 00:48:12,928 --> 00:48:20,188 I know it's absolute, but so much of the portfolio 1099 00:48:20,188 --> 00:48:24,060 metrics, I guess, are relative to the benchmark of the market 1100 00:48:24,060 --> 00:48:25,512 rather than [INAUDIBLE]. 1101 00:48:25,512 --> 00:48:26,480 So I don't know. 1102 00:48:26,480 --> 00:48:28,360 ANDREW LO: OK, so we're going to get to that. 1103 00:48:28,360 --> 00:48:30,485 We're going to talk about benchmarks because you're 1104 00:48:30,485 --> 00:48:34,370 right, that most investments today are all benchmarked 1105 00:48:34,370 --> 00:48:36,470 against something, right? 1106 00:48:36,470 --> 00:48:39,830 And you're probably wondering how that got to be. 1107 00:48:39,830 --> 00:48:45,810 That whole direction of analysis and performance attribution, 1108 00:48:45,810 --> 00:48:47,805 that came out of this. 1109 00:48:47,805 --> 00:48:49,180 In other words, it was because of 1110 00:48:49,180 --> 00:48:51,040 this particular academic framework 1111 00:48:51,040 --> 00:48:53,560 that was developed by Harry Markowitz, and Bill 1112 00:48:53,560 --> 00:48:56,380 Sharpe, and others, that indexation 1113 00:48:56,380 --> 00:48:58,494 and benchmarking came to be. 1114 00:48:58,494 --> 00:48:59,660 So I'm going to get to that. 1115 00:48:59,660 --> 00:49:02,050 Let me put that off for another lecture or so. 1116 00:49:02,050 --> 00:49:05,260 After we derive the implications of everybody wanting 1117 00:49:05,260 --> 00:49:07,210 to hold the tangency portfolio, it's 1118 00:49:07,210 --> 00:49:10,510 going to turn out that that tangency portfolio happens 1119 00:49:10,510 --> 00:49:11,740 to be the benchmark. 1120 00:49:11,740 --> 00:49:13,020 So we'll get to that. 1121 00:49:13,020 --> 00:49:14,639 Yeah? 1122 00:49:14,639 --> 00:49:16,014 AUDIENCE: Is there any assumption 1123 00:49:16,014 --> 00:49:19,402 behind all this analysis that would say, 1124 00:49:19,402 --> 00:49:21,338 if you have these preferences, then you 1125 00:49:21,338 --> 00:49:24,242 should choose this portfolio. 1126 00:49:24,242 --> 00:49:28,320 If everyone did that, does that change [INAUDIBLE] 1127 00:49:28,320 --> 00:49:30,980 ANDREW LO: So you would think that it would, but in fact, 1128 00:49:30,980 --> 00:49:33,610 I'm going to show you that there is exactly one case where it 1129 00:49:33,610 --> 00:49:34,167 doesn't. 1130 00:49:34,167 --> 00:49:35,750 And that's the case of the equilibrium 1131 00:49:35,750 --> 00:49:37,240 that I'm about describe. 1132 00:49:37,240 --> 00:49:39,010 So let me turn to that right now. 1133 00:49:39,010 --> 00:49:40,120 There any other questions? 1134 00:49:40,120 --> 00:49:42,369 AUDIENCE: We've been using risk and standard deviation 1135 00:49:42,369 --> 00:49:44,260 kind of interchangeably, whereas I 1136 00:49:44,260 --> 00:49:47,438 think of risk as the risk of not making anything. 1137 00:49:47,438 --> 00:49:49,188 Is there a way to mathematically translate 1138 00:49:49,188 --> 00:49:50,987 from a standard deviation in your portfolio 1139 00:49:50,987 --> 00:49:52,570 to the risk of not making [INAUDIBLE]? 1140 00:49:52,570 --> 00:49:54,580 ANDREW LO: Well, there is. 1141 00:49:54,580 --> 00:49:56,410 Although, I would have to say that if you 1142 00:49:56,410 --> 00:50:01,010 have a preference about the downside, 1143 00:50:01,010 --> 00:50:03,240 so not making anything as you point out. 1144 00:50:03,240 --> 00:50:05,240 Then that changes this analysis. 1145 00:50:05,240 --> 00:50:09,370 So this analysis really requires that you use standard deviation 1146 00:50:09,370 --> 00:50:13,270 as the sum total of your perception 1147 00:50:13,270 --> 00:50:15,310 of the risk of a portfolio. 1148 00:50:15,310 --> 00:50:17,890 If you have other kinds of sensitivities, 1149 00:50:17,890 --> 00:50:20,170 then you need to bring them into the analysis, 1150 00:50:20,170 --> 00:50:22,668 and that will change these outcomes. 1151 00:50:22,668 --> 00:50:25,720 AUDIENCE: The way we do this, if we 1152 00:50:25,720 --> 00:50:30,998 measure the risk of a company, if they're historically-- 1153 00:50:30,998 --> 00:50:31,962 Nevermind. 1154 00:50:31,962 --> 00:50:34,372 I guess if they were historically, 1155 00:50:34,372 --> 00:50:37,060 they varied higher, if that wasn't strictly normal, 1156 00:50:37,060 --> 00:50:39,820 and they end up being higher in market [INAUDIBLE] lower, 1157 00:50:39,820 --> 00:50:42,647 they would still have a larger deviation 1158 00:50:42,647 --> 00:50:46,383 so you're correlating that with companies that are also 1159 00:50:46,383 --> 00:50:47,784 [INAUDIBLE]. 1160 00:50:47,784 --> 00:50:48,730 Does that make sens? 1161 00:50:48,730 --> 00:50:50,438 ANDREW LO: That's true, but again, you've 1162 00:50:50,438 --> 00:50:52,684 made an assumption there that I'm not making. 1163 00:50:52,684 --> 00:50:54,850 Which is you're assuming companies are outperforming 1164 00:50:54,850 --> 00:50:56,230 or underperforming. 1165 00:50:56,230 --> 00:50:58,300 I'm assuming that the data are given, 1166 00:50:58,300 --> 00:51:00,790 and I'm not making a bet on whether any companies are 1167 00:51:00,790 --> 00:51:02,830 likely to succeed or fail. 1168 00:51:02,830 --> 00:51:06,040 I'm merely looking at companies as investment opportunities 1169 00:51:06,040 --> 00:51:08,410 that provide certain expected returns, 1170 00:51:08,410 --> 00:51:10,682 volatilities, and covariances. 1171 00:51:10,682 --> 00:51:12,640 You want to go down the path of Warren Buffett, 1172 00:51:12,640 --> 00:51:14,290 and I'm resisting that because I don't have 1173 00:51:14,290 --> 00:51:15,581 the skills of a Warren Buffett. 1174 00:51:15,581 --> 00:51:17,200 So I don't know what's a good value 1175 00:51:17,200 --> 00:51:18,820 and what's not a good value. 1176 00:51:18,820 --> 00:51:21,180 And the case in point is a discussion we just had today. 1177 00:51:21,180 --> 00:51:22,805 You tell me what is a good value today? 1178 00:51:22,805 --> 00:51:25,300 Do you really believe that Campbell's Soup or Wal-Mart 1179 00:51:25,300 --> 00:51:27,370 should be the companies you invest in today? 1180 00:51:27,370 --> 00:51:28,171 I don't know. 1181 00:51:28,171 --> 00:51:29,920 I mean, another argument is entertainment. 