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 --> 00:00:17,820 at ocw.mit.edu. 8 00:00:21,240 --> 00:00:24,520 ANDREW LO: Well, if you remember last time where we left off, 9 00:00:24,520 --> 00:00:27,270 we were talking about risk and return. 10 00:00:27,270 --> 00:00:29,340 And we said that we were going to make 11 00:00:29,340 --> 00:00:32,759 the following simplifying assumption, which 12 00:00:32,759 --> 00:00:37,230 is that we're going to assume that investors 13 00:00:37,230 --> 00:00:42,630 like expected return, and they do not like risk 14 00:00:42,630 --> 00:00:44,970 as measured by volatility. 15 00:00:44,970 --> 00:00:46,020 All right? 16 00:00:46,020 --> 00:00:49,920 And so the way that we depict it graphically 17 00:00:49,920 --> 00:00:54,150 is to use a graph where the x-axis is 18 00:00:54,150 --> 00:00:58,770 standard deviation of your entire portfolio, 19 00:00:58,770 --> 00:01:03,600 and the y-axis is the expected return of that portfolio. 20 00:01:03,600 --> 00:01:09,030 And the question is, where on this graph 21 00:01:09,030 --> 00:01:12,720 can we get to, given the securities that we have access 22 00:01:12,720 --> 00:01:16,790 to, that will maximize our level of happiness. 23 00:01:16,790 --> 00:01:18,960 Where happiness, again, is assumed 24 00:01:18,960 --> 00:01:23,550 to mean higher expected rate of return and lower risk, 25 00:01:23,550 --> 00:01:26,860 as measured by variance or standard deviation. 26 00:01:26,860 --> 00:01:29,940 So for example, if you take a look at this simple graph 27 00:01:29,940 --> 00:01:33,400 and you ask the question, where on the graph do you want to be, 28 00:01:33,400 --> 00:01:35,940 you would like to be always going 29 00:01:35,940 --> 00:01:38,050 in the northwest direction. 30 00:01:38,050 --> 00:01:38,550 Right? 31 00:01:38,550 --> 00:01:42,210 Because north means higher expected return and west 32 00:01:42,210 --> 00:01:45,120 means lower risk. 33 00:01:45,120 --> 00:01:52,040 So obviously, if we could, we'd love to be on this axis 34 00:01:52,040 --> 00:01:54,420 all the way, way up. 35 00:01:54,420 --> 00:01:56,130 Right? 36 00:01:56,130 --> 00:01:58,970 No risk, lots of return. 37 00:01:58,970 --> 00:02:01,040 That's an example of an arbitrage. 38 00:02:01,040 --> 00:02:03,050 And we know that that can't happen 39 00:02:03,050 --> 00:02:06,300 very easily because otherwise everybody would be there. 40 00:02:06,300 --> 00:02:09,740 And pretty soon it would wipe out that opportunity. 41 00:02:09,740 --> 00:02:13,370 So the question from the portfolio construction 42 00:02:13,370 --> 00:02:17,300 perspective is now a little bit sharper than it was last week 43 00:02:17,300 --> 00:02:19,200 when we started down this path. 44 00:02:19,200 --> 00:02:23,250 Now we want to construct a portfolio, 45 00:02:23,250 --> 00:02:26,090 we want to take a collection of securities 46 00:02:26,090 --> 00:02:31,550 and weight them in order to be as happy as possible. 47 00:02:31,550 --> 00:02:36,840 Meaning, we want to be as northwest as possible. 48 00:02:36,840 --> 00:02:39,050 So let's see how we go about doing that. 49 00:02:39,050 --> 00:02:42,570 One thing we could do is just pick an individual stock. 50 00:02:42,570 --> 00:02:46,160 So if you have these four stocks to pick from, then 51 00:02:46,160 --> 00:02:48,950 to go as northwest as possible, you're 52 00:02:48,950 --> 00:02:55,760 sort of looking at Merck as, you know, the extreme. 53 00:02:55,760 --> 00:02:58,400 But it's not at all clear whether or not that's something 54 00:02:58,400 --> 00:02:59,360 that you really want. 55 00:02:59,360 --> 00:03:02,840 Because, for example, General Motors, 56 00:03:02,840 --> 00:03:07,400 while it has a lower expected return than Merck, it does 57 00:03:07,400 --> 00:03:09,200 have a bit of a lower risk. 58 00:03:09,200 --> 00:03:14,310 And for some people, that might actually be preferred. 59 00:03:14,310 --> 00:03:19,550 So at this point, we don't have a lot of hard recommendations 60 00:03:19,550 --> 00:03:22,780 to provide you with, without any further analysis. 61 00:03:22,780 --> 00:03:25,120 So we're going to do some further analysis today. 62 00:03:25,120 --> 00:03:29,900 And the analysis is to ask the question, all right, what 63 00:03:29,900 --> 00:03:36,330 are the properties of mean and variance for a given portfolio, 64 00:03:36,330 --> 00:03:39,620 not just for an individual security. 65 00:03:39,620 --> 00:03:42,440 So it turns out that there's a relatively straightforward way 66 00:03:42,440 --> 00:03:43,410 of answering this. 67 00:03:43,410 --> 00:03:45,260 And let me just go through the calculations 68 00:03:45,260 --> 00:03:47,090 and then we can see what that implies 69 00:03:47,090 --> 00:03:51,770 for where we want to be in that mean-standard deviation graph. 70 00:03:51,770 --> 00:03:54,680 So the mean of a particular stock I'm 71 00:03:54,680 --> 00:03:59,910 going to write as expectation of Ri, or mu i, for short. 72 00:03:59,910 --> 00:04:01,760 The variance of a particular stock 73 00:04:01,760 --> 00:04:04,550 I'm going to write as sigma squared i, 74 00:04:04,550 --> 00:04:07,970 or the standard deviation is then just sigma i. 75 00:04:07,970 --> 00:04:09,320 OK? 76 00:04:09,320 --> 00:04:13,430 And it turns out, you can show this rigorously. 77 00:04:13,430 --> 00:04:15,860 I won't do that, but you can take a look at that 78 00:04:15,860 --> 00:04:17,750 if you are unconvinced. 79 00:04:17,750 --> 00:04:21,260 It turns out that if you construct a weighted average 80 00:04:21,260 --> 00:04:25,310 of stocks so that the return of the portfolio 81 00:04:25,310 --> 00:04:28,940 is given by that top line, R sub p, 82 00:04:28,940 --> 00:04:33,170 then when you take the expected value of that top line, what 83 00:04:33,170 --> 00:04:36,390 you get is the middle line. 84 00:04:36,390 --> 00:04:41,360 In other words, the expected return of a portfolio 85 00:04:41,360 --> 00:04:44,450 is just equal to the exact same weighted averages 86 00:04:44,450 --> 00:04:48,600 of the expected rates of return of the individual components. 87 00:04:48,600 --> 00:04:51,140 So you understand the difference between the second 88 00:04:51,140 --> 00:04:53,180 and the first line in that red box? 89 00:04:53,180 --> 00:04:55,230 That's a very important distinction. 90 00:04:55,230 --> 00:04:59,360 The top line is basically an accounting identity. 91 00:04:59,360 --> 00:05:02,540 It says that when you want to compute the actual realized 92 00:05:02,540 --> 00:05:04,640 return in your portfolio, you just 93 00:05:04,640 --> 00:05:07,460 take a weighted average of what you did on each stock, what 94 00:05:07,460 --> 00:05:08,920 your return is on each stock. 95 00:05:08,920 --> 00:05:11,210 That's an accounting identity. 96 00:05:11,210 --> 00:05:14,480 The second line is not an accounting identity. 97 00:05:14,480 --> 00:05:16,070 It comes from an accounting identity. 98 00:05:16,070 --> 00:05:20,050 But what it says is that on average, 99 00:05:20,050 --> 00:05:22,780 the rate of return of your portfolio 100 00:05:22,780 --> 00:05:28,060 is equal to a weighted average of the average rates of return 101 00:05:28,060 --> 00:05:30,890 on each of the components of your portfolio. 102 00:05:30,890 --> 00:05:31,390 OK? 103 00:05:31,390 --> 00:05:34,580 That's a very important principle. 104 00:05:34,580 --> 00:05:38,100 Any questions about that before we move on? 105 00:05:38,100 --> 00:05:38,600 OK. 106 00:05:38,600 --> 00:05:42,250 So I'm going to write that as a shorthand for the portfolio, mu 107 00:05:42,250 --> 00:05:43,270 sub p. 108 00:05:43,270 --> 00:05:45,560 So mu sub p is just equal to this whole expression 109 00:05:45,560 --> 00:05:47,180 right here. 110 00:05:47,180 --> 00:05:47,720 All right. 111 00:05:47,720 --> 00:05:53,210 So what we've now deduced is that the mean of my portfolio 112 00:05:53,210 --> 00:05:56,150 is simply the weighted average of the means of each 113 00:05:56,150 --> 00:05:59,660 of the securities in the portfolio. 114 00:05:59,660 --> 00:06:02,060 What I'm going to turn to next is a much more complicated 115 00:06:02,060 --> 00:06:08,060 calculation, which is, what is the variance of my portfolio. 116 00:06:08,060 --> 00:06:11,040 It turns out that the variance of the portfolio 117 00:06:11,040 --> 00:06:15,060 is not a simple weighted average of the variances 118 00:06:15,060 --> 00:06:16,510 of my individual securities. 119 00:06:16,510 --> 00:06:17,940 This is where it gets complicated, 120 00:06:17,940 --> 00:06:20,460 and also where it gets really interesting and valuable 121 00:06:20,460 --> 00:06:22,391 from the investor's perspective. 122 00:06:22,391 --> 00:06:22,890 OK. 123 00:06:22,890 --> 00:06:24,220 Let's do the calculation. 124 00:06:24,220 --> 00:06:29,220 You're going to have to dredge up your old DMD knowledge here 125 00:06:29,220 --> 00:06:33,560 of how to compute variances of sums of random variables. 126 00:06:33,560 --> 00:06:37,230 The variance of my portfolio return, R sub p, 127 00:06:37,230 --> 00:06:41,430 is simply equal to the expected value of the excess return 128 00:06:41,430 --> 00:06:44,450 of that portfolio, in excess of its mean, 129 00:06:44,450 --> 00:06:47,760 right, squared, and then take the expectation of that. 130 00:06:47,760 --> 00:06:51,660 And remember that the return of the portfolio 131 00:06:51,660 --> 00:06:53,520 is just a weighted average of the returns 132 00:06:53,520 --> 00:06:55,050 of the individual securities. 133 00:06:55,050 --> 00:06:58,470 And the mean of the portfolio is just the same weighted average 134 00:06:58,470 --> 00:07:00,510 of the means of the securities. 135 00:07:00,510 --> 00:07:03,690 So when you plug those relationships in, what you get 136 00:07:03,690 --> 00:07:06,060 is that the variance is simply equal to the expected 137 00:07:06,060 --> 00:07:12,781 value of the square of this long weighted average. 138 00:07:12,781 --> 00:07:14,905 So you've got a weighted average, a bunch of terms, 139 00:07:14,905 --> 00:07:15,840 right? 140 00:07:15,840 --> 00:07:18,340 And then you square that and then 141 00:07:18,340 --> 00:07:20,540 you take the expected value. 142 00:07:20,540 --> 00:07:26,680 Well, if you've got n terms in that weighted average, 143 00:07:26,680 --> 00:07:30,379 and you square that n terms, how many terms 144 00:07:30,379 --> 00:07:31,420 comes out of that square? 145 00:07:35,860 --> 00:07:37,490 Anybody? 146 00:07:37,490 --> 00:07:40,910 n terms, and you square those n terms, 147 00:07:40,910 --> 00:07:44,290 so those n terms multiplied by itself, 148 00:07:44,290 --> 00:07:46,660 how many terms do you get when you do that? 149 00:07:46,660 --> 00:07:47,160 Yeah? 150 00:07:47,160 --> 00:07:48,880 AUDIENCE: [INAUDIBLE] 151 00:07:48,880 --> 00:07:52,740 ANDREW LO: Well, why not just n squared? 152 00:07:52,740 --> 00:07:54,900 You're thinking about the unique elements, maybe, 153 00:07:54,900 --> 00:07:55,830 the off diagonal. 154 00:07:55,830 --> 00:07:57,690 I'm talking about all of them. 155 00:07:57,690 --> 00:08:00,100 So with n terms, when you square it, 156 00:08:00,100 --> 00:08:06,870 n terms multiplied by n terms, you get n times n terms, 157 00:08:06,870 --> 00:08:09,710 n squared terms. 158 00:08:09,710 --> 00:08:15,130 And so these n squared terms all look like this. 159 00:08:15,130 --> 00:08:18,290 They all look like omega i times omega 160 00:08:18,290 --> 00:08:22,180 j multiplied by the excess return of i 161 00:08:22,180 --> 00:08:23,920 times the excess return of j. 162 00:08:23,920 --> 00:08:26,190 Sometimes i equals j. 163 00:08:26,190 --> 00:08:30,450 And when that happens, you get omega i squared 164 00:08:30,450 --> 00:08:34,049 times the variance of security i. 165 00:08:34,049 --> 00:08:37,169 But when i does not equal to j, then you're 166 00:08:37,169 --> 00:08:42,059 going to get omega i times omega j times the covariance 167 00:08:42,059 --> 00:08:43,960 between the return on i and j. 168 00:08:47,840 --> 00:08:51,020 And another way of writing that covariance 169 00:08:51,020 --> 00:08:53,840 is equal to the correlation between i 170 00:08:53,840 --> 00:08:56,940 and j multiplied by the standard deviation of i 171 00:08:56,940 --> 00:09:00,410 and the standard deviation of j. 172 00:09:00,410 --> 00:09:04,762 The point is that when we look at the variance of a portfolio, 173 00:09:04,762 --> 00:09:06,470 it's not just the simple weighted average 174 00:09:06,470 --> 00:09:09,290 of the variances of the component stocks. 175 00:09:09,290 --> 00:09:14,150 It's actually a weighted average of all the cross products, 176 00:09:14,150 --> 00:09:20,140 where the weights are also the cross products of the weights. 177 00:09:20,140 --> 00:09:21,380 So it's these. 178 00:09:21,380 --> 00:09:22,567 It's all of these. 179 00:09:22,567 --> 00:09:23,900 And how many of these are there? 180 00:09:23,900 --> 00:09:29,240 Well, i goes from 1 to n, j goes from 1 to n, n times n, 181 00:09:29,240 --> 00:09:30,500 you get n squared of these. 182 00:09:30,500 --> 00:09:34,840 So this actually has a nice representation. 183 00:09:34,840 --> 00:09:37,130 It actually comes out of a table. 184 00:09:37,130 --> 00:09:37,880 Right? 185 00:09:37,880 --> 00:09:42,260 So you can think of all of the portfolio weights multiplied 186 00:09:42,260 --> 00:09:44,990 by the excess returns on one dimension 187 00:09:44,990 --> 00:09:49,340 of the table, the columns, and the exact same entries 188 00:09:49,340 --> 00:09:50,480 in the rows. 189 00:09:50,480 --> 00:09:54,290 You've got n columns, n rows, n times n, 190 00:09:54,290 --> 00:09:57,650 or n squared elements that make up 191 00:09:57,650 --> 00:10:01,620 the variance of your portfolio. 192 00:10:01,620 --> 00:10:04,280 So this is where it gets really complicated. 193 00:10:04,280 --> 00:10:06,530 But this is why we have computers and spreadsheets 194 00:10:06,530 --> 00:10:08,550 and things like that. 195 00:10:08,550 --> 00:10:12,200 So in order to figure out the variance of your portfolio, 196 00:10:12,200 --> 00:10:15,200 you've got to basically add all of the elements in this table. 