1182 00:51:29,920 --> 00:51:31,300 Why don't you invest in movie theaters? 1183 00:51:31,300 --> 00:51:33,730 Lots of people now are going to see the James Bond movie, 1184 00:51:33,730 --> 00:51:35,362 and they want to escape from reality. 1185 00:51:35,362 --> 00:51:36,820 Wouldn't that be a growth industry, 1186 00:51:36,820 --> 00:51:37,780 given market conditions? 1187 00:51:37,780 --> 00:51:39,363 Well, that's true, but how many people 1188 00:51:39,363 --> 00:51:40,594 have $12 to spend on a movie? 1189 00:51:40,594 --> 00:51:42,760 Plus, you've got to get the popcorn and the bonbons, 1190 00:51:42,760 --> 00:51:43,343 and all those. 1191 00:51:43,343 --> 00:51:46,570 And by the time you're done, it's like a $60 evening. 1192 00:51:46,570 --> 00:51:47,770 I mean, I don't know. 1193 00:51:47,770 --> 00:51:49,720 So the point is that unless you are 1194 00:51:49,720 --> 00:51:52,630 willing to make predictions, this 1195 00:51:52,630 --> 00:51:55,960 is the only alternative that provides a disciplined 1196 00:51:55,960 --> 00:52:01,630 approach to investing in so-called good portfolios. 1197 00:52:01,630 --> 00:52:05,690 So it's a different approach. 1198 00:52:05,690 --> 00:52:10,220 So now, let me turn to the next lectures. 1199 00:52:10,220 --> 00:52:13,040 Lectures 15 through 17, where we're now going 1200 00:52:13,040 --> 00:52:15,480 to talk about equilibrium. 1201 00:52:15,480 --> 00:52:18,800 We've already identified that all of us in this room, 1202 00:52:18,800 --> 00:52:22,370 assuming we have mean variance preferences, that's 1203 00:52:22,370 --> 00:52:24,800 an important assumption, I grant you, 1204 00:52:24,800 --> 00:52:26,420 but it's not an unreasonable one. 1205 00:52:26,420 --> 00:52:29,895 It's just, it is an important assumption. 1206 00:52:29,895 --> 00:52:31,270 We've all agreed that we're going 1207 00:52:31,270 --> 00:52:37,900 to take on portfolios that lie on that line, and therefore, 1208 00:52:37,900 --> 00:52:41,470 the portfolio that is the tangency portfolio, 1209 00:52:41,470 --> 00:52:43,930 I'm going to give it a special name. 1210 00:52:43,930 --> 00:52:50,990 I'm going to call it M, portfolio M. What we now 1211 00:52:50,990 --> 00:52:56,780 know is that, given a choice between holding n 1212 00:52:56,780 --> 00:53:04,940 securities and T-Bills, versus holding 1213 00:53:04,940 --> 00:53:11,030 T-Bills and a single portfolio, all of you 1214 00:53:11,030 --> 00:53:13,710 would be indifferent between those two choices, 1215 00:53:13,710 --> 00:53:21,470 if that single portfolio were M. The tangency portfolio. 1216 00:53:21,470 --> 00:53:24,270 Do we agree on that? 1217 00:53:24,270 --> 00:53:28,230 So therefore, I could, in principle, 1218 00:53:28,230 --> 00:53:35,110 construct a mutual fund called M. This mutual fund holds 1219 00:53:35,110 --> 00:53:39,610 stocks in exact proportion to the weights given 1220 00:53:39,610 --> 00:53:41,740 by that tangency portfolio. 1221 00:53:41,740 --> 00:53:46,800 In other words, it is the tangency portfolio. 1222 00:53:46,800 --> 00:53:51,730 So what that suggests is that all of you in this room 1223 00:53:51,730 --> 00:53:57,010 would be absolutely indifferent between investing among the n 1224 00:53:57,010 --> 00:53:59,470 stocks and T-Bills on the one hand, 1225 00:53:59,470 --> 00:54:02,530 versus investing in two securities on the other. 1226 00:54:02,530 --> 00:54:05,380 One security is T-Bills, and the other security 1227 00:54:05,380 --> 00:54:12,560 is shares of mutual fund M. Do we agree on that? 1228 00:54:12,560 --> 00:54:14,540 Any controversy there? 1229 00:54:14,540 --> 00:54:17,480 I know I've made a number of assumptions to get us here, 1230 00:54:17,480 --> 00:54:20,510 but given mean variance preferences, which is not 1231 00:54:20,510 --> 00:54:22,460 an unreasonable assumption, and given 1232 00:54:22,460 --> 00:54:24,830 that we've assumed these parameters are 1233 00:54:24,830 --> 00:54:27,516 stable over time, that's where we are. 1234 00:54:27,516 --> 00:54:28,016 Rami? 1235 00:54:28,016 --> 00:54:29,682 AUDIENCE: Somebody might have said this, 1236 00:54:29,682 --> 00:54:33,402 but you assume all fees are trading fees? 1237 00:54:33,402 --> 00:54:34,610 ANDREW LO: Forget about fees. 1238 00:54:34,610 --> 00:54:36,239 There are fees no matter what you do. 1239 00:54:36,239 --> 00:54:38,030 So for now, I'm going to forget about fees. 1240 00:54:38,030 --> 00:54:40,817 I'll put fees back in later, and if I do that, 1241 00:54:40,817 --> 00:54:42,650 then it's going to look even more compelling 1242 00:54:42,650 --> 00:54:47,870 for you to want to invest in mutual fund M, versus n stocks. 1243 00:54:47,870 --> 00:54:50,700 I don't know how many of you have traded individual stocks, 1244 00:54:50,700 --> 00:54:54,620 but if you ever try to manage a portfolio of 1,000 stocks, 1245 00:54:54,620 --> 00:54:58,092 it's actually fairly time consuming, right? 1246 00:54:58,092 --> 00:55:00,300 And by the way, there are more than 1,000 securities. 1247 00:55:00,300 --> 00:55:03,890 I mean the S&P 500 you can think of as being M, 1248 00:55:03,890 --> 00:55:05,360 but that's an approximation, right? 1249 00:55:05,360 --> 00:55:08,480 There's probably 7,000 or 8,000 securities that trade today. 1250 00:55:08,480 --> 00:55:10,910 Probably only 2,000 or 3,000 that you would really 1251 00:55:10,910 --> 00:55:13,520 take seriously, and probably only 1,500 1252 00:55:13,520 --> 00:55:16,530 that you really need from a diversification perspective. 1253 00:55:16,530 --> 00:55:17,676 1,500 stocks. 1254 00:55:17,676 --> 00:55:19,550 Would you want to trade in that, or would you 1255 00:55:19,550 --> 00:55:21,838 want to trade in one mutual fund? 1256 00:55:21,838 --> 00:55:22,338 Yeah? 1257 00:55:22,338 --> 00:55:26,024 AUDIENCE: Can I ask, I mean knowing that with M you're 1258 00:55:26,024 --> 00:55:30,340 trying to get on that tangent portfolio? 1259 00:55:30,340 --> 00:55:32,232 And you said, for example, Warren Buffet 1260 00:55:32,232 --> 00:55:33,232 beats it the whole time. 1261 00:55:33,232 --> 00:55:35,549 Why don't you just buy one share of Berkshire Hathaway, 1262 00:55:35,549 --> 00:55:37,090 and you'd have a higher Sharpe ratio? 1263 00:55:37,090 --> 00:55:38,740 ANDREW LO: Because Warren Buffett beat it in the past, 1264 00:55:38,740 --> 00:55:40,786 do you think he's going to beat it in the future? 1265 00:55:40,786 --> 00:55:42,160 AUDIENCE: I would [INAUDIBLE] it. 1266 00:55:42,160 --> 00:55:43,160 ANDREW LO: I don't know. 1267 00:55:43,160 --> 00:55:43,960 That's right. 1268 00:55:43,960 --> 00:55:45,181 Good question, good question. 1269 00:55:45,181 --> 00:55:47,680 I mean, if you're thinking about Warren Buffett as a 10 year 1270 00:55:47,680 --> 00:55:51,190 investment, I think I might short that. 1271 00:55:51,190 --> 00:55:53,140 I mean, you know, he seems healthy, 1272 00:55:53,140 --> 00:55:56,950 but you know those Cherry Cokes have to have an impact. 1273 00:55:56,950 --> 00:55:58,340 I'm sorry. 1274 00:55:58,340 --> 00:56:01,180 You know, you eat enough steaks at that Omaha restaurant, 1275 00:56:01,180 --> 00:56:03,250 I don't know what it is, and those Cherry Cokes, 1276 00:56:03,250 --> 00:56:06,250 I don't know. 1277 00:56:06,250 --> 00:56:06,824 OK, so fine. 1278 00:56:06,824 --> 00:56:08,990 Let's not do Warren Buffett, let's do somebody else. 1279 00:56:08,990 --> 00:56:09,490 Fine. 1280 00:56:09,490 --> 00:56:11,435 You tell me who that is? 1281 00:56:11,435 --> 00:56:13,060 Tell me who the next Warren Buffett is? 1282 00:56:13,060 --> 00:56:14,770 Can anybody tell me? 1283 00:56:14,770 --> 00:56:17,260 I'll be happy to do that, I'll be happy to invest in them. 1284 00:56:17,260 --> 00:56:18,397 Who is it? 1285 00:56:18,397 --> 00:56:21,260 AUDIENCE: Andrew Lo. 1286 00:56:21,260 --> 00:56:24,550 ANDREW LO: Thank you, but those who can't do teach, 1287 00:56:24,550 --> 00:56:26,130 those who can't teach, teach gym. 1288 00:56:26,130 --> 00:56:28,600 And at least I don't teach gym. 1289 00:56:31,360 --> 00:56:33,504 The point is that we don't know who the next Warren 1290 00:56:33,504 --> 00:56:35,170 Buffett is going to be, and I don't want 1291 00:56:35,170 --> 00:56:36,130 to have to figure that out. 1292 00:56:36,130 --> 00:56:37,546 I mean, that's a pretty tall order 1293 00:56:37,546 --> 00:56:39,910 to tell an investor that they've got to figure out who 1294 00:56:39,910 --> 00:56:41,410 the next investment genius is. 1295 00:56:41,410 --> 00:56:44,140 If they knew, they wouldn't have to ask them to invest. 1296 00:56:44,140 --> 00:56:46,210 They'd invest themselves, right? 1297 00:56:46,210 --> 00:56:49,540 So what I'm showing you is a simple way of investing 1298 00:56:49,540 --> 00:56:51,389 that may not be as good as Warren Buffett, 1299 00:56:51,389 --> 00:56:52,930 but it's certainly better than trying 1300 00:56:52,930 --> 00:56:54,160 to pick the next Warren Buffett if you 1301 00:56:54,160 --> 00:56:55,060 don't know what you're doing. 1302 00:56:55,060 --> 00:56:55,745 Jen? 1303 00:56:55,745 --> 00:56:58,357 AUDIENCE: Is it easier to kind of figure 1304 00:56:58,357 --> 00:57:01,682 out the future covariances of the different than it 1305 00:57:01,682 --> 00:57:03,774 is to pick the next Warren Buffet? 1306 00:57:03,774 --> 00:57:06,190 ANDREW LO: Thank you, that's another way of looking at it. 1307 00:57:06,190 --> 00:57:10,140 If you ask the question, is it easier to try-- 1308 00:57:10,140 --> 00:57:13,650 is the historical covariances and variances 1309 00:57:13,650 --> 00:57:16,920 and expected returns more predictive of the future 1310 00:57:16,920 --> 00:57:20,320 than your ability to find the next Warren Buffett, 1311 00:57:20,320 --> 00:57:22,030 then yes, that's another good argument. 1312 00:57:22,030 --> 00:57:23,488 That in other words, this framework 1313 00:57:23,488 --> 00:57:29,340 relies on less ability to forecast. 1314 00:57:29,340 --> 00:57:32,130 It doesn't completely rule it out because, as I said, 1315 00:57:32,130 --> 00:57:34,184 these parameters, they change over time. 1316 00:57:34,184 --> 00:57:35,850 And you have to think about that impact. 1317 00:57:35,850 --> 00:57:38,700 So it's not totally trivial, but from 1318 00:57:38,700 --> 00:57:42,210 the theoretical perspective, it seems 1319 00:57:42,210 --> 00:57:45,490 like it's a very internally consistent approach. 1320 00:57:45,490 --> 00:57:47,160 Now, let me go on for a little while 1321 00:57:47,160 --> 00:57:50,389 longer because if it were just this, then 1322 00:57:50,389 --> 00:57:52,180 this would be an interesting rule of thumb. 1323 00:57:52,180 --> 00:57:55,920 But this is not a theory of financial markets just yet. 1324 00:57:55,920 --> 00:58:00,130 I haven't really done anything truly astounding 1325 00:58:00,130 --> 00:58:02,400 because you're still left with the question of, 1326 00:58:02,400 --> 00:58:04,770 what's the appropriate risk-reward trade-off? 1327 00:58:04,770 --> 00:58:07,650 What should I use for my discount rate? 1328 00:58:07,650 --> 00:58:09,319 A lot of financial decision making 1329 00:58:09,319 --> 00:58:11,610 is not just picking stocks and making good investments. 1330 00:58:11,610 --> 00:58:14,640 But it's whether or not should I invest in nanotechnology 1331 00:58:14,640 --> 00:58:18,780 as a corporate officer of a particular tech company, 1332 00:58:18,780 --> 00:58:21,130 or should I invest in green technologies? 1333 00:58:21,130 --> 00:58:22,707 What discount rate should I use? 1334 00:58:22,707 --> 00:58:25,290 How should I engage in capital budgeting or project financing? 1335 00:58:25,290 --> 00:58:28,050 All of these questions seem like they have nothing 1336 00:58:28,050 --> 00:58:29,009 to do with investments. 1337 00:58:29,009 --> 00:58:31,299 So I don't want to make this course into an investments 1338 00:58:31,299 --> 00:58:31,877 course. 1339 00:58:31,877 --> 00:58:33,960 There's a lot about corporate financial management 1340 00:58:33,960 --> 00:58:38,160 that relies on being able to understand these markets. 1341 00:58:38,160 --> 00:58:40,080 So let me show you where we go next, 1342 00:58:40,080 --> 00:58:44,340 because we're very close now to the big payoff. 