197 00:10:15,200 --> 00:10:17,690 You've got n squared things to add up 198 00:10:17,690 --> 00:10:21,740 to get the variance of your portfolio. 199 00:10:21,740 --> 00:10:25,830 And the insight of modern finance theory-- 200 00:10:25,830 --> 00:10:28,310 the reason that Harry Markowitz won the Nobel Prize 201 00:10:28,310 --> 00:10:30,890 in economics for this idea-- 202 00:10:30,890 --> 00:10:35,810 is that when you add up all of these different elements, 203 00:10:35,810 --> 00:10:39,110 you can get results that are very different from just 204 00:10:39,110 --> 00:10:41,390 looking at one or two stocks. 205 00:10:41,390 --> 00:10:43,700 Because, in particular, there are 206 00:10:43,700 --> 00:10:46,970 some cases where these cross products 207 00:10:46,970 --> 00:10:51,190 are either small or maybe even negative. 208 00:10:51,190 --> 00:10:52,890 And in that case, it actually helps 209 00:10:52,890 --> 00:10:58,590 you to reduce the overall riskiness of your portfolio. 210 00:10:58,590 --> 00:11:00,180 This is the intuition. 211 00:11:00,180 --> 00:11:02,700 This is the mathematics underlying, 212 00:11:02,700 --> 00:11:04,890 don't put all your eggs in one basket. 213 00:11:04,890 --> 00:11:08,040 When you don't put all your eggs in one basket, what you're 214 00:11:08,040 --> 00:11:10,320 getting, what you're benefiting from, 215 00:11:10,320 --> 00:11:13,410 are these cross products that can either 216 00:11:13,410 --> 00:11:18,110 be small or negative, and will ultimately 217 00:11:18,110 --> 00:11:22,702 reduce the fluctuations of your portfolio. 218 00:11:22,702 --> 00:11:25,160 Another way of thinking about it is that some stocks go up, 219 00:11:25,160 --> 00:11:26,550 some stocks go down. 220 00:11:26,550 --> 00:11:30,380 And if they don't go up and down in the exact same way, 221 00:11:30,380 --> 00:11:34,010 if they are unrelated or less related, 222 00:11:34,010 --> 00:11:37,460 then when you put it all together in a portfolio 223 00:11:37,460 --> 00:11:43,880 it dampens the riskiness of your overall holdings. 224 00:11:43,880 --> 00:11:45,100 OK. 225 00:11:45,100 --> 00:11:48,520 So there are n squared elements that you 226 00:11:48,520 --> 00:11:52,360 have to add up in order to get the portfolio variance. 227 00:11:52,360 --> 00:11:56,140 It's literally the sum of all of the entries in this table. 228 00:11:56,140 --> 00:12:00,280 And the point of the next few bullet points 229 00:12:00,280 --> 00:12:03,430 is that there are a lot more covariances 230 00:12:03,430 --> 00:12:05,920 than there are variances. 231 00:12:05,920 --> 00:12:08,680 The variances are along the diagonal of that table, right? 232 00:12:08,680 --> 00:12:10,930 That's the variance of the first stock, 233 00:12:10,930 --> 00:12:12,533 the variance of the second stock, dot, 234 00:12:12,533 --> 00:12:15,530 dot, dot, the variance of the n stock. 235 00:12:15,530 --> 00:12:18,210 Those are how volatile the individual stocks are. 236 00:12:18,210 --> 00:12:20,640 But there are only n of those. 237 00:12:20,640 --> 00:12:24,690 There are n squared minus n of the other stuff. 238 00:12:24,690 --> 00:12:26,955 That's where you were thinking about n minus 1, right? 239 00:12:26,955 --> 00:12:32,520 It's n squared minus n, or n times n minus 1. 240 00:12:32,520 --> 00:12:35,100 Other non-diagonal entries. 241 00:12:35,100 --> 00:12:38,940 So what this suggests is that the covariances are a lot more 242 00:12:38,940 --> 00:12:43,470 important in determining the riskiness of your portfolio 243 00:12:43,470 --> 00:12:46,180 than the riskiness of the individual stocks. 244 00:12:46,180 --> 00:12:48,600 You know, Intel looks like a scary stock 245 00:12:48,600 --> 00:12:50,280 because it's really volatile. 246 00:12:50,280 --> 00:12:53,702 But when you've got n Intels in your portfolio, 247 00:12:53,702 --> 00:12:55,410 even though each of the individual stocks 248 00:12:55,410 --> 00:12:57,900 is scary, what you have to keep your eye on 249 00:12:57,900 --> 00:13:00,180 is how they are correlated. 250 00:13:00,180 --> 00:13:03,840 Because the correlations in a portfolio of n 251 00:13:03,840 --> 00:13:06,450 stocks is actually more important 252 00:13:06,450 --> 00:13:08,010 than the individual variances. 253 00:13:08,010 --> 00:13:10,830 The individual variances matter, but they 254 00:13:10,830 --> 00:13:13,860 don't matter nearly as much as the covariances. 255 00:13:13,860 --> 00:13:15,000 OK? 256 00:13:15,000 --> 00:13:16,950 So now I'm going to do an example. 257 00:13:16,950 --> 00:13:21,150 It's hard to get intuition for n by n matrices, 258 00:13:21,150 --> 00:13:25,110 unless you're Rain Man, or some incredible genius. 259 00:13:25,110 --> 00:13:27,760 The way that I think about this is, let's look at two assets. 260 00:13:27,760 --> 00:13:28,260 OK? 261 00:13:28,260 --> 00:13:31,110 If I can understand two assets, I can sort generalize and think 262 00:13:31,110 --> 00:13:32,460 about n assets. 263 00:13:32,460 --> 00:13:34,930 So I gave you the general result just 264 00:13:34,930 --> 00:13:36,820 to tell you that this is how you do it. 265 00:13:36,820 --> 00:13:38,400 But now to understand it, really, 266 00:13:38,400 --> 00:13:40,220 let's focus just on two assets. 267 00:13:40,220 --> 00:13:41,640 OK? 268 00:13:41,640 --> 00:13:44,160 Suppose we only have two stocks, a and b. 269 00:13:44,160 --> 00:13:49,140 And I'm going to calculate the expected return and standard 270 00:13:49,140 --> 00:13:52,080 deviation of a portfolio with just those two stocks. 271 00:13:52,080 --> 00:13:53,730 So the portfolio weights are going 272 00:13:53,730 --> 00:13:58,020 to be omega a for stock a, and omega b for stock b. 273 00:13:58,020 --> 00:14:00,357 And omega a plus omega b adds up to 1. 274 00:14:00,357 --> 00:14:01,440 So just keep that in mind. 275 00:14:01,440 --> 00:14:03,990 I don't put it there just because I 276 00:14:03,990 --> 00:14:05,895 want to save a little space on the slide. 277 00:14:05,895 --> 00:14:07,770 But these are the weights for the two stocks. 278 00:14:07,770 --> 00:14:09,660 And those are the only two things you hold. 279 00:14:09,660 --> 00:14:11,070 All right? 280 00:14:11,070 --> 00:14:12,660 The expected value is as we said. 281 00:14:12,660 --> 00:14:14,910 It's a weighted average of your holdings of 282 00:14:14,910 --> 00:14:19,350 and b, right, multiplied by the mean of a and the mean of b. 283 00:14:19,350 --> 00:14:24,120 Now the variance, the variance of these two securities 284 00:14:24,120 --> 00:14:26,430 when they are put into a portfolio 285 00:14:26,430 --> 00:14:29,370 is going to be simply omega a squared 286 00:14:29,370 --> 00:14:31,820 times the variance of a plus omega 287 00:14:31,820 --> 00:14:33,990 b squared times the variance of b. 288 00:14:33,990 --> 00:14:36,540 And now you only have one cross product 289 00:14:36,540 --> 00:14:38,760 to worry about because you've only got two stocks. 290 00:14:38,760 --> 00:14:39,540 Right? 291 00:14:39,540 --> 00:14:44,250 And that is 2 times omega a times omega 292 00:14:44,250 --> 00:14:47,580 b times the covariance between a and b. 293 00:14:47,580 --> 00:14:50,580 Now you can do this in a 2 by 2 table, just like the n 294 00:14:50,580 --> 00:14:51,670 by n table. 295 00:14:51,670 --> 00:14:53,190 And with a 2 by 2 table, you know 296 00:14:53,190 --> 00:14:57,070 that you've got two diagonal elements and two off 297 00:14:57,070 --> 00:14:58,450 diagonal elements. 298 00:14:58,450 --> 00:15:02,380 And that's where you get that number 2 for the 2 times 299 00:15:02,380 --> 00:15:05,950 omega a, omega b times the covariance. 300 00:15:05,950 --> 00:15:07,150 OK? 301 00:15:07,150 --> 00:15:09,910 And I rewrite it in terms of correlations 302 00:15:09,910 --> 00:15:13,131 because it's easier to think in terms of correlations. 303 00:15:13,131 --> 00:15:13,630 Why? 304 00:15:13,630 --> 00:15:15,004 Because correlations we know have 305 00:15:15,004 --> 00:15:16,950 to be between minus 1 and 1. 306 00:15:16,950 --> 00:15:20,170 And so if I tell you a stock has a correlation with another 307 00:15:20,170 --> 00:15:23,260 of 30%, you can actually get your arms around that. 308 00:15:23,260 --> 00:15:24,831 You can get intuition for that. 309 00:15:24,831 --> 00:15:25,330 All right? 310 00:15:25,330 --> 00:15:27,310 So it's a little easier to interpret. 311 00:15:27,310 --> 00:15:29,440 But either way is fine. 312 00:15:29,440 --> 00:15:32,620 So we now have an expression for the mean 313 00:15:32,620 --> 00:15:35,500 and the variance of a portfolio of two assets. 314 00:15:35,500 --> 00:15:38,630 Any questions about this? 315 00:15:38,630 --> 00:15:40,700 Everybody understand this and how I got it? 316 00:15:40,700 --> 00:15:41,790 No tricks up my sleeve. 317 00:15:41,790 --> 00:15:45,110 It's really meant to be relatively straightforward. 318 00:15:45,110 --> 00:15:47,510 But the implications will be dramatic, 319 00:15:47,510 --> 00:15:51,030 as I'm going to show you in a minute. 320 00:15:51,030 --> 00:15:52,720 OK. 321 00:15:52,720 --> 00:15:55,516 So let me go through at least one numerical example. 322 00:15:55,516 --> 00:15:57,640 And then I'm going to show you why this is actually 323 00:15:57,640 --> 00:15:59,380 so important. 324 00:15:59,380 --> 00:16:03,400 From 1946 to 2001, Motorola had an average monthly return 325 00:16:03,400 --> 00:16:09,130 of 1.75%, and a standard deviation of 9.73%. 326 00:16:09,130 --> 00:16:12,670 General Motors had an average return of 1.08%, 327 00:16:12,670 --> 00:16:16,030 and a standard deviation of 6.23%. 328 00:16:16,030 --> 00:16:18,700 That has obviously since gone up a great deal, 329 00:16:18,700 --> 00:16:21,010 the standard deviation, that is. 330 00:16:21,010 --> 00:16:26,230 And their correlation is 0.37, or 37%. 331 00:16:26,230 --> 00:16:29,590 How would a portfolio of the two stocks perform? 332 00:16:29,590 --> 00:16:30,510 Well, it depends. 333 00:16:30,510 --> 00:16:31,840 What are the weights? 334 00:16:31,840 --> 00:16:32,685 Right? 335 00:16:32,685 --> 00:16:34,060 If you change the weights, you're 336 00:16:34,060 --> 00:16:35,393 going to change the performance. 337 00:16:35,393 --> 00:16:36,820 So let's just take a look. 338 00:16:36,820 --> 00:16:39,940 If you put all your weight on General Motors and nothing 339 00:16:39,940 --> 00:16:42,010 on Motorola, not surprisingly, you're 340 00:16:42,010 --> 00:16:44,882 going to get General Motors' return. 341 00:16:44,882 --> 00:16:47,340 At the other end of the extreme, if you put all your weight 342 00:16:47,340 --> 00:16:49,650 on Motorola and nothing on General Motors, 343 00:16:49,650 --> 00:16:52,331 you're going to get Motorola's characteristics. 344 00:16:52,331 --> 00:16:52,830 All right? 345 00:16:52,830 --> 00:16:56,050 Nothing mysterious about that. 346 00:16:56,050 --> 00:17:00,420 However, if you weight the two, so if you put 50-50 347 00:17:00,420 --> 00:17:03,390 on General Motors and Motorola, you're 348 00:17:03,390 --> 00:17:08,220 going to get a return that is 50-50 of each of those returns. 349 00:17:08,220 --> 00:17:12,721 But you're going to get a risk that is not 50-50. 350 00:17:12,721 --> 00:17:13,220 Right? 351 00:17:13,220 --> 00:17:16,579 Because the risks, the risks don't 352 00:17:16,579 --> 00:17:18,560 aggregate in a linear way. 353 00:17:18,560 --> 00:17:22,400 The risks actually aggregate non-linearly 354 00:17:22,400 --> 00:17:25,520 because of those covariances or correlations. 355 00:17:25,520 --> 00:17:26,829 OK. 356 00:17:26,829 --> 00:17:29,350 So what you see here is that by taking 357 00:17:29,350 --> 00:17:32,590 a weighted average of the two, what you can do 358 00:17:32,590 --> 00:17:36,970 is you can boost your return to 1.42%, 359 00:17:36,970 --> 00:17:40,510 but the extra risk you're taking is 360 00:17:40,510 --> 00:17:46,820 from 6.23% to 6.68% per month. 361 00:17:46,820 --> 00:17:48,880 That's not a big increase in risk, 362 00:17:48,880 --> 00:17:53,140 but that is a pretty significant increase in return. 363 00:17:53,140 --> 00:17:55,780 Now again, I'm not telling you that's what you all should do, 364 00:17:55,780 --> 00:17:57,790 because it depends on your risk preferences. 365 00:17:57,790 --> 00:17:59,587 And for the next lecture or two, I'm 366 00:17:59,587 --> 00:18:01,420 going to be saying that over and over again. 367 00:18:01,420 --> 00:18:03,070 I'm going to be saying it depends upon your risk 368 00:18:03,070 --> 00:18:03,580 preferences. 369 00:18:03,580 --> 00:18:05,999 But at the end of the next two lectures, 370 00:18:05,999 --> 00:18:08,290 I'm going to be able to tell you something that doesn't 371 00:18:08,290 --> 00:18:09,550 depend on your risk preference. 372 00:18:09,550 --> 00:18:11,830 I'm going to be able to tell you something that all of you 373 00:18:11,830 --> 00:18:13,630 should be willing to do if you're rational. 374 00:18:13,630 --> 00:18:16,330 And if you're not, I will trade with you personally 375 00:18:16,330 --> 00:18:19,440 to help you learn rationality. 376 00:18:19,440 --> 00:18:20,440 OK? 377 00:18:20,440 --> 00:18:22,600 But for now, I don't have a story 378 00:18:22,600 --> 00:18:25,540 to tell you about which of these rows you ought to take. 379 00:18:25,540 --> 00:18:27,130 There is no ought to here. 380 00:18:27,130 --> 00:18:30,850 It's just a matter of what you prefer, risk versus return. 381 00:18:30,850 --> 00:18:31,960 Right? 382 00:18:31,960 --> 00:18:34,990 Now interestingly, I put here one more case 383 00:18:34,990 --> 00:18:39,640 which lies outside of the band of 0, 1. 384 00:18:39,640 --> 00:18:44,020 This is a case where you've put 125% of your wealth 385 00:18:44,020 --> 00:18:45,760 into Motorola. 386 00:18:45,760 --> 00:18:49,570 And you've put negative 25% of your wealth 387 00:18:49,570 --> 00:18:50,582 into General Motors. 