1343 00:58:44,340 --> 00:58:46,320 We've already identified the tangency portfolio 1344 00:58:46,320 --> 00:58:47,490 as being special. 1345 00:58:47,490 --> 00:58:49,450 I'm going to call that portfolio M, 1346 00:58:49,450 --> 00:58:52,200 and I'm going to argue that everybody in their right minds 1347 00:58:52,200 --> 00:58:55,560 are going to be indifferent between picking among these two 1348 00:58:55,560 --> 00:58:57,960 investment opportunities, T-Bills and M, 1349 00:58:57,960 --> 00:59:02,340 versus the n plus 1 investment opportunities of all stocks, 1350 00:59:02,340 --> 00:59:04,800 plus T-Bills. 1351 00:59:04,800 --> 00:59:10,170 It turns out that portfolio M, therefore, 1352 00:59:10,170 --> 00:59:14,290 has to be a very specific portfolio. 1353 00:59:17,370 --> 00:59:20,510 And it turns out that that portfolio 1354 00:59:20,510 --> 00:59:28,080 is the portfolio of all assets in the entire economy, 1355 00:59:28,080 --> 00:59:32,990 in proportion to their market capitalizations. 1356 00:59:32,990 --> 00:59:36,140 Now what I just said is an incredibly deep result, 1357 00:59:36,140 --> 00:59:39,010 so I don't expect you to just get it. 1358 00:59:39,010 --> 00:59:40,000 Let me say it again. 1359 00:59:40,000 --> 00:59:42,170 First of all, I want you to understand it, 1360 00:59:42,170 --> 00:59:44,628 and then I'm going to try to give you the intuition for it. 1361 00:59:46,810 --> 00:59:51,130 If it's true that everybody, not only in this room, 1362 00:59:51,130 --> 00:59:55,520 but in the world, if everybody in the world 1363 00:59:55,520 --> 01:00:00,110 is indifferent between investing in those n plus 1 securities, 1364 01:00:00,110 --> 01:00:09,350 and in two, then we can argue that those two securities 1365 01:00:09,350 --> 01:00:11,630 play a very special role. 1366 01:00:11,630 --> 01:00:18,100 In particular, think about what that mutual fund M has to be. 1367 01:00:18,100 --> 01:00:22,670 Everybody in the world wants to hold M. 1368 01:00:22,670 --> 01:00:27,344 So, let's make the leap of faith that everybody does hold M. So 1369 01:00:27,344 --> 01:00:29,510 in other words, now we're in a world where everybody 1370 01:00:29,510 --> 01:00:33,290 is already mean variance optimizers, 1371 01:00:33,290 --> 01:00:39,350 and they already hold two assets in their portfolio. 1372 01:00:39,350 --> 01:00:44,440 The treasury bill asset, and the mutual fund M. So you hold M, 1373 01:00:44,440 --> 01:00:47,100 you hold M, you hold M, you hold M, you hold M, 1374 01:00:47,100 --> 01:00:51,490 everybody holds M. We hold different amounts of it, 1375 01:00:51,490 --> 01:00:53,680 so as a hedge fund manager, you're 1376 01:00:53,680 --> 01:00:56,170 holding a large amount of M. In fact, you're 1377 01:00:56,170 --> 01:00:59,350 holding twice as much M as your wealth allows, 1378 01:00:59,350 --> 01:01:01,582 and you're borrowing T-Bills to do so. 1379 01:01:01,582 --> 01:01:03,040 Somebody who's very conservative is 1380 01:01:03,040 --> 01:01:05,140 holding a very tiny little bit of M. 1381 01:01:05,140 --> 01:01:08,770 Mostly, that person is invested in T-Bills. 1382 01:01:08,770 --> 01:01:11,410 But the point is that every single person's portfolio 1383 01:01:11,410 --> 01:01:13,630 you look at, when you look at their portfolio, 1384 01:01:13,630 --> 01:01:20,940 it's M. If that's true, if what I just said is true, 1385 01:01:20,940 --> 01:01:24,610 what portfolio does M have to be? 1386 01:01:24,610 --> 01:01:27,670 There's only one that it can possibly be. 1387 01:01:27,670 --> 01:01:31,960 And that is the portfolio of all equities 1388 01:01:31,960 --> 01:01:36,530 in the marketplace, held in proportion to their market 1389 01:01:36,530 --> 01:01:38,410 value. 1390 01:01:38,410 --> 01:01:40,570 Do you see the beauty of that? 1391 01:01:40,570 --> 01:01:42,520 Now, let me try to explain it. 1392 01:01:42,520 --> 01:01:45,220 I hope you understand it, let me explain it. 1393 01:01:45,220 --> 01:01:47,050 Why does that have to be? 1394 01:01:47,050 --> 01:01:48,970 This has to do with supply equaling demand. 1395 01:01:48,970 --> 01:01:51,460 Now, I'm going to make an argument about equilibrium. 1396 01:01:51,460 --> 01:01:53,369 I haven't done so up until now. 1397 01:01:53,369 --> 01:01:54,910 Up until now, I haven't said anything 1398 01:01:54,910 --> 01:01:59,540 about supply equaling demand, but I'm about to do so. 1399 01:01:59,540 --> 01:02:03,250 If everybody is holding this portfolio M, 1400 01:02:03,250 --> 01:02:04,990 that's the demand side, right? 1401 01:02:04,990 --> 01:02:10,810 Everybody is demanding M. On the supply side, 1402 01:02:10,810 --> 01:02:16,460 I'm assuming that all stocks that are being supplied 1403 01:02:16,460 --> 01:02:17,780 are held. 1404 01:02:17,780 --> 01:02:23,410 If all stocks that are being supplied are held by somebody, 1405 01:02:23,410 --> 01:02:27,490 but if everybody in the world is holding the same portfolio 1406 01:02:27,490 --> 01:02:31,044 M, when you aggregate all of the demands. 1407 01:02:31,044 --> 01:02:33,210 So I'm going to add up your demand, and your demand, 1408 01:02:33,210 --> 01:02:33,940 and your demand, and you're. 1409 01:02:33,940 --> 01:02:34,910 We're going to go through the class, 1410 01:02:34,910 --> 01:02:36,451 and go through the world, we're going 1411 01:02:36,451 --> 01:02:38,070 to add up everybody's demand. 1412 01:02:38,070 --> 01:02:43,690 In every single case, your weights are identical. 1413 01:02:43,690 --> 01:02:46,400 You're holding the same portfolio M. 1414 01:02:46,400 --> 01:02:49,200 So when I aggregate the entire world, 1415 01:02:49,200 --> 01:02:53,900 and I get the portfolio M, what does it have to equal? 1416 01:02:53,900 --> 01:02:57,740 It can only equal the sum total of all assets in the world, 1417 01:02:57,740 --> 01:02:58,370 right? 1418 01:02:58,370 --> 01:03:01,800 Supply equals demand. 1419 01:03:01,800 --> 01:03:06,120 And therefore, when I aggregate all of your holdings of M 1420 01:03:06,120 --> 01:03:09,780 into one big fat M, that big fat M 1421 01:03:09,780 --> 01:03:11,700 can only be equal to one thing, which is 1422 01:03:11,700 --> 01:03:14,530 all the equities in the world. 