388 00:18:50,582 --> 00:18:52,790 Remember, we talked about this in the very beginning. 389 00:18:52,790 --> 00:18:55,660 You're using short selling to short sell 390 00:18:55,660 --> 00:18:58,360 General Motors, which you might think 391 00:18:58,360 --> 00:19:02,485 is not a bad idea nowadays, given how troubled they are. 392 00:19:02,485 --> 00:19:06,730 You're going to short General Motors, take the proceeds, 393 00:19:06,730 --> 00:19:09,330 and put that plus the 100% that you started with, 394 00:19:09,330 --> 00:19:11,530 you're going to put that in Motorola. 395 00:19:11,530 --> 00:19:15,700 So you really want to make a big bet on communications 396 00:19:15,700 --> 00:19:18,370 and microprocessors. 397 00:19:18,370 --> 00:19:19,930 And you're going to leverage that 398 00:19:19,930 --> 00:19:23,530 bet by putting a negative bet on the auto industry. 399 00:19:23,530 --> 00:19:26,710 That's what that last row implies. 400 00:19:26,710 --> 00:19:31,360 And what you get is a much, much higher rate 401 00:19:31,360 --> 00:19:35,781 of return than anything else in the other examples. 402 00:19:35,781 --> 00:19:36,280 Right? 403 00:19:36,280 --> 00:19:37,540 Much higher rate of return. 404 00:19:37,540 --> 00:19:39,430 But look at the risk. 405 00:19:39,430 --> 00:19:42,132 Now the risk is close to double what 406 00:19:42,132 --> 00:19:44,590 you would have gotten had you put all your money in General 407 00:19:44,590 --> 00:19:45,089 Motors. 408 00:19:45,089 --> 00:19:46,930 Are you willing to do that? 409 00:19:46,930 --> 00:19:48,430 Well, you can easily imagine there 410 00:19:48,430 --> 00:19:51,910 are people in this class that would be delighted to do that, 411 00:19:51,910 --> 00:19:54,790 and there are others of you that would be scared to death. 412 00:19:54,790 --> 00:19:56,480 That's way too much volatility. 413 00:19:56,480 --> 00:19:58,310 These are monthly numbers, by the way. 414 00:19:58,310 --> 00:20:00,760 So if you want to calculate the annual equivalent, 415 00:20:00,760 --> 00:20:02,060 how would you do that? 416 00:20:02,060 --> 00:20:04,480 How would you get an annual standard deviation 417 00:20:04,480 --> 00:20:08,530 from a monthly standard deviation? 418 00:20:08,530 --> 00:20:11,290 Any idea? 419 00:20:11,290 --> 00:20:13,270 Well, let's do the return first. 420 00:20:13,270 --> 00:20:17,860 How do you annualize the monthly return to get an annual return? 421 00:20:17,860 --> 00:20:20,350 Let's forget about compounding. 422 00:20:20,350 --> 00:20:21,340 What would you do? 423 00:20:21,340 --> 00:20:22,256 AUDIENCE: [INAUDIBLE]. 424 00:20:22,256 --> 00:20:24,040 ANDREW LO: What? 425 00:20:24,040 --> 00:20:25,059 Multiply by 12. 426 00:20:25,059 --> 00:20:25,600 That's right. 427 00:20:25,600 --> 00:20:27,640 If you forget about confounding, you multiply by 12. 428 00:20:27,640 --> 00:20:28,890 What if you didn't forget about compounding? 429 00:20:28,890 --> 00:20:29,901 Then what do you do? 430 00:20:29,901 --> 00:20:32,060 AUDIENCE: [INAUDIBLE] 431 00:20:32,060 --> 00:20:34,860 ANDREW LO: Add 1, raise it to the 12th power, 432 00:20:34,860 --> 00:20:35,670 then subtract 1. 433 00:20:35,670 --> 00:20:36,750 That's right. 434 00:20:36,750 --> 00:20:37,440 OK. 435 00:20:37,440 --> 00:20:39,770 What about risk? 436 00:20:39,770 --> 00:20:40,270 Ah, see? 437 00:20:40,270 --> 00:20:41,200 This is tricky. 438 00:20:41,200 --> 00:20:47,810 What's the variance of a 12-month return 439 00:20:47,810 --> 00:20:51,260 in relationship to the variance of a one-month return? 440 00:20:51,260 --> 00:20:53,560 We haven't talked about that. 441 00:20:53,560 --> 00:20:54,760 What is it? 442 00:20:54,760 --> 00:20:55,700 Yeah? 443 00:20:55,700 --> 00:20:58,417 AUDIENCE: [INAUDIBLE] 444 00:20:58,417 --> 00:20:59,000 ANDREW LO: No. 445 00:20:59,000 --> 00:20:59,500 No. 446 00:20:59,500 --> 00:21:01,670 It doesn't compound in that way. 447 00:21:01,670 --> 00:21:04,010 Variance is actually a little simpler, 448 00:21:04,010 --> 00:21:05,460 under certain assumptions. 449 00:21:05,460 --> 00:21:06,120 Yeah? 450 00:21:06,120 --> 00:21:11,332 AUDIENCE: [INAUDIBLE] 451 00:21:11,332 --> 00:21:12,040 ANDREW LO: Right. 452 00:21:12,040 --> 00:21:14,900 There's no correlation between one month to the next. 453 00:21:14,900 --> 00:21:19,920 AUDIENCE: [INAUDIBLE] 454 00:21:19,920 --> 00:21:20,920 ANDREW LO: That's right. 455 00:21:20,920 --> 00:21:21,419 Agreed. 456 00:21:21,419 --> 00:21:23,980 So let's assume that away. 457 00:21:23,980 --> 00:21:25,480 Then it's constant. 458 00:21:25,480 --> 00:21:28,390 But the monthly variance is constant. 459 00:21:28,390 --> 00:21:30,250 But what's the annual variance? 460 00:21:30,250 --> 00:21:32,537 What's the riskiness of a 12-month return 461 00:21:32,537 --> 00:21:34,870 if you know what the riskiness of a one-month return is? 462 00:21:34,870 --> 00:21:35,506 Yeah? 463 00:21:35,506 --> 00:21:37,990 AUDIENCE: [INAUDIBLE] 464 00:21:37,990 --> 00:21:40,810 ANDREW LO: Well, it can, depending on your assumptions. 465 00:21:40,810 --> 00:21:43,580 But I want a simple set of assumptions. 466 00:21:43,580 --> 00:21:44,080 Andy? 467 00:21:44,080 --> 00:21:46,330 AUDIENCE: [INAUDIBLE] 468 00:21:46,330 --> 00:21:49,060 ANDREW LO: Variance, yes. 469 00:21:49,060 --> 00:21:50,840 Multiply by 12. 470 00:21:50,840 --> 00:21:52,130 Now why is that? 471 00:21:52,130 --> 00:21:53,740 It's because-- and you know all this. 472 00:21:53,740 --> 00:21:55,345 You know this from your DMD. 473 00:21:55,345 --> 00:21:57,490 At least I think you do. 474 00:21:57,490 --> 00:21:59,770 You may not think you do, but you do. 475 00:21:59,770 --> 00:22:04,000 The variance of a plus b is equal to what? 476 00:22:04,000 --> 00:22:05,680 Who can tell me? 477 00:22:05,680 --> 00:22:07,390 What's the variance of a plus b? 478 00:22:07,390 --> 00:22:08,470 Two random variables. 479 00:22:12,930 --> 00:22:18,930 Variance of a plus variance of b plus 2 times 480 00:22:18,930 --> 00:22:20,310 the covariance of a and b. 481 00:22:20,310 --> 00:22:23,824 If the covariance is 0, then it is the variance 482 00:22:23,824 --> 00:22:24,990 of a plus the variance of b. 483 00:22:24,990 --> 00:22:26,700 Absolutely, that's right. 484 00:22:26,700 --> 00:22:29,070 So if we assume there are no cycles, 485 00:22:29,070 --> 00:22:31,680 there are no predictabilities, there's no regularities, 486 00:22:31,680 --> 00:22:33,540 there's no correlation, every month 487 00:22:33,540 --> 00:22:35,560 is independent of every other month, 488 00:22:35,560 --> 00:22:39,630 than the variance of a 12-month return 489 00:22:39,630 --> 00:22:43,680 is literally just 12 times the variance of a one-month return, 490 00:22:43,680 --> 00:22:46,650 assuming the variances stay constant throughout the month. 491 00:22:46,650 --> 00:22:49,270 The monthly variance doesn't go up or down. 492 00:22:49,270 --> 00:22:50,740 OK? 493 00:22:50,740 --> 00:22:51,784 Now, that's the variance. 494 00:22:51,784 --> 00:22:53,200 What about the standard deviation? 495 00:22:53,200 --> 00:22:55,674 The standard deviation is the square root of the variance. 496 00:22:55,674 --> 00:22:58,090 So that means that the standard deviation on an annualized 497 00:22:58,090 --> 00:23:03,840 basis is equal to square root of 12 times the standard deviation 498 00:23:03,840 --> 00:23:04,420 of a monthly. 499 00:23:04,420 --> 00:23:04,920 OK. 500 00:23:04,920 --> 00:23:06,390 So finally, we're almost there. 501 00:23:06,390 --> 00:23:08,070 What's the square root of 12? 502 00:23:08,070 --> 00:23:11,760 Something between 3 and 4, right? 503 00:23:11,760 --> 00:23:15,180 So I don't know, call it 3 and 1/2, whatever. 504 00:23:15,180 --> 00:23:20,850 We have to take this number and multiply it by about 3 and 1/2. 505 00:23:20,850 --> 00:23:23,220 And that will get us an approximate annual standard 506 00:23:23,220 --> 00:23:24,370 deviation. 507 00:23:24,370 --> 00:23:27,600 So a 12% volatility multiplied by 3 and 1/2, 508 00:23:27,600 --> 00:23:31,650 you're talking about some crazy volatile stock, or portfolio. 509 00:23:31,650 --> 00:23:33,277 But actually, nowadays, you know, 510 00:23:33,277 --> 00:23:34,860 we're used to that kind of volatility. 511 00:23:34,860 --> 00:23:36,310 That's no big deal. 512 00:23:36,310 --> 00:23:39,090 But relative to the individual stock, 513 00:23:39,090 --> 00:23:40,830 it's quite a bit more volatile. 514 00:23:40,830 --> 00:23:42,300 OK? 515 00:23:42,300 --> 00:23:45,120 So this is an example of how you compute 516 00:23:45,120 --> 00:23:50,160 the riskiness and the return of a portfolio of two stocks. 517 00:23:50,160 --> 00:23:53,979 And when you do this, you get a whole bunch of possibilities. 518 00:23:53,979 --> 00:23:55,770 And the idea behind modern portfolio theory 519 00:23:55,770 --> 00:23:58,760 is you pick the one that you like the best. 520 00:23:58,760 --> 00:24:00,320 And that's optimal for you. 521 00:24:03,480 --> 00:24:10,500 Now let me graph this in that mean-standard deviation graph, 522 00:24:10,500 --> 00:24:14,020 and you'll see something really interesting. 523 00:24:14,020 --> 00:24:19,550 So here's General Motors and there is Motorola. 524 00:24:19,550 --> 00:24:22,520 And what I've graphed with the red dots 525 00:24:22,520 --> 00:24:26,570 is just the different combinations of 25/75, 526 00:24:26,570 --> 00:24:30,820 50/50, 75/25, and so on. 527 00:24:30,820 --> 00:24:35,860 And so the red dots that are strictly between 0 and 1, 528 00:24:35,860 --> 00:24:38,830 where the weights are between 0 and 1, those are the dots 529 00:24:38,830 --> 00:24:44,566 contained in that arc between GM and Motorola. 530 00:24:44,566 --> 00:24:45,940 So the first point I want to make 531 00:24:45,940 --> 00:24:52,370 is that when you graph the risk/reward trade-off, 532 00:24:52,370 --> 00:24:55,430 as I told you, it's not linear. 533 00:24:55,430 --> 00:24:56,540 It's non-linear. 534 00:24:56,540 --> 00:24:58,470 In fact, it's curved. 535 00:24:58,470 --> 00:25:00,950 It looks like a bullet. 536 00:25:00,950 --> 00:25:01,450 OK? 537 00:25:04,870 --> 00:25:06,670 That tells you right away that there's 538 00:25:06,670 --> 00:25:09,970 something more subtle about portfolio theory 539 00:25:09,970 --> 00:25:12,370 than just taking weighted averages. 540 00:25:12,370 --> 00:25:17,620 The means are weighted averages, but the variances 541 00:25:17,620 --> 00:25:21,027 are not because of the covariance between them. 542 00:25:21,027 --> 00:25:22,610 And then when you take the square root 543 00:25:22,610 --> 00:25:26,200 to get the standard deviation, it looks a little bit more 544 00:25:26,200 --> 00:25:29,120 complicated, yet again. 545 00:25:29,120 --> 00:25:30,920 It looks like this. 546 00:25:30,920 --> 00:25:35,570 Now that red dot that goes beyond Motorola, 547 00:25:35,570 --> 00:25:39,500 that red dot is the case where you're shorting General Motors 548 00:25:39,500 --> 00:25:44,290 and you're using the proceeds to take an extra large position 549 00:25:44,290 --> 00:25:45,350 in Motorola. 550 00:25:45,350 --> 00:25:46,490 OK? 551 00:25:46,490 --> 00:25:52,869 So you're going beyond the Motorola risk/reward point. 552 00:25:52,869 --> 00:25:54,660 So I need you to spend a little bit of time 553 00:25:54,660 --> 00:25:57,990 now twisting your brain in a way that 554 00:25:57,990 --> 00:26:00,700 makes this more understandable. 555 00:26:00,700 --> 00:26:04,420 What I mean by that is that you have 556 00:26:04,420 --> 00:26:07,330 to think about the weights of the various different 557 00:26:07,330 --> 00:26:08,380 portfolios. 558 00:26:08,380 --> 00:26:11,050 But the weights don't show up on this graph. 559 00:26:11,050 --> 00:26:12,550 But I want you to keep those weights 560 00:26:12,550 --> 00:26:13,870 in the back of your head. 561 00:26:13,870 --> 00:26:16,320 So while you're looking at this graph, 562 00:26:16,320 --> 00:26:18,840 you have to remember that as you move along, 563 00:26:18,840 --> 00:26:23,430 that bullet, the omegas, the weights are changing. 564 00:26:23,430 --> 00:26:28,860 So at the General Motors dot, your weight for General Motors 565 00:26:28,860 --> 00:26:32,590 is a 100% and your weight for Motorola is 0. 566 00:26:32,590 --> 00:26:37,470 At the Motorola dot, your weight for General Motors is 0 567 00:26:37,470 --> 00:26:41,360 and your weight for Motorola is 100%. 568 00:26:41,360 --> 00:26:47,390 And as you vary the weights, you trace this arc, this curve. 569 00:26:50,120 --> 00:26:53,000 And as you go beyond either General Motors or Motorola-- 570 00:26:53,000 --> 00:26:54,620 in other words, as you short sell 571 00:26:54,620 --> 00:26:58,310 one to invest more than 100% in the other, 572 00:26:58,310 --> 00:27:04,280 you then go outside of the two points in either direction. 573 00:27:04,280 --> 00:27:06,800 Right? 574 00:27:06,800 --> 00:27:09,140 Any questions about this graph? 575 00:27:09,140 --> 00:27:09,922 Yeah? 576 00:27:09,922 --> 00:27:16,190 AUDIENCE: [INAUDIBLE] 577 00:27:16,190 --> 00:27:16,920 ANDREW LO: OK. 578 00:27:16,920 --> 00:27:18,586 So how do you calculate the correlation? 579 00:27:18,586 --> 00:27:19,740 That's a good question. 580 00:27:19,740 --> 00:27:21,810 Historically, the way you would do it 581 00:27:21,810 --> 00:27:23,790 is exactly the way that these correlations 582 00:27:23,790 --> 00:27:25,530 look like they're defined. 583 00:27:25,530 --> 00:27:29,460 So in other words, you would calculate the covariance 584 00:27:29,460 --> 00:27:33,090 by essentially taking a historical average 585 00:27:33,090 --> 00:27:35,650 of these cross products. 586 00:27:35,650 --> 00:27:38,970 So going back to the first slide in this particular lecture, 587 00:27:38,970 --> 00:27:41,291 I gave you a formula that shows you how to do that. 588 00:27:41,291 --> 00:27:42,790 Let me go back and take a look at it 589 00:27:42,790 --> 00:27:44,123 so that you refresh your memory. 590 00:27:46,711 --> 00:27:47,210 Let's see. 591 00:27:47,210 --> 00:27:48,570 Oh, it was not in this lecture. 592 00:27:48,570 --> 00:27:49,890 It was in the previous lecture. 593 00:27:49,890 --> 00:27:51,264 In a previous lecture, I gave you 594 00:27:51,264 --> 00:27:53,190 a formula that shows you how to calculate 595 00:27:53,190 --> 00:27:55,920 the weighted average of this, or the historical average. 596 00:27:55,920 --> 00:27:58,320 It's basically just taking historical data 597 00:27:58,320 --> 00:28:02,530 and then estimating this quantity right here, 598 00:28:02,530 --> 00:28:03,780 this particular cross product. 599 00:28:03,780 --> 00:28:05,820 The omega weights come out, because that's 600 00:28:05,820 --> 00:28:07,080 what you get to decide. 601 00:28:07,080 --> 00:28:11,580 But the inside stuff, how to calculate that is simply 602 00:28:11,580 --> 00:28:17,820 to take, historically, your actual return minus the mean 603 00:28:17,820 --> 00:28:26,520 of i, Rjt minus the mean of j, and then take 1 over t, 604 00:28:26,520 --> 00:28:33,350 and that's your estimator for the sigma hat ij. 605 00:28:33,350 --> 00:28:36,620 So you get historical data and you would estimate the mean, 606 00:28:36,620 --> 00:28:39,600 estimate the mean, take the cross products over time. 607 00:28:39,600 --> 00:28:44,990 This t goes from 1 to t, and that gets you an estimator. 608 00:28:44,990 --> 00:28:47,000 Is that the best estimator? 609 00:28:47,000 --> 00:28:50,150 Not necessarily, because things change over time. 610 00:28:50,150 --> 00:28:51,720 So you have to worry about that. 611 00:28:51,720 --> 00:28:53,870 And that's what you learn in 15433, 612 00:28:53,870 --> 00:28:57,960 how to actually implement a lot of these ideas using data. 613 00:28:57,960 --> 00:28:58,556 Yeah? 614 00:28:58,556 --> 00:29:01,210 AUDIENCE: [INAUDIBLE] 615 00:29:01,210 --> 00:29:01,960 ANDREW LO: OK. 616 00:29:01,960 --> 00:29:06,860 AUDIENCE: [INAUDIBLE] 617 00:29:06,860 --> 00:29:08,870 ANDREW LO: OK. 618 00:29:08,870 --> 00:29:11,630 So the question is, is there any recommended length of time 619 00:29:11,630 --> 00:29:14,480 for which you would go about estimating the data, 620 00:29:14,480 --> 00:29:16,590 estimating these quantities using the data? 621 00:29:16,590 --> 00:29:17,450 The answer is no. 622 00:29:17,450 --> 00:29:21,020 There's no recommended time, simply because you 623 00:29:21,020 --> 00:29:23,270 have to trade off two things. 624 00:29:23,270 --> 00:29:25,670 The longer time you have, the more accurate 625 00:29:25,670 --> 00:29:27,530 your estimate is going to be. 626 00:29:27,530 --> 00:29:29,660 In the limit, as t goes to infinity, 627 00:29:29,660 --> 00:29:31,660 you'll get a perfect estimator. 628 00:29:31,660 --> 00:29:33,050 OK? 629 00:29:33,050 --> 00:29:36,610 However, that's under the assumption 630 00:29:36,610 --> 00:29:38,470 that nothing ever changes. 631 00:29:38,470 --> 00:29:42,370 And I'm pretty sure that beyond some t, 632 00:29:42,370 --> 00:29:45,970 over let's say 150 or 200 years, you know, 633 00:29:45,970 --> 00:29:50,360 we didn't exist in terms of stock markets and data. 634 00:29:50,360 --> 00:29:52,690 I don't think that, for example, General Motors 635 00:29:52,690 --> 00:29:54,310 goes back 200 years. 636 00:29:54,310 --> 00:29:57,432 So beyond some t, this is going to look very boring. 637 00:29:57,432 --> 00:29:59,140 But the other thing you have to trade off 638 00:29:59,140 --> 00:30:02,080 is the fact that market conditions change. 639 00:30:02,080 --> 00:30:05,440 So General Motors today is not what General Motors was, 640 00:30:05,440 --> 00:30:06,860 even 10 years ago. 641 00:30:06,860 --> 00:30:08,410 So if you use lots of data, you're 642 00:30:08,410 --> 00:30:10,720 going to build into your results what 643 00:30:10,720 --> 00:30:12,910 are called non-stationarities. 644 00:30:12,910 --> 00:30:15,400 And the bottom line is you have to balance off 645 00:30:15,400 --> 00:30:17,920 the non-stationarities against the error 646 00:30:17,920 --> 00:30:20,470 that you introduce by not using enough data. 647 00:30:20,470 --> 00:30:24,260 And that really is where 15433 comes in. 648 00:30:24,260 --> 00:30:26,380 So there are methods that we have developed 649 00:30:26,380 --> 00:30:27,880 for balancing those two. 650 00:30:27,880 --> 00:30:30,370 But the bottom line is that there isn't one single answer. 651 00:30:30,370 --> 00:30:33,130 So it depends on how unstable the markets are 652 00:30:33,130 --> 00:30:38,400 and how volatile the underlying estimates are, given the data. 653 00:30:38,400 --> 00:30:39,380 Yeah? 654 00:30:39,380 --> 00:30:54,204 AUDIENCE: [INAUDIBLE] 655 00:30:54,204 --> 00:30:55,120 ANDREW LO: Well, sure. 656 00:30:55,120 --> 00:30:58,085 You could try to include a full cycle. 657 00:30:58,085 --> 00:30:59,710 But then you're left with the question, 658 00:30:59,710 --> 00:31:01,690 the full cycle of what? 659 00:31:01,690 --> 00:31:03,730 For example, there are these things 660 00:31:03,730 --> 00:31:06,610 called Kondratiev cycles that claim to be 661 00:31:06,610 --> 00:31:09,530 something like 50 year periods. 662 00:31:09,530 --> 00:31:11,320 So if you want to include one of those, 663 00:31:11,320 --> 00:31:13,810 you've got to include 50 years of data. 664 00:31:13,810 --> 00:31:15,700 But do you really think that General Motors 665 00:31:15,700 --> 00:31:18,160 is the same company over those 50 years? 666 00:31:18,160 --> 00:31:21,160 You're building in a lot of bias in terms 667 00:31:21,160 --> 00:31:24,820 of the non-stationarity of one aspect, 668 00:31:24,820 --> 00:31:26,950 while incorporating the stationarities 669 00:31:26,950 --> 00:31:28,570 of another aspect. 670 00:31:28,570 --> 00:31:30,880 So how do you balance off those two, right? 671 00:31:30,880 --> 00:31:33,280 This is where it gets more complex. 672 00:31:33,280 --> 00:31:36,130 And you have to take a stand on the kind of non-stationarities 673 00:31:36,130 --> 00:31:38,180 that are in the data. 674 00:31:38,180 --> 00:31:39,213 Other questions? 675 00:31:39,213 --> 00:31:40,139 Yeah? 676 00:31:40,139 --> 00:31:52,542 AUDIENCE: [INAUDIBLE] 677 00:31:52,542 --> 00:31:54,500 ANDREW LO: Well, first of all, we're not trying 678 00:31:54,500 --> 00:31:55,770 to predict the stock market. 679 00:31:55,770 --> 00:31:57,311 Remember, we're not trying to predict 680 00:31:57,311 --> 00:31:58,880 where markets are going to go. 681 00:31:58,880 --> 00:32:01,040 What we're suggesting is that there 682 00:32:01,040 --> 00:32:06,030 are underlying parameters of the data that are stable over time. 683 00:32:06,030 --> 00:32:08,570 So in other words, the stocks can go up or down, right? 684 00:32:08,570 --> 00:32:12,740 But on average, they have some level of return 685 00:32:12,740 --> 00:32:15,330 that people expect, given their risk. 686 00:32:15,330 --> 00:32:17,060 That's what we're assuming is stable. 687 00:32:17,060 --> 00:32:19,190 Similarly, covariances. 688 00:32:19,190 --> 00:32:21,470 Stocks can go up or down in relationship 689 00:32:21,470 --> 00:32:22,940 to other stocks going down or up. 690 00:32:22,940 --> 00:32:24,940 We don't know what's going to happen day to day. 691 00:32:24,940 --> 00:32:28,310 But over a period of time, it seems 692 00:32:28,310 --> 00:32:32,240 like there's a pattern where most stocks on the New York 693 00:32:32,240 --> 00:32:34,580 and American stock exchanges and NASDAQ, 694 00:32:34,580 --> 00:32:36,620 they seem to go up together and they seem to go 695 00:32:36,620 --> 00:32:38,684 down together, on average. 696 00:32:38,684 --> 00:32:40,100 So that's what, really, what we're 697 00:32:40,100 --> 00:32:42,590 trying to distill from the data. 698 00:32:42,590 --> 00:32:44,870 It's very different than trying to forecast 699 00:32:44,870 --> 00:32:47,516 what's going to happen with the stock market next week. 700 00:32:47,516 --> 00:32:49,640 And this, I told you, is the fundamental difference 701 00:32:49,640 --> 00:32:52,310 between academic finance and Warren Buffett. 702 00:32:52,310 --> 00:32:55,430 Warren Buffett is all about prediction. 703 00:32:55,430 --> 00:32:58,670 He's not about creating a good portfolio that 704 00:32:58,670 --> 00:33:01,130 will be worth something reasonable over a period 705 00:33:01,130 --> 00:33:01,670 of time. 706 00:33:01,670 --> 00:33:03,920 What he wants to do is he wants to beat the market. 707 00:33:03,920 --> 00:33:06,429 He wants to find undervalued stocks, invest in them, 708 00:33:06,429 --> 00:33:08,720 and then sell them when they actually reach equilibrium 709 00:33:08,720 --> 00:33:10,010 or become overvalued. 710 00:33:10,010 --> 00:33:10,760 Right? 711 00:33:10,760 --> 00:33:12,966 That's a very different approach and perspective 712 00:33:12,966 --> 00:33:14,090 than what we're doing here. 713 00:33:14,090 --> 00:33:17,534 So that's a good question, and it highlights that distinction. 714 00:33:17,534 --> 00:33:18,502 Yeah? 715 00:33:18,502 --> 00:33:25,239 AUDIENCE: [INAUDIBLE] 716 00:33:25,239 --> 00:33:26,280 ANDREW LO: Yes, there is. 717 00:33:26,280 --> 00:33:29,880 And there have been many studies that have been done on it. 718 00:33:29,880 --> 00:33:32,720 And that is beyond the scope of this course. 719 00:33:32,720 --> 00:33:35,520 So I'm going to refer you, again, to 15433. 720 00:33:35,520 --> 00:33:37,292 But I will tell you, at the end of this, 721 00:33:37,292 --> 00:33:39,750 once I get through this and make sure everybody is with me, 722 00:33:39,750 --> 00:33:42,600 I will tell you how this works in practice. 723 00:33:42,600 --> 00:33:43,440 OK? 724 00:33:43,440 --> 00:33:45,150 So I'll talk a bit about applications. 725 00:33:45,150 --> 00:33:46,900 But first I want to get through the theory 726 00:33:46,900 --> 00:33:48,480 to make sure we all understand it. 727 00:33:48,480 --> 00:33:49,500 All right. 728 00:33:49,500 --> 00:33:52,620 So we now have this bullet. 729 00:33:52,620 --> 00:33:58,830 And the bullet obviously depends on all the parameters. 730 00:33:58,830 --> 00:34:02,220 But in particular, it depends upon the correlation. 731 00:34:02,220 --> 00:34:05,630 The reason that there is a bullet shape 732 00:34:05,630 --> 00:34:08,840 is because there is a correlation between these two 733 00:34:08,840 --> 00:34:09,719 stocks. 734 00:34:09,719 --> 00:34:12,620 And by the way, this correlation and this bullet shape 735 00:34:12,620 --> 00:34:13,940 is really important. 736 00:34:13,940 --> 00:34:14,960 OK? 737 00:34:14,960 --> 00:34:18,290 I'll tell you why it's really important. 738 00:34:18,290 --> 00:34:21,860 Take a look at a vertical slice, a vertical line 739 00:34:21,860 --> 00:34:24,110 going through the GM dot. 740 00:34:27,320 --> 00:34:33,110 That vertical line is the riskiness of GM. 741 00:34:33,110 --> 00:34:33,610 Right? 742 00:34:33,610 --> 00:34:36,730 It's the standard deviation of GM. 743 00:34:36,730 --> 00:34:41,650 One of the things that you'll notice about this graph 744 00:34:41,650 --> 00:34:45,130 is that if you only had two stocks available to you, GM 745 00:34:45,130 --> 00:34:51,100 and Motorola, the least amount of risk that you could possibly 746 00:34:51,100 --> 00:34:56,219 create for yourself is if you put 100% in General Motors. 747 00:34:56,219 --> 00:34:59,540 In other words, if all you cared about was going west, 748 00:34:59,540 --> 00:35:02,190 the most extreme west you can go other 749 00:35:02,190 --> 00:35:03,810 than putting your money in T-bills 750 00:35:03,810 --> 00:35:08,630 over here, the most extreme west you can go is General Motors. 751 00:35:08,630 --> 00:35:10,870 You can't get any less risky than that. 752 00:35:10,870 --> 00:35:11,390 Yeah? 753 00:35:11,390 --> 00:35:12,700 Question? 754 00:35:12,700 --> 00:35:23,600 AUDIENCE: [INAUDIBLE] 755 00:35:23,600 --> 00:35:29,870 ANDREW LO: What I'm saying is, if you 756 00:35:29,870 --> 00:35:32,690 are trying to get less risky as possible, 757 00:35:32,690 --> 00:35:35,990 and you only had General Motors or Motorola, 758 00:35:35,990 --> 00:35:39,300 then the most west you can go is General Motors. 759 00:35:39,300 --> 00:35:39,800 Right? 760 00:35:39,800 --> 00:35:41,200 That's all. 761 00:35:41,200 --> 00:35:46,280 However, however, this is the important point. 762 00:35:46,280 --> 00:35:49,510 If now I let you take weighted averages of the two, 763 00:35:49,510 --> 00:35:53,050 if I give you the right to form a portfolio, 764 00:35:53,050 --> 00:36:00,370 then you can get a dot, which is this dot right here. 765 00:36:00,370 --> 00:36:04,710 That dot, everybody in this room should prefer 766 00:36:04,710 --> 00:36:06,660 that dot to General Motors. 767 00:36:06,660 --> 00:36:07,740 Why? 768 00:36:07,740 --> 00:36:12,110 Because it's less risk than General Motors, 769 00:36:12,110 --> 00:36:14,767 but it's also higher return. 770 00:36:14,767 --> 00:36:16,850 There's no downside, at least from the perspective 771 00:36:16,850 --> 00:36:18,141 of mean and standard deviation. 772 00:36:18,141 --> 00:36:20,690 There may be other reasons you don't like that dot. 773 00:36:20,690 --> 00:36:23,330 But if all you care about is mean and standard deviation, 774 00:36:23,330 --> 00:36:27,320 that dot is strictly preferred. 775 00:36:27,320 --> 00:36:29,150 So I just made all of you better off 776 00:36:29,150 --> 00:36:31,250 with this piece of knowledge. 777 00:36:31,250 --> 00:36:33,440 Just by telling you how to weight these two 778 00:36:33,440 --> 00:36:36,560 stocks and the fact that they've got some kind of correlation, 779 00:36:36,560 --> 00:36:40,670 I've given you a mechanism of reducing your risk and, 780 00:36:40,670 --> 00:36:44,790 and increasing your expected rate of return, both. 781 00:36:44,790 --> 00:36:45,290 OK? 782 00:36:45,290 --> 00:36:46,756 So if you were really risk averse, 783 00:36:46,756 --> 00:36:48,380 if you said to yourself, you know what? 784 00:36:48,380 --> 00:36:50,240 I don't want all this fancy stock picking. 785 00:36:50,240 --> 00:36:54,420 Just give me the least risky stock of the two. 786 00:36:54,420 --> 00:36:56,487 Then you'd be at General Motors. 