1423 01:03:14,530 --> 01:03:18,231 And the weightings are just simply their market caps, 1424 01:03:18,231 --> 01:03:18,730 right? 1425 01:03:18,730 --> 01:03:20,740 There's only so much of General Motors. 1426 01:03:20,740 --> 01:03:22,640 Take the entire sum total of that, 1427 01:03:22,640 --> 01:03:25,690 that's the global investment in General Motors. 1428 01:03:25,690 --> 01:03:27,670 And then you do that for every single stock, 1429 01:03:27,670 --> 01:03:33,260 and you divide that by the total market capital of all stocks, 1430 01:03:33,260 --> 01:03:37,240 you get the market portfolio, M. 1431 01:03:37,240 --> 01:03:43,680 So this shockingly, simple, but extraordinarily powerful 1432 01:03:43,680 --> 01:03:46,670 result is due to Bill Sharpe. 1433 01:03:46,670 --> 01:03:50,310 Harry Markowitz came up with portfolio optimization. 1434 01:03:50,310 --> 01:03:52,380 He applied mean variance analysis 1435 01:03:52,380 --> 01:03:55,094 to portfolio optimization and argued that everybody 1436 01:03:55,094 --> 01:03:56,010 has to be on the line. 1437 01:03:56,010 --> 01:03:59,030 Bill Sharpe looked at this and said, aha. 1438 01:03:59,030 --> 01:04:00,619 If everybody's on that line, that 1439 01:04:00,619 --> 01:04:02,660 means that everybody's going to be either holding 1440 01:04:02,660 --> 01:04:06,050 M or T-Bills, or both, and therefore, 1441 01:04:06,050 --> 01:04:09,260 the only thing that M could possibly be 1442 01:04:09,260 --> 01:04:11,680 is the market portfolio. 1443 01:04:11,680 --> 01:04:14,620 And now we have a proxy for the market portfolio, the Russell 1444 01:04:14,620 --> 01:04:15,700 2000. 1445 01:04:15,700 --> 01:04:17,320 Or the S&P 500. 1446 01:04:17,320 --> 01:04:19,944 Both of those are very well diversified stock that have 1447 01:04:19,944 --> 01:04:21,610 lot-- they don't have everything in it-- 1448 01:04:21,610 --> 01:04:24,950 but they have a lot of things in it, that proxy for everything. 1449 01:04:24,950 --> 01:04:29,770 The Russell 2000 has 2,000 stocks weighted by market cap. 1450 01:04:29,770 --> 01:04:33,310 That's as close as you're going to get to everything that you 1451 01:04:33,310 --> 01:04:35,712 care about. 1452 01:04:35,712 --> 01:04:37,920 So now, you'll see we're benchmarking is coming from, 1453 01:04:37,920 --> 01:04:40,290 but I'm going to get back to that in more detail. 1454 01:04:40,290 --> 01:04:47,380 So this equilibrium result that says supply equals demand, 1455 01:04:47,380 --> 01:04:52,030 identifies this portfolio M. And what it says 1456 01:04:52,030 --> 01:04:56,590 is that if everybody does this, if everybody takes finance 1457 01:04:56,590 --> 01:04:59,980 here and learns how to do this, it's 1458 01:04:59,980 --> 01:05:01,810 not going to kill the idea. 1459 01:05:01,810 --> 01:05:05,320 It's going to lead to a very well-defined portfolio 1460 01:05:05,320 --> 01:05:09,445 M. Now, let me take it one step farther, 1461 01:05:09,445 --> 01:05:13,830 then I want to ask you to ask questions. 1462 01:05:13,830 --> 01:05:16,940 If I know what that portfolio M is, 1463 01:05:16,940 --> 01:05:20,945 then I've got an equation for this line. 1464 01:05:23,840 --> 01:05:28,040 I can write down a relationship between the expected return 1465 01:05:28,040 --> 01:05:31,280 and risk of a portfolio on this line. 1466 01:05:31,280 --> 01:05:33,190 And this is it. 1467 01:05:33,190 --> 01:05:36,700 The expected rate of return of an efficient portfolio, 1468 01:05:36,700 --> 01:05:39,640 by efficient I mean a portfolio that's on that line. 1469 01:05:39,640 --> 01:05:43,100 Anything that's not on that line, if it's below that line, 1470 01:05:43,100 --> 01:05:45,470 it's inefficient, right? 1471 01:05:45,470 --> 01:05:47,870 You're not getting as much expected return per unit risk, 1472 01:05:47,870 --> 01:05:50,780 and you're not reducing your risk as much as you can, 1473 01:05:50,780 --> 01:05:52,880 per unit of expected return. 1474 01:05:52,880 --> 01:05:55,430 The expected return of an efficient portfolio 1475 01:05:55,430 --> 01:06:01,710 is equal to the risk-free rate, plus the ratio 1476 01:06:01,710 --> 01:06:05,610 of the standard deviation of that portfolio, divided 1477 01:06:05,610 --> 01:06:09,180 by the standard deviation of the tangency portfolio, 1478 01:06:09,180 --> 01:06:15,120 or the market, multiplied by the excess return of the market 1479 01:06:15,120 --> 01:06:15,780 portfolio. 1480 01:06:20,360 --> 01:06:28,340 This result is a risk-reward trade-off between risk 1481 01:06:28,340 --> 01:06:30,610 and expected return. 1482 01:06:30,610 --> 01:06:34,090 You see, what it says is really something quite astounding. 1483 01:06:34,090 --> 01:06:37,060 It's telling you that, here's the risk-free rate. 1484 01:06:37,060 --> 01:06:40,630 That's the base return for your portfolio. 1485 01:06:40,630 --> 01:06:42,520 And what this is telling you is that what 1486 01:06:42,520 --> 01:06:45,610 you should expect for your portfolio 1487 01:06:45,610 --> 01:06:50,650 is that base return, plus something extra. 1488 01:06:50,650 --> 01:06:55,800 And the extra is the market's excess return, 1489 01:06:55,800 --> 01:06:58,650 multiplied by a factor. 1490 01:06:58,650 --> 01:07:00,930 And the factor is simply how risky your portfolio 1491 01:07:00,930 --> 01:07:03,990 is relative to the market. 1492 01:07:03,990 --> 01:07:05,290 Let's do a simple example. 1493 01:07:05,290 --> 01:07:09,960 Suppose that your portfolio is the exact same risk 1494 01:07:09,960 --> 01:07:11,912 as the market. 1495 01:07:11,912 --> 01:07:13,370 Well, if that's the case, then what 1496 01:07:13,370 --> 01:07:15,216 is your expected rate of return? 1497 01:07:15,216 --> 01:07:16,150 AUDIENCE: The market. 1498 01:07:16,150 --> 01:07:17,274 ANDREW LO: It's the market. 1499 01:07:17,274 --> 01:07:21,750 So it's the risk-free rate, plus the market excess return, 1500 01:07:21,750 --> 01:07:24,122 which, when you add it together, is just the market. 1501 01:07:24,122 --> 01:07:25,830 Suppose you're holding a portfolio that's 1502 01:07:25,830 --> 01:07:29,040 more risky than the market. 