787 00:36:56,487 --> 00:36:58,570 And then if I came to you and said, you know what? 788 00:36:58,570 --> 00:36:59,861 I can do even better than that. 789 00:36:59,861 --> 00:37:01,800 I can get even less risk than General Motors. 790 00:37:01,800 --> 00:37:05,730 And at the same time, I'm going to give you higher return. 791 00:37:05,730 --> 00:37:10,770 So right there, the value of portfolio theory 792 00:37:10,770 --> 00:37:12,460 is pretty clear. 793 00:37:12,460 --> 00:37:14,908 It gives you options you did not have. 794 00:37:14,908 --> 00:37:15,864 Yeah? 795 00:37:15,864 --> 00:37:29,064 AUDIENCE: [INAUDIBLE] 796 00:37:29,064 --> 00:37:29,730 ANDREW LO: Yeah. 797 00:37:29,730 --> 00:37:36,010 AUDIENCE: [INAUDIBLE] 798 00:37:36,010 --> 00:37:37,010 ANDREW LO: That's right. 799 00:37:37,010 --> 00:37:37,551 That's right. 800 00:37:37,551 --> 00:37:40,034 When you have correlations that are unstable over time 801 00:37:40,034 --> 00:37:42,450 and you didn't account for them, you can get into trouble. 802 00:37:42,450 --> 00:37:43,210 OK? 803 00:37:43,210 --> 00:37:46,151 Again, let's come back to that after I go through all of this. 804 00:37:46,151 --> 00:37:46,650 All right? 805 00:37:46,650 --> 00:37:49,200 So, just if this is a clarifying question, I'll answer. 806 00:37:49,200 --> 00:37:53,130 But extensions, let's wait until we actually go through this. 807 00:37:53,130 --> 00:37:53,880 OK. 808 00:37:53,880 --> 00:37:59,190 So let me talk about the General Motors and Motorola example, 809 00:37:59,190 --> 00:38:02,640 but now where I am going to tell you that correlations 810 00:38:02,640 --> 00:38:04,360 are changing over time. 811 00:38:04,360 --> 00:38:06,709 So let's do the example that Lewis wants us to do. 812 00:38:06,709 --> 00:38:08,250 Let's actually assume the correlation 813 00:38:08,250 --> 00:38:11,200 is, I don't know, 0 or 1 or minus 1. 814 00:38:11,200 --> 00:38:13,570 Let's go through all three cases and see what happens. 815 00:38:13,570 --> 00:38:17,350 You'll see some remarkable things coming out of this. 816 00:38:17,350 --> 00:38:22,690 If the correlation is 0, then it's pretty easy for you 817 00:38:22,690 --> 00:38:26,080 to just plug in 0 for that correlation 818 00:38:26,080 --> 00:38:27,580 in that second equation. 819 00:38:27,580 --> 00:38:29,740 And all you get are the first two terms 820 00:38:29,740 --> 00:38:31,750 of that variance expression. 821 00:38:31,750 --> 00:38:32,440 Right? 822 00:38:32,440 --> 00:38:34,956 So when you assume that the-- 823 00:38:34,956 --> 00:38:36,580 when you assume that the correlation is 824 00:38:36,580 --> 00:38:39,940 0, when this thing is 0, what happens 825 00:38:39,940 --> 00:38:42,430 is that this entire term goes away 826 00:38:42,430 --> 00:38:45,220 and all you have are the first two terms. 827 00:38:45,220 --> 00:38:46,960 That's this column right over here. 828 00:38:46,960 --> 00:38:47,620 OK? 829 00:38:47,620 --> 00:38:49,900 So I've just produced the previous table 830 00:38:49,900 --> 00:38:54,600 but with the assumption of 0 correlation, and I get that. 831 00:38:54,600 --> 00:38:56,350 Now I'm not going to graph it for you yet. 832 00:38:56,350 --> 00:38:58,330 I want to just show you what happens when you 833 00:38:58,330 --> 00:39:00,320 choose different assumptions. 834 00:39:00,320 --> 00:39:01,750 So that's one assumption. 835 00:39:01,750 --> 00:39:04,000 On the other hand, if I choose a different assumption, 836 00:39:04,000 --> 00:39:06,730 if I choose the assumption that the correlation is, 837 00:39:06,730 --> 00:39:08,440 let's say 1-- 838 00:39:08,440 --> 00:39:11,020 so in other words, they are perfectly correlated-- 839 00:39:11,020 --> 00:39:13,945 then I'm going to get, I'm going to get this. 840 00:39:18,550 --> 00:39:22,600 I'm going to get this becoming 1, 841 00:39:22,600 --> 00:39:24,760 and then I'll get another simplification 842 00:39:24,760 --> 00:39:26,650 of this expression. 843 00:39:26,650 --> 00:39:30,130 And similarly, if I assume that the correlation is minus 1, 844 00:39:30,130 --> 00:39:32,680 then I'm going to get yet another simplification 845 00:39:32,680 --> 00:39:33,520 of this expression. 846 00:39:33,520 --> 00:39:35,930 It's an algebra that I'll show you in a minute. 847 00:39:35,930 --> 00:39:38,410 But this is what I'm doing in these three columns. 848 00:39:38,410 --> 00:39:41,260 I'm assuming different values of correlation between General 849 00:39:41,260 --> 00:39:44,410 Motors and Motorola, and then computing 850 00:39:44,410 --> 00:39:47,390 the standard deviation of the portfolio using this formula. 851 00:39:47,390 --> 00:39:47,890 OK? 852 00:39:47,890 --> 00:39:51,130 So you can all do this at home, do this in a spreadsheet. 853 00:39:51,130 --> 00:39:55,650 It's going to be very easy to check my work. 854 00:39:55,650 --> 00:39:58,570 But I want to show you the graph because that's really, 855 00:39:58,570 --> 00:40:02,530 I think, the insightful intuition here. 856 00:40:02,530 --> 00:40:04,250 Look at the graphs. 857 00:40:04,250 --> 00:40:08,410 These are the graphs of the three different cases 858 00:40:08,410 --> 00:40:13,210 of zero correlation, perfect correlation, 859 00:40:13,210 --> 00:40:15,820 and perfect negative correlation. 860 00:40:15,820 --> 00:40:17,830 I want to go to each one of these with you. 861 00:40:17,830 --> 00:40:20,710 So the case of perfect positive correlation, 862 00:40:20,710 --> 00:40:25,000 it turns out that the risk/reward trade off is 863 00:40:25,000 --> 00:40:27,410 actually just a straight line. 864 00:40:27,410 --> 00:40:29,230 It gets really, really simple. 865 00:40:29,230 --> 00:40:31,350 It's just a straight line. 866 00:40:31,350 --> 00:40:35,089 So in particular, there is no non-linearity. 867 00:40:35,089 --> 00:40:37,380 Because if they're perfectly correlated, you know what? 868 00:40:37,380 --> 00:40:40,182 You basically have virtually the same stock. 869 00:40:40,182 --> 00:40:41,640 The only difference is that they're 870 00:40:41,640 --> 00:40:43,180 different scales of each other. 871 00:40:43,180 --> 00:40:43,950 Right? 872 00:40:43,950 --> 00:40:45,706 But if they're perfectly correlated, 873 00:40:45,706 --> 00:40:48,330 that means that there's a linear relationship between those two 874 00:40:48,330 --> 00:40:49,320 stocks. 875 00:40:49,320 --> 00:40:52,820 And so when you do this mean variance analysis, 876 00:40:52,820 --> 00:40:54,900 in mean variance space, you basically 877 00:40:54,900 --> 00:40:58,790 get no nonlinear-- you don't get that little bump, 878 00:40:58,790 --> 00:41:01,470 the little bullet that we saw before. 879 00:41:01,470 --> 00:41:01,970 Right? 880 00:41:01,970 --> 00:41:05,580 So there's no way you can get less risk than General Motors 881 00:41:05,580 --> 00:41:08,580 unless you end up shorting Motorola and putting it 882 00:41:08,580 --> 00:41:09,330 in General Motors. 883 00:41:09,330 --> 00:41:13,350 And then your expected return goes down, 884 00:41:13,350 --> 00:41:15,941 as well as your risk. 885 00:41:15,941 --> 00:41:16,440 All right. 886 00:41:16,440 --> 00:41:22,040 Now let's take the case where you've got no correlation. 887 00:41:22,040 --> 00:41:24,590 It turns out that if you have no correlation, 888 00:41:24,590 --> 00:41:26,870 you get the bullet. 889 00:41:26,870 --> 00:41:31,430 But the bullet is even more wedge shaped. 890 00:41:31,430 --> 00:41:34,970 What that means is that you can reduce even more risk 891 00:41:34,970 --> 00:41:36,220 without getting rid of return. 892 00:41:36,220 --> 00:41:39,080 Remember the red dot that we saw in the previous slide? 893 00:41:39,080 --> 00:41:40,190 This red dot? 894 00:41:40,190 --> 00:41:43,710 Well, in the case where you've got 0 correlation-- 895 00:41:43,710 --> 00:41:46,980 this is the case where the correlation is 0.37, 37%. 896 00:41:46,980 --> 00:41:47,480 Right? 897 00:41:47,480 --> 00:41:48,980 That's what the data tells us. 898 00:41:48,980 --> 00:41:52,000 But if you assume that the correlation was zero, 899 00:41:52,000 --> 00:41:56,900 you would get even more of a savings of volatility 900 00:41:56,900 --> 00:41:59,100 for a given level of expected return. 901 00:41:59,100 --> 00:41:59,600 OK? 902 00:41:59,600 --> 00:42:01,310 So it would look like this. 903 00:42:01,310 --> 00:42:02,840 You see how this bullet sticks out 904 00:42:02,840 --> 00:42:05,210 a lot more than the previous red dot that 905 00:42:05,210 --> 00:42:07,070 was somewhere over here? 906 00:42:07,070 --> 00:42:09,220 OK? 907 00:42:09,220 --> 00:42:11,170 Now let's continue. 908 00:42:11,170 --> 00:42:14,560 Suppose the correlation is minus 50%. 909 00:42:14,560 --> 00:42:19,600 Then you get an even wider bullet, 910 00:42:19,600 --> 00:42:23,110 that sticks out even more, that saves you 911 00:42:23,110 --> 00:42:24,310 more standard deviation. 912 00:42:24,310 --> 00:42:26,690 You're getting to the west even more. 913 00:42:26,690 --> 00:42:30,110 And finally, and here is where you get a really remarkable 914 00:42:30,110 --> 00:42:36,700 result. If the correlation is minus 100%, 915 00:42:36,700 --> 00:42:38,310 you know what you get? 916 00:42:38,310 --> 00:42:41,510 You get a piecewise linear trade off. 917 00:42:41,510 --> 00:42:46,060 You get this and that. 918 00:42:46,060 --> 00:42:48,100 It doesn't turn into a wedge. 919 00:42:48,100 --> 00:42:52,630 It turns into a triangle that actually hits the x-axis. 920 00:42:52,630 --> 00:42:54,850 And the reason this is such a startling result 921 00:42:54,850 --> 00:42:58,030 is it tells us that there exists a way 922 00:42:58,030 --> 00:43:02,710 to construct a portfolio that gives us a rate of return 923 00:43:02,710 --> 00:43:09,410 of like 1.39% with no risk. 924 00:43:09,410 --> 00:43:12,920 Zero risk. 925 00:43:12,920 --> 00:43:14,390 1.39%. 926 00:43:14,390 --> 00:43:18,200 Let's just call it 1.3% to be conservative. 927 00:43:18,200 --> 00:43:21,530 Multiply that by 12 and you're going 928 00:43:21,530 --> 00:43:23,600 to get something like what? 929 00:43:23,600 --> 00:43:26,980 16% a year? 930 00:43:26,980 --> 00:43:29,950 You tell me if you know of any investment opportunities that 931 00:43:29,950 --> 00:43:33,490 gives you a return of 16% a year with no risk, 932 00:43:33,490 --> 00:43:36,070 and I'll examine that for you carefully. 933 00:43:36,070 --> 00:43:38,290 OK? 934 00:43:38,290 --> 00:43:40,480 It doesn't exist, of course. 935 00:43:40,480 --> 00:43:42,250 The reason it doesn't exist is because you 936 00:43:42,250 --> 00:43:48,860 can't find two assets that have perfect negative correlation. 937 00:43:48,860 --> 00:43:51,890 If you could, there are wondrous things 938 00:43:51,890 --> 00:43:55,310 you could achieve with that combination. 939 00:43:55,310 --> 00:43:58,400 And portfolio theory basically tells us how, right? 940 00:43:58,400 --> 00:44:03,930 This is a recipe book for how to exploit correlation. 941 00:44:03,930 --> 00:44:04,430 All right? 942 00:44:04,430 --> 00:44:07,040 Now again, I can't tell you where 943 00:44:07,040 --> 00:44:09,110 you should be on this curve other 944 00:44:09,110 --> 00:44:13,030 than if it's really minus 1, then I would be here. 945 00:44:13,030 --> 00:44:13,790 OK? 946 00:44:13,790 --> 00:44:16,020 Lots of return, no risk. 947 00:44:16,020 --> 00:44:17,990 That's an arbitrage. 948 00:44:17,990 --> 00:44:19,780 We know that that can't possibly happen. 949 00:44:19,780 --> 00:44:20,696 There's no free lunch. 950 00:44:20,696 --> 00:44:24,650 There's no arbitrage, on average, over circumstances. 951 00:44:24,650 --> 00:44:28,220 However, if the bullet is minus 0.5, 952 00:44:28,220 --> 00:44:30,890 then you've got lots of opportunity 953 00:44:30,890 --> 00:44:35,240 to create really attractive portfolios that doesn't require 954 00:44:35,240 --> 00:44:36,890 Warren Buffett's skills. 955 00:44:36,890 --> 00:44:38,431 There's no forecasting here. 956 00:44:38,431 --> 00:44:38,930 Right? 957 00:44:38,930 --> 00:44:40,264 We're not trying to pick stocks. 958 00:44:40,264 --> 00:44:42,180 We're not trying to see how the market's going 959 00:44:42,180 --> 00:44:43,070 to do next month. 960 00:44:43,070 --> 00:44:44,540 Who knows? 961 00:44:44,540 --> 00:44:47,480 All we're assuming is that means and variances 962 00:44:47,480 --> 00:44:50,990 are stable over time, and the correlation 963 00:44:50,990 --> 00:44:51,980 is stable over time. 964 00:44:51,980 --> 00:44:55,070 Those are nontrivial assumptions, I grant you. 965 00:44:55,070 --> 00:44:56,750 But if you believe in those assumptions 966 00:44:56,750 --> 00:44:58,700 more than you believe in your ability 967 00:44:58,700 --> 00:45:01,100 to forecast what's going to happen to General Motors 968 00:45:01,100 --> 00:45:03,710 six months from now, then this might be a good way 969 00:45:03,710 --> 00:45:05,330 to construct a portfolio. 970 00:45:05,330 --> 00:45:07,970 But I haven't told you where on this curve you ought to be. 971 00:45:07,970 --> 00:45:10,620 I just told you how to construct that curve. 972 00:45:10,620 --> 00:45:11,120 All right? 973 00:45:11,120 --> 00:45:13,310 It's your job to look at the curve and say, 974 00:45:13,310 --> 00:45:16,080 I like this point, or I like that point, 975 00:45:16,080 --> 00:45:18,230 based upon your own personal preferences 976 00:45:18,230 --> 00:45:19,640 for risk and reward. 977 00:45:19,640 --> 00:45:23,180 So we're not there yet where I can tell you how to behave. 978 00:45:23,180 --> 00:45:25,750 I will get there in about a lecture and a half. 979 00:45:25,750 --> 00:45:27,890 But we've got to build up the infrastructure 980 00:45:27,890 --> 00:45:29,420 to be able to get us there. 981 00:45:29,420 --> 00:45:30,170 OK. 982 00:45:30,170 --> 00:45:32,180 So what this tells us is that we need 983 00:45:32,180 --> 00:45:35,420 to know what the correlation is in order to figure out 984 00:45:35,420 --> 00:45:38,600 where we are going to be on these different curves, which 985 00:45:38,600 --> 00:45:40,760 curve is going to apply. 