1503 01:07:29,040 --> 01:07:30,900 Is your rate of return greater or less 1504 01:07:30,900 --> 01:07:32,070 than the rate of return of the market? 1505 01:07:32,070 --> 01:07:32,760 AUDIENCE: Greater. 1506 01:07:32,760 --> 01:07:33,551 ANDREW LO: Greater. 1507 01:07:33,551 --> 01:07:36,570 Suppose that your portfolio has no risk. 1508 01:07:36,570 --> 01:07:39,476 Suppose that sigma p is 0, then what's your rate of return? 1509 01:07:39,476 --> 01:07:40,350 AUDIENCE: [INAUDIBLE] 1510 01:07:40,350 --> 01:07:41,310 ANDREW LO: Exactly. 1511 01:07:41,310 --> 01:07:42,270 Makes sense, right? 1512 01:07:42,270 --> 01:07:44,370 This is very intuitive. 1513 01:07:44,370 --> 01:07:48,000 What this tells us, now, is that we can figure out 1514 01:07:48,000 --> 01:07:52,170 what the fair rate of return is for an efficient portfolio. 1515 01:07:52,170 --> 01:07:54,420 For any portfolio on this line, I 1516 01:07:54,420 --> 01:07:56,400 can tell you what my fair rate of return is, 1517 01:07:56,400 --> 01:07:58,620 and it's an objective measure. 1518 01:07:58,620 --> 01:08:00,060 It's not just theory now. 1519 01:08:00,060 --> 01:08:01,551 Now, I can go into the marketplace, 1520 01:08:01,551 --> 01:08:03,550 I can measure the expected return of the market. 1521 01:08:03,550 --> 01:08:05,050 You know what that is, historically? 1522 01:08:05,050 --> 01:08:06,945 Not including the last few months. 1523 01:08:06,945 --> 01:08:08,126 AUDIENCE: 7%. 1524 01:08:08,126 --> 01:08:09,750 ANDREW LO: It's about 7%, historically. 1525 01:08:09,750 --> 01:08:12,180 Over the last 100 years, 7%, the expected 1526 01:08:12,180 --> 01:08:14,730 rate of return of the market. 1527 01:08:14,730 --> 01:08:16,316 Sorry, the expected risk premium, 1528 01:08:16,316 --> 01:08:17,399 the excess rate of return. 1529 01:08:17,399 --> 01:08:18,414 About 7%. 1530 01:08:18,414 --> 01:08:20,080 What about the volatility of the market? 1531 01:08:20,080 --> 01:08:22,229 It's been about 15% historically. 1532 01:08:22,229 --> 01:08:24,899 So according to this relationship, 1533 01:08:24,899 --> 01:08:26,910 I've already figured out what this number is. 1534 01:08:26,910 --> 01:08:28,036 It's like 7%. 1535 01:08:28,036 --> 01:08:29,910 I've already figured out what this number is. 1536 01:08:29,910 --> 01:08:31,410 It's like 15%. 1537 01:08:31,410 --> 01:08:34,890 So now, you should be able to get a benchmark for what 1538 01:08:34,890 --> 01:08:38,670 to expect when you've got a particular level of risk 1539 01:08:38,670 --> 01:08:41,069 in an efficient portfolio. 1540 01:08:41,069 --> 01:08:42,510 You've got all the ingredients. 1541 01:08:42,510 --> 01:08:43,593 What about risk-free rate? 1542 01:08:43,593 --> 01:08:45,450 Well, it depends on what risk-free rate, 1543 01:08:45,450 --> 01:08:47,460 but let's talk about over a one year period. 1544 01:08:47,460 --> 01:08:48,960 Right now we're looking at somewhere 1545 01:08:48,960 --> 01:08:52,109 between, I don't know, 1%, 2%, 3%, 1546 01:08:52,109 --> 01:08:55,290 depending on what day of the week you're looking at. 1547 01:08:55,290 --> 01:08:58,938 So one year T-Bill rate is about 1% or so, yeah? 1548 01:08:58,938 --> 01:09:00,688 AUDIENCE: I think it was the last class we 1549 01:09:00,688 --> 01:09:02,064 talked about unsystematic risk. 1550 01:09:02,064 --> 01:09:02,689 ANDREW LO: Yes. 1551 01:09:02,689 --> 01:09:06,870 AUDIENCE: Is that defined by [INAUDIBLE] in this case? 1552 01:09:06,870 --> 01:09:08,580 ANDREW LO: No, the unsystematic risk 1553 01:09:08,580 --> 01:09:10,890 is risk that is not measured by sigma p, 1554 01:09:10,890 --> 01:09:12,960 so we're going to come back to that. 1555 01:09:12,960 --> 01:09:14,729 Let me hold off on that for now, because I 1556 01:09:14,729 --> 01:09:16,930 want to come back to it after I finish developing this. 1557 01:09:16,930 --> 01:09:18,626 There's going to be a connection between 1558 01:09:18,626 --> 01:09:20,250 systematic and unsystematic risk that's 1559 01:09:20,250 --> 01:09:22,990 going to come right out of this relationship. 1560 01:09:22,990 --> 01:09:23,490 Yeah, Brian? 1561 01:09:23,490 --> 01:09:28,310 AUDIENCE: So if you take the S&P 500 as M here, 1562 01:09:28,310 --> 01:09:31,640 the market portfolio, and the capitalization is the weight, 1563 01:09:31,640 --> 01:09:34,855 so you've got non-zero weights for all the different stocks 1564 01:09:34,855 --> 01:09:35,430 there. 1565 01:09:35,430 --> 01:09:38,760 Does that imply that there's no stocks in the S&P 500 1566 01:09:38,760 --> 01:09:41,160 that are Southeast of any others? 1567 01:09:41,160 --> 01:09:42,819 ANDREW LO: No, no, there could be. 1568 01:09:42,819 --> 01:09:43,630 AUDIENCE: Why would you have them, 1569 01:09:43,630 --> 01:09:45,671 because we said those are strictly non-preferred? 1570 01:09:45,671 --> 01:09:47,910 ANDREW LO: Well, that's if you're looking 1571 01:09:47,910 --> 01:09:49,800 at a pairwise comparison. 1572 01:09:49,800 --> 01:09:52,200 If, now, you're trying to create an entire collection 1573 01:09:52,200 --> 01:09:53,850 of these portfolios of securities, 1574 01:09:53,850 --> 01:09:55,129 that's a different story. 1575 01:09:55,129 --> 01:09:57,420 That's why I answered in response to Justin's question. 1576 01:09:57,420 --> 01:09:59,850 Justin said, why not just trade off those two? 1577 01:09:59,850 --> 01:10:00,380 Why not? 1578 01:10:00,380 --> 01:10:03,090 It's because you can do far better by using all of them 1579 01:10:03,090 --> 01:10:04,290 in this way. 1580 01:10:04,290 --> 01:10:07,200 You see, by looking at pairwise, you can no doubt do better. 1581 01:10:07,200 --> 01:10:10,644 But if I use all of them, I get this entire line. 1582 01:10:10,644 --> 01:10:12,060 And you can't get that entire line 1583 01:10:12,060 --> 01:10:13,780 just from looking at two of these stocks, 1584 01:10:13,780 --> 01:10:14,580 you need all of them. 1585 01:10:14,580 --> 01:10:16,250 AUDIENCE: So in this portfolio of three, 1586 01:10:16,250 --> 01:10:18,520 if you kick GM over to the right a little bit, 1587 01:10:18,520 --> 01:10:20,530 and made it strictly non-preferred to IBM, 1588 01:10:20,530 --> 01:10:24,280 then you still might have a positive portfolio 1589 01:10:24,280 --> 01:10:25,560 weight on GM? 1590 01:10:25,560 --> 01:10:27,360 ANDREW LO: You might, but more likely, 1591 01:10:27,360 --> 01:10:28,917 it'll be a negative portfolio weight. 