986 00:45:40,760 --> 00:45:43,220 When you have lots of correlation, 987 00:45:43,220 --> 00:45:46,670 a correlation of 1, there really isn't much of a risk 988 00:45:46,670 --> 00:45:49,280 savings per unit return. 989 00:45:49,280 --> 00:45:53,960 We can't get a lower risk for a given level of return. 990 00:45:53,960 --> 00:45:59,660 But we can if there is less correlation than perfect. 991 00:45:59,660 --> 00:46:03,680 And these are the different curves that illustrate that. 992 00:46:03,680 --> 00:46:05,180 OK. 993 00:46:05,180 --> 00:46:07,940 So there are some other examples that I'd like you 994 00:46:07,940 --> 00:46:09,450 to work through on your own. 995 00:46:09,450 --> 00:46:11,390 This is another portfolio calculation. 996 00:46:11,390 --> 00:46:14,690 Just go through the same calculations that we did here. 997 00:46:14,690 --> 00:46:17,510 And you know, you'll graph the different risk/reward trade off 998 00:46:17,510 --> 00:46:20,120 between these two, General Dynamics and Motorola. 999 00:46:20,120 --> 00:46:22,580 And you can get exactly the same analysis 1000 00:46:22,580 --> 00:46:26,450 with those two stocks, General Dynamics and Motorola. 1001 00:46:26,450 --> 00:46:29,200 OK? 1002 00:46:29,200 --> 00:46:33,080 Now what about if you got a risk free rate? 1003 00:46:33,080 --> 00:46:36,760 So suppose that the two assets that I want to look at 1004 00:46:36,760 --> 00:46:40,750 is not General Motors and Motorola, but rather the stock 1005 00:46:40,750 --> 00:46:44,700 market and treasury bills. 1006 00:46:44,700 --> 00:46:47,710 Then what does your risk/reward trade off look like? 1007 00:46:47,710 --> 00:46:51,180 Well, it turns out that in the case where treasury bills are 1008 00:46:51,180 --> 00:46:57,070 in question, the volatility of treasury bills 1009 00:46:57,070 --> 00:46:59,260 is virtually zero. 1010 00:46:59,260 --> 00:47:01,420 It's not exactly zero because there 1011 00:47:01,420 --> 00:47:04,450 may be some kind of randomness in the underlying rates 1012 00:47:04,450 --> 00:47:07,000 of return because of inflationary expectations. 1013 00:47:07,000 --> 00:47:11,050 But as an approximation, if it's a risk free rate 1014 00:47:11,050 --> 00:47:13,780 and you know that you're going to get that risk free rate, 1015 00:47:13,780 --> 00:47:16,450 then the volatility of that return 1016 00:47:16,450 --> 00:47:19,940 is in fact 0 over that period of time. 1017 00:47:19,940 --> 00:47:23,860 So in that case, the expected rate of return of a portfolio 1018 00:47:23,860 --> 00:47:25,990 between the stock market and T-bills, 1019 00:47:25,990 --> 00:47:27,520 that's the weighted average. 1020 00:47:27,520 --> 00:47:31,300 But the variance is going to be very simple because it's 1021 00:47:31,300 --> 00:47:34,150 going to be the variance of 1, which is 0, 1022 00:47:34,150 --> 00:47:36,850 plus the variance of the other, plus 2 1023 00:47:36,850 --> 00:47:41,260 times the weighted average times the covariance. 1024 00:47:41,260 --> 00:47:42,560 But there is no covariance. 1025 00:47:42,560 --> 00:47:43,060 Right? 1026 00:47:43,060 --> 00:47:45,250 There's no correlation because one of the things 1027 00:47:45,250 --> 00:47:47,290 is non-random. 1028 00:47:47,290 --> 00:47:51,670 And so when you work out the weights for the two 1029 00:47:51,670 --> 00:47:55,846 and you graph them, you get this. 1030 00:47:55,846 --> 00:48:00,860 This is a beautiful thing, nice and simple, no weird curves 1031 00:48:00,860 --> 00:48:03,320 or any kind of bullet shape. 1032 00:48:03,320 --> 00:48:05,370 You've got T-bills here. 1033 00:48:05,370 --> 00:48:07,460 You've got the stock market here. 1034 00:48:07,460 --> 00:48:10,640 And the weights, as you vary them, 1035 00:48:10,640 --> 00:48:15,740 will bring you anywhere along this line or possibly 1036 00:48:15,740 --> 00:48:17,090 up over here. 1037 00:48:17,090 --> 00:48:19,820 If you're in the middle of this line-- so literally, 1038 00:48:19,820 --> 00:48:24,770 if you have the same distance between here and here and here 1039 00:48:24,770 --> 00:48:25,760 and here-- 1040 00:48:25,760 --> 00:48:28,400 that actually gives you a 50-50 weighting 1041 00:48:28,400 --> 00:48:30,980 on those portfolio weights. 1042 00:48:30,980 --> 00:48:34,370 So geometrically, this actually corresponds 1043 00:48:34,370 --> 00:48:40,070 to a 50-50 weighting of T-bills and the S&P. OK? 1044 00:48:40,070 --> 00:48:43,940 Now question, what happens when you are over here? 1045 00:48:43,940 --> 00:48:45,710 Let's suppose you're at this point. 1046 00:48:45,710 --> 00:48:48,350 At this point, what would your portfolio weights look like? 1047 00:48:48,350 --> 00:48:51,490 How would you characterize that? 1048 00:48:51,490 --> 00:48:52,170 Yeah, Brian? 1049 00:48:52,170 --> 00:48:56,190 AUDIENCE: Short T-bills to buy into the stock market? 1050 00:48:56,190 --> 00:48:57,190 ANDREW LO: That's right. 1051 00:48:57,190 --> 00:48:57,970 Short T-bills. 1052 00:48:57,970 --> 00:49:01,510 What does it mean to short T-bills? 1053 00:49:01,510 --> 00:49:03,380 What are you doing? 1054 00:49:03,380 --> 00:49:04,482 You're borrowing. 1055 00:49:04,482 --> 00:49:05,440 You're borrowing money. 1056 00:49:05,440 --> 00:49:06,700 You're leveraging. 1057 00:49:06,700 --> 00:49:08,530 When you're shorting T-bills, you're 1058 00:49:08,530 --> 00:49:10,860 basically borrowing and getting cash upfront 1059 00:49:10,860 --> 00:49:13,670 you're to pay back later with interest. 1060 00:49:13,670 --> 00:49:16,390 So shorting T-bills is just borrowing. 1061 00:49:16,390 --> 00:49:18,880 If you're borrowing money and you're putting it 1062 00:49:18,880 --> 00:49:21,580 into the stock market in addition to 100% of your own 1063 00:49:21,580 --> 00:49:23,990 wealth-- you've borrowed additional money to put it 1064 00:49:23,990 --> 00:49:24,911 in the stock market-- 1065 00:49:24,911 --> 00:49:26,410 then you're going to be way up here. 1066 00:49:26,410 --> 00:49:31,360 Higher return, much higher up here than down here, 1067 00:49:31,360 --> 00:49:33,710 but you're going to get higher risk, as well. 1068 00:49:33,710 --> 00:49:38,270 So leverage, this idea of borrowing and putting 1069 00:49:38,270 --> 00:49:41,450 your money in the stock market, that increases 1070 00:49:41,450 --> 00:49:43,850 your expected rate of return. 1071 00:49:43,850 --> 00:49:45,980 But it also increases your risk. 1072 00:49:45,980 --> 00:49:47,210 OK? 1073 00:49:47,210 --> 00:49:49,575 Leverage increases your risk. 1074 00:49:49,575 --> 00:49:51,950 And now getting back to the question that Lewis asked us, 1075 00:49:51,950 --> 00:49:53,600 is this where we got into trouble 1076 00:49:53,600 --> 00:49:54,680 with the current crisis? 1077 00:49:54,680 --> 00:49:56,850 Yes, in a nutshell it is. 1078 00:49:56,850 --> 00:49:58,850 But it's more complicated because the underlying 1079 00:49:58,850 --> 00:50:00,530 securities are more complex. 1080 00:50:00,530 --> 00:50:04,160 But the basic idea is if you leverage up, 1081 00:50:04,160 --> 00:50:06,590 if you leverage up, way up here-- 1082 00:50:06,590 --> 00:50:08,570 you're up maybe out there-- 1083 00:50:08,570 --> 00:50:11,756 and all of a sudden there's a bump in the road 1084 00:50:11,756 --> 00:50:13,880 and what you are leveraging, this thing that you're 1085 00:50:13,880 --> 00:50:19,550 investing in is not nearly as smooth and as riskless 1086 00:50:19,550 --> 00:50:23,760 as you thought it was, it could wipe you out. 1087 00:50:23,760 --> 00:50:29,100 And one of those elements that could cause such a wipeout 1088 00:50:29,100 --> 00:50:32,550 is if you somehow forgot about the fact 1089 00:50:32,550 --> 00:50:35,350 that correlations can change. 1090 00:50:35,350 --> 00:50:39,810 So you thought that you were, I don't know, somewhere here, 1091 00:50:39,810 --> 00:50:43,380 and all of a sudden correlations go to 1 1092 00:50:43,380 --> 00:50:45,700 and now you're actually over here. 1093 00:50:45,700 --> 00:50:47,940 You see how risk can change really quickly? 1094 00:50:47,940 --> 00:50:51,790 Correlations don't have to be stable over time. 1095 00:50:51,790 --> 00:50:55,099 And that's the lesson that most people in industry, 1096 00:50:55,099 --> 00:50:57,390 who don't have a finance background, who've never taken 1097 00:50:57,390 --> 00:50:59,880 this course, they won't know. 1098 00:50:59,880 --> 00:51:01,590 These are physicists or mathematicians 1099 00:51:01,590 --> 00:51:02,370 or computer scientists. 1100 00:51:02,370 --> 00:51:03,490 They estimate the correlation. 1101 00:51:03,490 --> 00:51:05,580 It's a parameter, like the gravitational constant 1102 00:51:05,580 --> 00:51:06,690 or Avogadro's number. 1103 00:51:06,690 --> 00:51:08,350 Let's plug it in. 1104 00:51:08,350 --> 00:51:12,630 Hey, 9.08 times 10 the 23rd, that's what it should be. 1105 00:51:12,630 --> 00:51:16,650 And nobody ever told them that it could change. 1106 00:51:16,650 --> 00:51:20,820 And when it changes, bad things can happen really quickly. 1107 00:51:20,820 --> 00:51:22,320 So we're going to come back to that. 1108 00:51:22,320 --> 00:51:24,347 But let's get the standard theory down first, 1109 00:51:24,347 --> 00:51:26,180 and then we'll talk a bit about application. 1110 00:51:26,180 --> 00:51:26,679 Anon? 1111 00:51:26,679 --> 00:51:32,370 AUDIENCE: [INAUDIBLE] 1112 00:51:32,370 --> 00:51:33,158 ANDREW LO: Yeah. 1113 00:51:33,158 --> 00:51:40,544 AUDIENCE: [INAUDIBLE] 1114 00:51:40,544 --> 00:51:41,210 ANDREW LO: Yeah. 1115 00:51:41,210 --> 00:51:47,760 AUDIENCE: [INAUDIBLE] 1116 00:51:47,760 --> 00:51:48,930 ANDREW LO: Right here? 1117 00:51:48,930 --> 00:51:49,950 Of course. 1118 00:51:49,950 --> 00:51:52,440 Because you might want to get more return and you're 1119 00:51:52,440 --> 00:51:54,030 willing to take on risk. 1120 00:51:54,030 --> 00:51:55,170 Let's take a look at this. 1121 00:51:55,170 --> 00:51:57,690 This bullet point here says that you're 1122 00:51:57,690 --> 00:52:00,190 going to be at approximately, something-- 1123 00:52:00,190 --> 00:52:02,370 if the correlation-- let's be realistic about it. 1124 00:52:02,370 --> 00:52:02,520 OK? 1125 00:52:02,520 --> 00:52:04,440 The correlation is not going to be minus 0.5. 1126 00:52:04,440 --> 00:52:06,690 It's going to be more like 0.37. 1127 00:52:06,690 --> 00:52:09,530 So this bullet here is going to be like 1-- 1128 00:52:09,530 --> 00:52:12,570 I don't know, let's call it 1.3% just to make it easy. 1129 00:52:12,570 --> 00:52:14,760 And you've got a standard deviation of about 6%. 1130 00:52:14,760 --> 00:52:15,390 OK? 1131 00:52:15,390 --> 00:52:17,010 So on an annualized basis, that's 1132 00:52:17,010 --> 00:52:22,740 giving you a return, 1.3 times 12 is something like what, 16%? 1133 00:52:22,740 --> 00:52:27,600 16% return, but the risk on an annualized basis multiplied 1134 00:52:27,600 --> 00:52:29,790 by 3 and 1/2 is about, let's say 20%. 1135 00:52:29,790 --> 00:52:32,940 So 20%, 22% annual standard deviation 1136 00:52:32,940 --> 00:52:36,470 for a 16% rate of return. 1137 00:52:36,470 --> 00:52:38,290 Now some of you might like that. 1138 00:52:38,290 --> 00:52:41,680 But I have a few friends here for whom 1139 00:52:41,680 --> 00:52:44,546 that return is just boring. 1140 00:52:44,546 --> 00:52:46,420 That's just not going to get their attention. 1141 00:52:46,420 --> 00:52:48,520 What they want is they want to be up here, 1142 00:52:48,520 --> 00:52:50,680 you know, like at 2% rate of return. 1143 00:52:50,680 --> 00:52:54,350 2% rate of return per month is about 24% a year. 1144 00:52:54,350 --> 00:52:55,840 So if you're a hedge fund manager, 1145 00:52:55,840 --> 00:52:58,570 24% is when you start to begin to feel alive. 1146 00:52:58,570 --> 00:52:59,290 You know? 1147 00:52:59,290 --> 00:53:01,660 That's when things really start to happen for you. 1148 00:53:01,660 --> 00:53:05,080 And at 24% annual return, you can't 1149 00:53:05,080 --> 00:53:07,464 have volatility of like 15% or 20% 1150 00:53:07,464 --> 00:53:08,880 unless you're doing something, you 1151 00:53:08,880 --> 00:53:12,970 know, really different than the standard market portfolio. 1152 00:53:12,970 --> 00:53:15,760 So a hedge fund manager is going to say, Anon, give me a break. 1153 00:53:15,760 --> 00:53:17,551 You know, this is going to put me to sleep. 1154 00:53:17,551 --> 00:53:18,520 I want to be up here. 1155 00:53:18,520 --> 00:53:20,470 But you know, you've got a family, three kids, 1156 00:53:20,470 --> 00:53:23,290 you have to worry about making mortgage payments. 1157 00:53:23,290 --> 00:53:25,720 That kind of lifestyle is not for you. 1158 00:53:25,720 --> 00:53:28,630 So in both cases, though, in both cases, 1159 00:53:28,630 --> 00:53:33,290 we can agree that where we want to be is on this curve, right? 1160 00:53:33,290 --> 00:53:37,810 In other words, you would never want to be here, at this point. 1161 00:53:37,810 --> 00:53:38,500 Why? 1162 00:53:38,500 --> 00:53:41,140 Because if you were here, you can either, 1163 00:53:41,140 --> 00:53:43,830 for the same level of risk, increase your return. 1164 00:53:43,830 --> 00:53:47,100 Or for the same level of return, decrease your risk. 1165 00:53:47,100 --> 00:53:50,540 So what we can agree on, even though we 1166 00:53:50,540 --> 00:53:53,450 don't agree on where we want to be on that curve, 1167 00:53:53,450 --> 00:53:56,750 we can all agree we want to be on that curve as opposed 1168 00:53:56,750 --> 00:53:57,890 to inside the curve. 1169 00:53:57,890 --> 00:53:58,625 Right? 1170 00:53:58,625 --> 00:54:00,110 AUDIENCE: [INAUDIBLE] 1171 00:54:00,110 --> 00:54:00,920 ANDREW LO: Exactly. 1172 00:54:00,920 --> 00:54:01,679 Great point. 1173 00:54:01,679 --> 00:54:03,470 Why would we want to be on the bottom half? 1174 00:54:03,470 --> 00:54:04,610 Nobody would want to do that. 1175 00:54:04,610 --> 00:54:06,090 Because if were on the bottom half, 1176 00:54:06,090 --> 00:54:08,465 you can just as easily move up to the top, 1177 00:54:08,465 --> 00:54:10,340 and you have a higher expected rate of return 1178 00:54:10,340 --> 00:54:11,741 for the same level of risk. 1179 00:54:11,741 --> 00:54:12,740 So you're exactly right. 1180 00:54:12,740 --> 00:54:15,590 That bottom half of the curve, you can just throw it away. 1181 00:54:15,590 --> 00:54:16,452 All right? 1182 00:54:16,452 --> 00:54:18,410 The only people that are going to be down there 1183 00:54:18,410 --> 00:54:19,600 are knuckleheads. 1184 00:54:19,600 --> 00:54:20,600 All right? 1185 00:54:20,600 --> 00:54:22,520 So we don't want to be down there. 1186 00:54:22,520 --> 00:54:25,280 So in fact, the only part of the curve that really matters 1187 00:54:25,280 --> 00:54:28,400 is this point, going all the way up. 1188 00:54:28,400 --> 00:54:30,680 And we're going to give a name to that in a minute. 1189 00:54:30,680 --> 00:54:35,469 That's going to be called the efficient frontier because you 1190 00:54:35,469 --> 00:54:37,010 would never want to do anything else. 1191 00:54:37,010 --> 00:54:39,270 Actually, that's not quite true. 1192 00:54:39,270 --> 00:54:44,450 What everybody would like to do is they'd like to be up here. 1193 00:54:44,450 --> 00:54:47,740 Unfortunately, we can't get there. 1194 00:54:47,740 --> 00:54:51,970 We can't get there with just General Motors and Motorola. 1195 00:54:51,970 --> 00:54:53,860 In a few minutes, I'm going to show you 1196 00:54:53,860 --> 00:54:57,250 how you might be able to get there with other stocks. 1197 00:54:57,250 --> 00:54:59,590 If you introduce other assets, then you 1198 00:54:59,590 --> 00:55:00,820 might be able to get there. 1199 00:55:00,820 --> 00:55:03,190 But you can't get there right now. 1200 00:55:03,190 --> 00:55:04,360 OK. 1201 00:55:04,360 --> 00:55:09,280 So here we are with stocks and T-bills. 1202 00:55:09,280 --> 00:55:14,690 And we know that we'll get the linear combination here. 1203 00:55:14,690 --> 00:55:17,720 What I'd like to do next is to make this yet more complicated. 1204 00:55:17,720 --> 00:55:18,220 All right? 1205 00:55:18,220 --> 00:55:20,050 And before I do, let me just summarize 1206 00:55:20,050 --> 00:55:21,640 what we've done so far. 1207 00:55:21,640 --> 00:55:24,490 What we've done is to show that with two assets, 1208 00:55:24,490 --> 00:55:27,280 we can do all of this analytically, 1209 00:55:27,280 --> 00:55:31,330 and illustrate graphically, all of the intuition that holds 1210 00:55:31,330 --> 00:55:33,460 for the more general case. 1211 00:55:33,460 --> 00:55:35,910 With two assets, the expected rate of return 1212 00:55:35,910 --> 00:55:37,870 is just a simple weighted average. 1213 00:55:37,870 --> 00:55:40,870 But the variance is not just a simple weighted average. 1214 00:55:40,870 --> 00:55:43,480 It's more complicated and it depends, in particular, 1215 00:55:43,480 --> 00:55:44,830 on the correlation. 1216 00:55:44,830 --> 00:55:47,590 For different correlations, we get different shapes 1217 00:55:47,590 --> 00:55:50,260 of trade offs in mean-variance space. 1218 00:55:50,260 --> 00:55:53,320 And you have to understand what those trade offs are. 1219 00:55:53,320 --> 00:55:55,840 If it turns out that one of the two assets is T-bills, 1220 00:55:55,840 --> 00:55:58,990 for example, then the trade off is really straight line. 1221 00:55:58,990 --> 00:55:59,740 OK? 1222 00:55:59,740 --> 00:56:02,320 But if both assets are risky, then you 1223 00:56:02,320 --> 00:56:06,940 get the bullet shape until you either get minus 1 or 1 1224 00:56:06,940 --> 00:56:08,020 in terms of correlations. 1225 00:56:08,020 --> 00:56:11,350 And then you get straight lines of different stripes 1226 00:56:11,350 --> 00:56:12,950 in those two different cases. 1227 00:56:12,950 --> 00:56:13,450 OK. 1228 00:56:13,450 --> 00:56:15,670 Now we're ready to talk about the general case. 1229 00:56:15,670 --> 00:56:19,630 The general case works exactly the same as the two asset case. 1230 00:56:19,630 --> 00:56:21,460 You've got your means. 1231 00:56:21,460 --> 00:56:22,840 You've got your variances. 1232 00:56:22,840 --> 00:56:24,580 And you've got your covariances. 1233 00:56:24,580 --> 00:56:27,040 And you add up all of these different covariances 1234 00:56:27,040 --> 00:56:30,050 to get the total variance of the portfolio. 1235 00:56:30,050 --> 00:56:35,480 And if you consider a couple of simple cases, 1236 00:56:35,480 --> 00:56:38,520 like for example, an equal weighted portfolio-- 1237 00:56:38,520 --> 00:56:43,670 so you've got n assets and you put each of your-- 1238 00:56:43,670 --> 00:56:46,670 for each of your assets, you put 1 over n of your wealth 1239 00:56:46,670 --> 00:56:47,750 in them. 1240 00:56:47,750 --> 00:56:50,090 Then you can show that the variance 1241 00:56:50,090 --> 00:56:55,730 of your entire portfolio is equal to the average variance 1242 00:56:55,730 --> 00:57:01,940 plus n times n minus 1 times the average covariance. 1243 00:57:01,940 --> 00:57:05,390 When n gets large, then it turns out 1244 00:57:05,390 --> 00:57:09,110 that the average variance doesn't matter. 1245 00:57:09,110 --> 00:57:12,080 What is driving the risk of your portfolio 1246 00:57:12,080 --> 00:57:13,910 has nothing to do with the variance 1247 00:57:13,910 --> 00:57:15,560 of the individual components. 1248 00:57:15,560 --> 00:57:19,640 What it has to do with is what the average covariance is. 1249 00:57:19,640 --> 00:57:22,610 That's what's driving the risk of your portfolio. 1250 00:57:22,610 --> 00:57:24,380 Because you've got a lot more covariances 1251 00:57:24,380 --> 00:57:26,140 than you do variances. 1252 00:57:26,140 --> 00:57:27,110 OK? 1253 00:57:27,110 --> 00:57:28,760 So this is kind of a neat insight 1254 00:57:28,760 --> 00:57:31,220 because it says that it's really important how 1255 00:57:31,220 --> 00:57:33,110 things are related. 1256 00:57:33,110 --> 00:57:35,330 And as those relationships change, 1257 00:57:35,330 --> 00:57:38,585 the risk of your portfolio is going to change. 1258 00:57:38,585 --> 00:57:40,460 So getting back to Lewis's point about what's 1259 00:57:40,460 --> 00:57:42,590 going on in current markets and what 1260 00:57:42,590 --> 00:57:44,420 caught a lot of portfolio managers 1261 00:57:44,420 --> 00:57:47,540 by surprise with the subprime markets, 1262 00:57:47,540 --> 00:57:51,110 if the risk of your portfolio is approximately given 1263 00:57:51,110 --> 00:57:54,740 by the average covariance and you've been assuming 1264 00:57:54,740 --> 00:57:58,460 all along that you've got this big pool of mortgages, 1265 00:57:58,460 --> 00:58:01,730 and the mortgages are all uncorrelated, 1266 00:58:01,730 --> 00:58:05,600 you've essentially assumed that you've got virtually no risk. 1267 00:58:05,600 --> 00:58:08,480 Because the average covariance by assumption 1268 00:58:08,480 --> 00:58:13,180 and by historical analysis is close to zero. 1269 00:58:13,180 --> 00:58:17,020 But when the real state market goes down nationally, 1270 00:58:17,020 --> 00:58:20,410 then everybody starts to default. 1271 00:58:20,410 --> 00:58:24,410 And foreclosures become very highly correlated. 1272 00:58:24,410 --> 00:58:27,880 And so you can see how overnight, literally overnight, 1273 00:58:27,880 --> 00:58:29,980 your risks can shoot up. 1274 00:58:29,980 --> 00:58:32,530 And you're not prepared for that unless you 1275 00:58:32,530 --> 00:58:35,970 know that this is what's going on in your portfolio. 1276 00:58:35,970 --> 00:58:36,641 Yup? 1277 00:58:36,641 --> 00:58:54,037 AUDIENCE: [INAUDIBLE] 1278 00:58:54,037 --> 00:58:55,120 ANDREW LO: Great question. 1279 00:58:55,120 --> 00:58:58,670 Why should things change in terms of correlation? 1280 00:58:58,670 --> 00:58:59,740 Well, you know what? 1281 00:58:59,740 --> 00:59:00,940 I'll give you an example. 1282 00:59:00,940 --> 00:59:02,710 I'll give you a personal example? 1283 00:59:02,710 --> 00:59:05,680 Correlation is a function of human behavior, right? 1284 00:59:05,680 --> 00:59:09,790 I mean, prices are being formed by you and I, investors. 1285 00:59:09,790 --> 00:59:13,420 And so correlations simply means that all of us 1286 00:59:13,420 --> 00:59:16,030 end up doing the same thing around to same time. 1287 00:59:16,030 --> 00:59:18,270 Right? 1288 00:59:18,270 --> 00:59:21,850 I'm going to be heading to the airport later on this evening. 1289 00:59:21,850 --> 00:59:24,300 And I don't expect that there's going 1290 00:59:24,300 --> 00:59:27,300 to be much of a big deal getting on my flight. 1291 00:59:27,300 --> 00:59:29,970 I'll probably get to the airport about half an hour, 1292 00:59:29,970 --> 00:59:31,140 45 minutes ahead of time. 1293 00:59:31,140 --> 00:59:32,130 It's just a shuttle. 1294 00:59:32,130 --> 00:59:36,402 So I'm heading to Washington, so it's not a big deal. 1295 00:59:36,402 --> 00:59:38,360 What's going to happen in two weeks from today? 1296 00:59:41,320 --> 00:59:42,430 Anything? 1297 00:59:42,430 --> 00:59:45,500 Anything going on two weeks from today that you can think of? 1298 00:59:45,500 --> 00:59:46,580 Thanksgiving. 1299 00:59:46,580 --> 00:59:50,342 So if I went to the airport two weeks from today 1300 00:59:50,342 --> 00:59:52,550 and tried to get on that shuttle half an hour before, 1301 00:59:52,550 --> 00:59:55,460 you think I can get on the flight? 1302 00:59:55,460 --> 00:59:57,070 Why not? 1303 00:59:57,070 --> 00:59:59,770 Because everybody else is going to do that. 1304 00:59:59,770 --> 01:00:02,460 Well, why should everything be correlated on that day? 1305 01:00:02,460 --> 01:00:05,710 Isn't that a Wednesday like every other Wednesday? 1306 01:00:05,710 --> 01:00:06,740 Well, it's not. 1307 01:00:06,740 --> 01:00:09,400 It's because somehow we've all decided 1308 01:00:09,400 --> 01:00:11,830 that we're going to take off at the same time 1309 01:00:11,830 --> 01:00:14,560 on the same day in that year. 1310 01:00:14,560 --> 01:00:15,253 Right? 1311 01:00:15,253 --> 01:00:20,826 AUDIENCE: [INAUDIBLE] 1312 01:00:20,826 --> 01:00:23,200 ANDREW LO: Well, first of all, I don't think Thanksgiving 1313 01:00:23,200 --> 01:00:25,600 is that low a probability. 1314 01:00:25,600 --> 01:00:27,040 OK? 1315 01:00:27,040 --> 01:00:29,950 But to your point, is that the correlation changing? 1316 01:00:29,950 --> 01:00:31,150 Well, it is. 1317 01:00:31,150 --> 01:00:37,110 Correlation is a statistical measure of two objects. 1318 01:00:37,110 --> 01:00:40,020 And what we're trying to capture is when they move up or down 1319 01:00:40,020 --> 01:00:41,160 at the same time. 1320 01:00:41,160 --> 01:00:42,140 Right? 1321 01:00:42,140 --> 01:00:43,890 So what I'm trying to get at is that there 1322 01:00:43,890 --> 01:00:45,265 are certain periods of time where 1323 01:00:45,265 --> 01:00:49,750 human behavior, all of a sudden, becomes very highly correlated. 1324 01:00:49,750 --> 01:00:51,310 And there are many reasons for that. 1325 01:00:51,310 --> 01:00:53,100 In particular, Thanksgiving is a reason 1326 01:00:53,100 --> 01:00:56,340 we all decide that Thursday is this national holiday. 1327 01:00:56,340 --> 01:00:58,615 And therefore, on that Wednesday, we actually 1328 01:00:58,615 --> 01:01:00,240 have to travel in order to get to where 1329 01:01:00,240 --> 01:01:02,020 we need to go by Thursday. 1330 01:01:02,020 --> 01:01:02,850 Right? 1331 01:01:02,850 --> 01:01:05,430 But any arbitrary Wednesday is not 1332 01:01:05,430 --> 01:01:07,380 going to be necessarily highly correlated. 1333 01:01:07,380 --> 01:01:10,380 So when I go to the airport on a typical Wednesday, 1334 01:01:10,380 --> 01:01:12,030 I don't expect it to be so bad. 1335 01:01:12,030 --> 01:01:13,710 But on Wednesday before Thanksgiving, 1336 01:01:13,710 --> 01:01:15,510 it's going to be a madhouse. 1337 01:01:15,510 --> 01:01:16,290 Right? 1338 01:01:16,290 --> 01:01:18,240 That's an example of a changing correlation 1339 01:01:18,240 --> 01:01:20,730 because of a particular coordination 1340 01:01:20,730 --> 01:01:22,320 that we've all agreed upon. 1341 01:01:22,320 --> 01:01:24,322 Now that's a relatively artificial example. 1342 01:01:24,322 --> 01:01:25,530 But I did it to make a point. 1343 01:01:25,530 --> 01:01:26,904 Now let me give you a real answer 1344 01:01:26,904 --> 01:01:30,250 to your question about why things may be correlated. 1345 01:01:30,250 --> 01:01:34,390 When we are all scared about the value of our investments, 1346 01:01:34,390 --> 01:01:37,690 when our fear circuitry gets triggered, 1347 01:01:37,690 --> 01:01:41,440 what's the natural instinct for all of us because it's 1348 01:01:41,440 --> 01:01:43,640 hardwired into our brains? 1349 01:01:43,640 --> 01:01:46,060 It's to get to safety. 1350 01:01:46,060 --> 01:01:48,130 It's to get out of those bad assets 1351 01:01:48,130 --> 01:01:49,630 and get into the good assets. 1352 01:01:49,630 --> 01:01:50,170 Right? 1353 01:01:50,170 --> 01:01:53,710 You saw that three month T-bill yield at 10 basis points, or 5 1354 01:01:53,710 --> 01:01:54,880 basis points? 1355 01:01:54,880 --> 01:01:57,490 That's a sign that we're all scared to death 1356 01:01:57,490 --> 01:01:58,960 and we want to get to safety. 1357 01:01:58,960 --> 01:02:00,460 That's an example of a correlation. 1358 01:02:00,460 --> 01:02:00,960 Why? 1359 01:02:00,960 --> 01:02:04,280 Because everybody is going to be selling at the same time. 1360 01:02:04,280 --> 01:02:09,140 In a crowded theater, if you smell smoke and somebody shouts 1361 01:02:09,140 --> 01:02:12,250 fire, what are you going to do? 1362 01:02:12,250 --> 01:02:14,980 It's not rocket science to predict 1363 01:02:14,980 --> 01:02:17,360 that the four exits that are out there 1364 01:02:17,360 --> 01:02:20,180 will be a bit crowded after that. 1365 01:02:20,180 --> 01:02:21,640 Right? 1366 01:02:21,640 --> 01:02:24,590 So correlation is not a physical quantity. 1367 01:02:24,590 --> 01:02:27,180 That's the problem with physics and biology. 