1592 01:10:28,917 --> 01:10:30,750 It'll be negative, and you'll be shorting it 1593 01:10:30,750 --> 01:10:32,750 somewhere along the line here. 1594 01:10:32,750 --> 01:10:36,930 However, the tangency portfolio, by assumption, 1595 01:10:36,930 --> 01:10:41,490 if it's the market portfolio, cannot have negative weights. 1596 01:10:41,490 --> 01:10:45,570 And so there, what will happen, is that all of the stocks 1597 01:10:45,570 --> 01:10:47,580 will change in their relationship 1598 01:10:47,580 --> 01:10:50,230 based upon various different kinds of equilibrium, 1599 01:10:50,230 --> 01:10:52,440 so that you won't get into a lot of those situations 1600 01:10:52,440 --> 01:10:55,300 where you're going to be shorting these negative stocks. 1601 01:10:55,300 --> 01:10:55,800 Yeah? 1602 01:10:55,800 --> 01:10:59,132 AUDIENCE: So basically, according to the Sharpe theory, 1603 01:10:59,132 --> 01:11:04,380 every stock that the market, the capital is not 0 1604 01:11:04,380 --> 01:11:06,500 is worth holding in some portfolio? 1605 01:11:06,500 --> 01:11:07,500 ANDREW LO: That's right. 1606 01:11:07,500 --> 01:11:09,030 AUDIENCE: Diversifying your portfolio. 1607 01:11:09,030 --> 01:11:10,030 ANDREW LO: That's right. 1608 01:11:10,030 --> 01:11:12,990 Every stock has some benefit in adding 1609 01:11:12,990 --> 01:11:15,570 to this particular risk-reward trade-off, 1610 01:11:15,570 --> 01:11:19,660 and the sum total benefit is summarized by this line. 1611 01:11:19,660 --> 01:11:20,910 That's the ultimate objective. 1612 01:11:20,910 --> 01:11:23,390 AUDIENCE: If I don't hold a specific stock in the market 1613 01:11:23,390 --> 01:11:26,502 and I gain a diversification [INAUDIBLE]? 1614 01:11:26,502 --> 01:11:27,210 ANDREW LO: Sorry? 1615 01:11:27,210 --> 01:11:29,864 If you hold a specific stock? 1616 01:11:29,864 --> 01:11:31,530 AUDIENCE: If I don't hold it, because it 1617 01:11:31,530 --> 01:11:32,510 has market [INAUDIBLE]. 1618 01:11:32,510 --> 01:11:34,370 ANDREW LO: Oh, if you put 0 weight. 1619 01:11:34,370 --> 01:11:35,640 Yes. 1620 01:11:35,640 --> 01:11:38,730 What Sharpe would argue, based upon this theory, 1621 01:11:38,730 --> 01:11:41,190 is that you want to hold as many stocks 1622 01:11:41,190 --> 01:11:43,770 as you can to get the most diversification. 1623 01:11:43,770 --> 01:11:45,210 Now, that's the theory. 1624 01:11:45,210 --> 01:11:49,410 In practice, it may well be that the benefits do not 1625 01:11:49,410 --> 01:11:51,960 outweigh the costs, because when you hold multiple stocks, 1626 01:11:51,960 --> 01:11:54,900 you have to manage them, and so it may cost more. 1627 01:11:54,900 --> 01:11:57,330 So a mutual fund that has 3,000 stocks 1628 01:11:57,330 --> 01:12:01,080 may have a higher expense ratio than a mutual fund with 500. 1629 01:12:01,080 --> 01:12:01,800 It may not. 1630 01:12:01,800 --> 01:12:04,500 Nowadays, actually, the technology is so good that 1631 01:12:04,500 --> 01:12:05,580 probably it doesn't. 1632 01:12:05,580 --> 01:12:08,250 But 15 years ago, that was not true. 1633 01:12:08,250 --> 01:12:10,140 But apart from the transactions cost, 1634 01:12:10,140 --> 01:12:13,420 the theory suggests more is better. 1635 01:12:13,420 --> 01:12:16,124 Because it will always give you more opportunities, and it 1636 01:12:16,124 --> 01:12:18,540 can never hurt you because you could always put a 0 weight 1637 01:12:18,540 --> 01:12:19,831 on them if you don't like them. 1638 01:12:22,330 --> 01:12:30,040 Now, it turns out that this is a trade-off between the expected 1639 01:12:30,040 --> 01:12:35,820 return of an efficient portfolio, 1640 01:12:35,820 --> 01:12:37,860 and the risk of that portfolio. 1641 01:12:37,860 --> 01:12:41,460 In other words, this applies only 1642 01:12:41,460 --> 01:12:45,100 to portfolios on that tangency line. 1643 01:12:45,100 --> 01:12:48,040 What if you want to know what the expected rate of return 1644 01:12:48,040 --> 01:12:51,120 is for Wal-Mart? 1645 01:12:51,120 --> 01:12:54,510 We just said that no individual stock 1646 01:12:54,510 --> 01:12:58,170 is going to be likely to be on that efficient frontier. 1647 01:12:58,170 --> 01:13:00,690 And therefore, no individual stock 1648 01:13:00,690 --> 01:13:03,016 is likely to be on this line. 1649 01:13:03,016 --> 01:13:04,890 So this is great if what you're talking about 1650 01:13:04,890 --> 01:13:07,230 is investing in efficient portfolios, 1651 01:13:07,230 --> 01:13:09,500 but how does that help the corporate financial officer 1652 01:13:09,500 --> 01:13:11,041 that's trying to figure out how to do 1653 01:13:11,041 --> 01:13:13,650 capital budgeting for a particular pharmaceutical 1654 01:13:13,650 --> 01:13:15,399 project? 1655 01:13:15,399 --> 01:13:16,440 It turns out, it doesn't. 1656 01:13:16,440 --> 01:13:17,106 It doesn't help. 1657 01:13:17,106 --> 01:13:19,770 This doesn't answer that question. 1658 01:13:19,770 --> 01:13:24,630 It turns out, you need to have an additional piece of theory 1659 01:13:24,630 --> 01:13:28,590 that allows you to derive the same results, not 1660 01:13:28,590 --> 01:13:33,370 just for the efficient portfolios here, 1661 01:13:33,370 --> 01:13:35,590 but for any portfolio. 1662 01:13:35,590 --> 01:13:37,780 And this is another innovation of Bill Sharpe. 1663 01:13:37,780 --> 01:13:40,190 This is actually why Bill Sharpe won the Nobel Prize. 1664 01:13:40,190 --> 01:13:42,250 It was not for this little picture here, 1665 01:13:42,250 --> 01:13:45,310 but it was for this equation right here. 1666 01:13:45,310 --> 01:13:50,110 What Bill Sharpe discovered is after computing the equilibrium 1667 01:13:50,110 --> 01:13:53,810 relationships among various different securities, 1668 01:13:53,810 --> 01:13:58,670 he's demonstrated that there has to be a linear relationship 1669 01:13:58,670 --> 01:14:05,230 between any stock's expected return and the market risk 1670 01:14:05,230 --> 01:14:06,550 premium. 1671 01:14:06,550 --> 01:14:09,370 Just like here, where you've got the risk-free rate, 1672 01:14:09,370 --> 01:14:11,800 plus some extra premium. 1673 01:14:11,800 --> 01:14:14,620 So this is the premium, the second term. 1674 01:14:14,620 --> 01:14:18,790 But what Bill Sharpe showed was that if this portfolio is not 1675 01:14:18,790 --> 01:14:24,120 an efficient portfolio, if it's not on that line, 1676 01:14:24,120 --> 01:14:27,810 the linear relationship still holds. 1677 01:14:27,810 --> 01:14:31,080 But it turns out that this particular multiplier is 1678 01:14:31,080 --> 01:14:34,210 no longer the right one to use. 1679 01:14:34,210 --> 01:14:39,300 It turns out that the right parameter to plug in there, 1680 01:14:39,300 --> 01:14:42,240 is something called beta. 1681 01:14:42,240 --> 01:14:44,440 Now, you've heard all about beta, I'm sure. 1682 01:14:44,440 --> 01:14:47,500 But now, I'm telling you exactly what beta is. 1683 01:14:47,500 --> 01:14:51,010 Beta is the multiplier that is defined 1684 01:14:51,010 --> 01:14:54,640 by the covariance between the return on the market 1685 01:14:54,640 --> 01:14:57,760 and the return on the individual asset, divided 1686 01:14:57,760 --> 01:15:00,220 by the variance of that market return. 1687 01:15:03,860 --> 01:15:08,780 If the portfolio happens to be on that efficient frontier, 1688 01:15:08,780 --> 01:15:13,040 then this beta reduces to this previous measure. 1689 01:15:13,040 --> 01:15:17,090 So this is a special case of the more general relationship 1690 01:15:17,090 --> 01:15:20,560 where beta is used as the multiplier. 1691 01:15:20,560 --> 01:15:21,960 So let me repeat what beta is. 1692 01:15:21,960 --> 01:15:26,250 Beta is the ratio of the covariance between the return 1693 01:15:26,250 --> 01:15:28,860 on the particular asset or portfolio, that may or may not 1694 01:15:28,860 --> 01:15:33,350 be efficient, it's any asset, with the return 1695 01:15:33,350 --> 01:15:34,980 on the market portfolio. 1696 01:15:34,980 --> 01:15:39,710 So this numerator is a measure of the covariability 1697 01:15:39,710 --> 01:15:42,200 between the particular asset that you're 1698 01:15:42,200 --> 01:15:45,410 trying to measure the expected return of, 1699 01:15:45,410 --> 01:15:48,890 and that tangency portfolio, divided 1700 01:15:48,890 --> 01:15:53,390 by the variance of that tangency portfolio. 1701 01:15:56,010 --> 01:16:00,960 Beta, it turns out, is the right measure of risk, 1702 01:16:00,960 --> 01:16:05,190 in the sense that it is the beta that determines what 1703 01:16:05,190 --> 01:16:09,480 the multiplier is going to be on the market risk premium, which 1704 01:16:09,480 --> 01:16:13,290 is to be added to your asset's expected rate of return, 1705 01:16:13,290 --> 01:16:15,810 above and beyond the risk-free rate. 1706 01:16:15,810 --> 01:16:19,920 That's how the cost of capital is determined for your asset. 1707 01:16:24,030 --> 01:16:27,420 So I think you all saw how I derived this, 1708 01:16:27,420 --> 01:16:29,070 but I didn't derive this. 1709 01:16:29,070 --> 01:16:30,600 I'm just telling you this is really 1710 01:16:30,600 --> 01:16:35,700 where Sharpe's ideas became extraordinarily compelling. 1711 01:16:35,700 --> 01:16:37,800 And in order to understand how to derive 1712 01:16:37,800 --> 01:16:41,017 that, I'm going to refer you to 433, because 1713 01:16:41,017 --> 01:16:42,600 in that investment's course, we really 1714 01:16:42,600 --> 01:16:46,082 delve into the underpinnings of that kind of calculation. 1715 01:16:46,082 --> 01:16:47,540 It's a little bit more involved, it 1716 01:16:47,540 --> 01:16:49,470 involves some matrix algebra. 1717 01:16:49,470 --> 01:16:52,502 But it's not terribly difficult or challenging, 1718 01:16:52,502 --> 01:16:54,960 and certainly be happy to give you references if any of you 1719 01:16:54,960 --> 01:16:55,710 are interested. 1720 01:16:55,710 --> 01:16:59,370 I believe it's in Brealey, Myers, and Allen. 1721 01:16:59,370 --> 01:17:01,380 But the bottom line is that this gives you 1722 01:17:01,380 --> 01:17:04,490 an extraordinarily important conclusion now 1723 01:17:04,490 --> 01:17:09,930 to the several weeks that we've been working towards this goal. 1724 01:17:09,930 --> 01:17:14,100 Which is now, finally, after eight or nine weeks, 1725 01:17:14,100 --> 01:17:17,850 I can tell you how to come up with the appropriate discount 1726 01:17:17,850 --> 01:17:21,120 rate for various NPV calculations. 1727 01:17:21,120 --> 01:17:23,640 The answer is the expected rate of return, 1728 01:17:23,640 --> 01:17:26,940 the appropriate fair rate of return, 1729 01:17:26,940 --> 01:17:30,300 or the market equilibrium rate of return, 1730 01:17:30,300 --> 01:17:33,960 is simply given by the beta of that security, multiplied 1731 01:17:33,960 --> 01:17:39,140 by the expected excess return on the market portfolio. 1732 01:17:39,140 --> 01:17:41,729 So now, this has a lot of assumptions, granted. 1733 01:17:41,729 --> 01:17:43,520 We're going to talk about those assumptions 1734 01:17:43,520 --> 01:17:46,430 over the next couple of lectures. 1735 01:17:46,430 --> 01:17:50,740 But what we've done today is move the theory forward 1736 01:17:50,740 --> 01:17:55,030 by quite a bit, because we've identified a particular method 1737 01:17:55,030 --> 01:17:58,150 for coming up with the appropriate cost of capital 1738 01:17:58,150 --> 01:17:59,440 as a function of the risk. 1739 01:17:59,440 --> 01:18:03,340 Where the risk is measured, not by volatility anymore, 1740 01:18:03,340 --> 01:18:09,710 but by the covariance between an asset and the market portfolio. 1741 01:18:09,710 --> 01:18:11,960 And next time, I'm going to try to give you 1742 01:18:11,960 --> 01:18:14,300 some intuition for why this should be, 1743 01:18:14,300 --> 01:18:20,120 why this makes sense, and why, in a mean variance efficient 1744 01:18:20,120 --> 01:18:23,840 set of portfolios, why it reduces to something that we 1745 01:18:23,840 --> 01:18:24,650 know and love. 1746 01:18:27,930 --> 01:18:30,670 Any questions? 1747 01:18:30,670 --> 01:18:31,170 OK. 1748 01:18:31,170 --> 01:18:34,010 I'll stop here, and I'll see you on Wednesday.