1368 01:02:27,180 --> 01:02:30,770 Physics has parameters that don't change over time. 1369 01:02:30,770 --> 01:02:32,720 I wish we could have that in finance. 1370 01:02:32,720 --> 01:02:33,840 We don't. 1371 01:02:33,840 --> 01:02:35,632 We have parameters that are not parameters. 1372 01:02:35,632 --> 01:02:36,673 They're random variables. 1373 01:02:36,673 --> 01:02:38,090 They depend on a lot of things. 1374 01:02:38,090 --> 01:02:39,780 Correlation is one of them. 1375 01:02:39,780 --> 01:02:42,410 And until we start recognizing that correlations are really 1376 01:02:42,410 --> 01:02:44,935 part and partial of human interactions, 1377 01:02:44,935 --> 01:02:47,060 we're going to continue to make mistakes like we've 1378 01:02:47,060 --> 01:02:49,270 made over the last 10 years. 1379 01:02:49,270 --> 01:02:52,370 And that's a simple example of that. 1380 01:02:52,370 --> 01:02:53,122 Other question? 1381 01:02:53,122 --> 01:02:53,622 Yeah. 1382 01:02:53,622 --> 01:02:57,865 AUDIENCE: [INAUDIBLE] 1383 01:02:57,865 --> 01:02:58,490 ANDREW LO: Yes. 1384 01:02:58,490 --> 01:03:07,640 AUDIENCE: [INAUDIBLE] 1385 01:03:07,640 --> 01:03:09,410 ANDREW LO: Yeah. 1386 01:03:09,410 --> 01:03:10,040 That's true. 1387 01:03:10,040 --> 01:03:11,250 That's possible. 1388 01:03:11,250 --> 01:03:14,720 So you have to decide whether or not what you're looking at 1389 01:03:14,720 --> 01:03:17,120 is an aberration, or whether it's 1390 01:03:17,120 --> 01:03:18,590 something that's systematic. 1391 01:03:18,590 --> 01:03:21,440 So for example, if I didn't know anything about Thanksgiving 1392 01:03:21,440 --> 01:03:24,740 and I happened to travel on Wednesdays on a regular basis 1393 01:03:24,740 --> 01:03:28,520 to Washington, then you know, in two weeks, 1394 01:03:28,520 --> 01:03:29,922 I'll get-- it's really crowded. 1395 01:03:29,922 --> 01:03:31,880 And then it won't be crowded, won't be crowded, 1396 01:03:31,880 --> 01:03:32,930 won't be crowded. 1397 01:03:32,930 --> 01:03:34,970 And pretty soon, after a while, I'll 1398 01:03:34,970 --> 01:03:37,730 say, well, that was just a 1% event. 1399 01:03:37,730 --> 01:03:40,040 Of course, next year it will happen again. 1400 01:03:40,040 --> 01:03:43,470 And then I'll say, gee, you know, well, that's another 1%, 1401 01:03:43,470 --> 01:03:43,970 kind of. 1402 01:03:43,970 --> 01:03:46,730 But pretty soon I'm going to realize that, gee, it seems 1403 01:03:46,730 --> 01:03:48,260 like there's a pattern here. 1404 01:03:48,260 --> 01:03:53,620 And that 1%, it's not just so simple as a 1% error. 1405 01:03:53,620 --> 01:03:57,500 Statistics, remember, is a mathematical quantification 1406 01:03:57,500 --> 01:03:58,460 of our stupidity. 1407 01:03:58,460 --> 01:03:59,960 Right? 1408 01:03:59,960 --> 01:04:01,150 I mean, what we don't know. 1409 01:04:01,150 --> 01:04:03,650 Well, we don't know why things happen the way they happen, 1410 01:04:03,650 --> 01:04:05,300 so we put a distribution on it. 1411 01:04:05,300 --> 01:04:06,650 We say that it's normal. 1412 01:04:06,650 --> 01:04:09,650 And we say that there's 5% this way, 5% that way. 1413 01:04:09,650 --> 01:04:11,150 But that just is a representation 1414 01:04:11,150 --> 01:04:12,170 of our ignorance. 1415 01:04:12,170 --> 01:04:15,699 The more we know, the less we have to rely on statistics. 1416 01:04:15,699 --> 01:04:17,240 So I don't need statistics to tell me 1417 01:04:17,240 --> 01:04:18,698 that two weeks from now, it's going 1418 01:04:18,698 --> 01:04:19,970 to be really crowded at Logan. 1419 01:04:19,970 --> 01:04:22,070 But if I didn't know about Thanksgiving, 1420 01:04:22,070 --> 01:04:25,460 if I came from Mars and I was doing a study of airport 1421 01:04:25,460 --> 01:04:27,590 congestion, it would take me a while 1422 01:04:27,590 --> 01:04:29,480 to get enough data to figure out that 1423 01:04:29,480 --> 01:04:32,220 once a year on the Wednesday before the Thursday 1424 01:04:32,220 --> 01:04:35,240 in, you know, November, it gets crowded. 1425 01:04:35,240 --> 01:04:39,710 So the challenge for all of you is how much intelligence 1426 01:04:39,710 --> 01:04:41,820 can you bring to the analysis. 1427 01:04:41,820 --> 01:04:44,730 What I'm showing you is very simple mathematics. 1428 01:04:44,730 --> 01:04:46,820 Mathematics is not enough. 1429 01:04:46,820 --> 01:04:48,620 If you just had the mathematics, you 1430 01:04:48,620 --> 01:04:50,990 would be losing money, you know, continuously. 1431 01:04:50,990 --> 01:04:53,990 Because there's just much more to financial markets than just 1432 01:04:53,990 --> 01:04:54,570 the math. 1433 01:04:54,570 --> 01:04:55,070 OK? 1434 01:04:55,070 --> 01:04:55,903 The math is trivial. 1435 01:04:55,903 --> 01:04:57,660 This is high school algebra. 1436 01:04:57,660 --> 01:04:59,690 But the real key is to put together 1437 01:04:59,690 --> 01:05:02,120 the framework of economic and financial analysis 1438 01:05:02,120 --> 01:05:03,830 with the mathematics. 1439 01:05:03,830 --> 01:05:06,500 So let me continue doing that. 1440 01:05:06,500 --> 01:05:11,840 This is another example of computing the average variance, 1441 01:05:11,840 --> 01:05:13,730 and then looking at the volatility 1442 01:05:13,730 --> 01:05:17,130 of your particular holdings, given different correlations. 1443 01:05:17,130 --> 01:05:19,342 So I want you to do lots of exercises 1444 01:05:19,342 --> 01:05:21,050 where you look at different correlations. 1445 01:05:21,050 --> 01:05:23,240 I don't ever want you to take correlation 1446 01:05:23,240 --> 01:05:25,700 as a parameter that is fixed over time, 1447 01:05:25,700 --> 01:05:26,870 when you apply the stuff. 1448 01:05:26,870 --> 01:05:29,750 When you're doing problem sets in final exams, that's fine. 1449 01:05:29,750 --> 01:05:31,850 You know, the correlation is whatever it is. 1450 01:05:31,850 --> 01:05:33,950 But recognize that in practice, these things 1451 01:05:33,950 --> 01:05:36,200 change a lot over time. 1452 01:05:36,200 --> 01:05:37,820 OK? 1453 01:05:37,820 --> 01:05:41,030 Now the idea behind correlation is 1454 01:05:41,030 --> 01:05:42,590 that they help you reduce the ups 1455 01:05:42,590 --> 01:05:45,920 and the downs of the variance of your portfolio. 1456 01:05:45,920 --> 01:05:49,160 But it turns out that there's a limit to how much of the risk 1457 01:05:49,160 --> 01:05:50,420 you can reduce. 1458 01:05:50,420 --> 01:05:53,420 And this graph basically shows that as you 1459 01:05:53,420 --> 01:05:55,370 add more and more securities-- 1460 01:05:55,370 --> 01:05:58,190 even though the correlations are bouncing around-- 1461 01:05:58,190 --> 01:06:00,230 as you add more and more securities, 1462 01:06:00,230 --> 01:06:03,170 there ends up being some kind of a steady state 1463 01:06:03,170 --> 01:06:07,310 limit to what the variance of your overall portfolio is. 1464 01:06:07,310 --> 01:06:12,210 After 20, 30, 40, 50, 100 stocks, 1465 01:06:12,210 --> 01:06:14,400 you'll notice that the variance of your portfolio 1466 01:06:14,400 --> 01:06:17,190 doesn't go down anymore. 1467 01:06:17,190 --> 01:06:21,480 And it turns out that that limit, whatever that is, 1468 01:06:21,480 --> 01:06:27,120 is what we consider to be the systematic risk that 1469 01:06:27,120 --> 01:06:29,220 is implicit in the economy. 1470 01:06:29,220 --> 01:06:32,490 In other words, that's the risk that no matter how well 1471 01:06:32,490 --> 01:06:37,110 diversified you are, you've got to bear that risk. 1472 01:06:37,110 --> 01:06:38,720 Everybody, all of us, we have to, 1473 01:06:38,720 --> 01:06:41,690 we can't get any less risky than that 1474 01:06:41,690 --> 01:06:44,120 unless we start putting our money in the mattress 1475 01:06:44,120 --> 01:06:45,350 or in T-bills. 1476 01:06:45,350 --> 01:06:46,250 OK? 1477 01:06:46,250 --> 01:06:50,210 That limit is known as the systematic, or market risk, 1478 01:06:50,210 --> 01:06:51,740 of a portfolio. 1479 01:06:51,740 --> 01:06:55,370 So, see here is the graph of your portfolio variance 1480 01:06:55,370 --> 01:06:58,010 as you increase more and more securities. 1481 01:06:58,010 --> 01:07:01,980 But at some point, it asymptotes to this level, 1482 01:07:01,980 --> 01:07:04,040 which is what I'm going to be focusing on, 1483 01:07:04,040 --> 01:07:06,440 as undiversifiable risk. 1484 01:07:06,440 --> 01:07:09,110 It's the risk that's left over after all 1485 01:07:09,110 --> 01:07:11,840 of the various correlations have done what they 1486 01:07:11,840 --> 01:07:16,040 can to dampen the ups and the downs of your collection 1487 01:07:16,040 --> 01:07:17,780 of securities. 1488 01:07:17,780 --> 01:07:20,360 And obviously, there's going to be 1489 01:07:20,360 --> 01:07:22,550 some calculations you'll need to do to figure out 1490 01:07:22,550 --> 01:07:23,849 what that number is. 1491 01:07:23,849 --> 01:07:25,640 But I want to give you the intuition first, 1492 01:07:25,640 --> 01:07:27,167 and then we'll do the calculations. 1493 01:07:27,167 --> 01:07:29,000 And then we're going to study the properties 1494 01:07:29,000 --> 01:07:30,750 of this hard limit. 1495 01:07:30,750 --> 01:07:32,850 OK? 1496 01:07:32,850 --> 01:07:39,260 Now, I'm about to proceed to the next stage of our analysis 1497 01:07:39,260 --> 01:07:41,810 where we start asking the question, 1498 01:07:41,810 --> 01:07:46,431 how do we pick the very best possible portfolio. 1499 01:07:46,431 --> 01:07:47,930 But before I do, I want to make sure 1500 01:07:47,930 --> 01:07:49,880 that everybody is comfortable with the analytics 1501 01:07:49,880 --> 01:07:50,546 we've developed. 1502 01:07:50,546 --> 01:07:53,180 If you have any questions, now would be a good time to ask. 1503 01:07:53,180 --> 01:07:54,721 Because from this point on, I'm going 1504 01:07:54,721 --> 01:07:57,980 to assume that you understand how portfolio theory works, 1505 01:07:57,980 --> 01:08:00,410 the mechanics of portfolio weights, 1506 01:08:00,410 --> 01:08:03,890 and how to compute means, variances, covariances, 1507 01:08:03,890 --> 01:08:06,790 and what they imply for the portfolio. 1508 01:08:06,790 --> 01:08:08,160 Question, yeah? 1509 01:08:08,160 --> 01:08:10,240 AUDIENCE: Earlier, you said that we 1510 01:08:10,240 --> 01:08:12,860 start from this, ensuring that the markets are pretty much 1511 01:08:12,860 --> 01:08:13,730 stable. 1512 01:08:13,730 --> 01:08:17,140 But I'm wondering, how reasonable is that assumption? 1513 01:08:17,140 --> 01:08:23,140 And why would nobody ever pick a lower part of on the curve, 1514 01:08:23,140 --> 01:08:28,170 for example, you know, if there was a big scandal tomorrow, 1515 01:08:28,170 --> 01:08:31,390 like a corruption, like I'm thinking about Enron 1516 01:08:31,390 --> 01:08:35,080 or [INAUDIBLE] in Italy. 1517 01:08:35,080 --> 01:08:39,710 Why would I not want to short sell, you know, 1518 01:08:39,710 --> 01:08:46,466 the higher return stock and buy the lower return stock, 1519 01:08:46,466 --> 01:08:48,960 and take the lower part of the curve? 1520 01:08:48,960 --> 01:08:53,160 ANDREW LO: You might, if there are other elements that 1521 01:08:53,160 --> 01:08:55,410 are in this analysis. 1522 01:08:55,410 --> 01:08:58,020 So as I said, this analysis assumes 1523 01:08:58,020 --> 01:09:02,319 there are only two dimensions, mean and standard deviation. 1524 01:09:02,319 --> 01:09:03,359 Right? 1525 01:09:03,359 --> 01:09:05,189 You've now introduced a third, which 1526 01:09:05,189 --> 01:09:08,850 is corporate responsibility or corporate governance 1527 01:09:08,850 --> 01:09:10,439 or fraud or something like that. 1528 01:09:10,439 --> 01:09:12,240 Then you would need three dimensions. 1529 01:09:12,240 --> 01:09:16,680 And there'd be something sticking outside the screen. 1530 01:09:16,680 --> 01:09:19,470 And then you'd have to choose among the three possibilities. 1531 01:09:19,470 --> 01:09:21,420 So it's possible that you want to be 1532 01:09:21,420 --> 01:09:25,890 down here because this company has a better reputation 1533 01:09:25,890 --> 01:09:28,470 than this one up here. 1534 01:09:28,470 --> 01:09:30,210 But then what you're telling me is 1535 01:09:30,210 --> 01:09:34,439 that reputation matters to you beyond standard deviation 1536 01:09:34,439 --> 01:09:37,600 and expected return. 1537 01:09:37,600 --> 01:09:38,715 What's that? 1538 01:09:38,715 --> 01:09:40,456 AUDIENCE: Market conditions. 1539 01:09:40,456 --> 01:09:42,330 ANDREW LO: Stable means that these parameters 1540 01:09:42,330 --> 01:09:45,857 are likely to stay the same over time. 1541 01:09:45,857 --> 01:09:47,732 AUDIENCE: There's a crisis every three years. 1542 01:09:47,732 --> 01:09:50,040 There's an industry crisis every two years. 1543 01:09:50,040 --> 01:09:51,450 ANDREW LO: Right. 1544 01:09:51,450 --> 01:09:53,850 Well, the argument that a financial economist 1545 01:09:53,850 --> 01:09:55,100 would make-- 1546 01:09:55,100 --> 01:09:56,850 and I'm not saying that I believe in this. 1547 01:09:56,850 --> 01:09:58,350 I'm going to tell you what I believe 1548 01:09:58,350 --> 01:09:59,430 at the end of this class. 1549 01:09:59,430 --> 01:10:01,770 But I'm telling you now what the party line is, 1550 01:10:01,770 --> 01:10:03,510 and what's in your textbook. 1551 01:10:03,510 --> 01:10:05,850 What's in your textbook and what the party line is 1552 01:10:05,850 --> 01:10:09,090 is that when you use historical amounts of data, 1553 01:10:09,090 --> 01:10:12,730 those kind of crisis periods are in there already. 1554 01:10:12,730 --> 01:10:14,430 So if I estimate the correlations, 1555 01:10:14,430 --> 01:10:19,560 including October '87, including March 2000, the bursting 1556 01:10:19,560 --> 01:10:22,770 of the internet bubble, August 1998, LTCM, 1557 01:10:22,770 --> 01:10:25,960 if I include all of those in my data, 1558 01:10:25,960 --> 01:10:27,360 then it's captured in there. 1559 01:10:27,360 --> 01:10:28,720 Right? 1560 01:10:28,720 --> 01:10:30,530 It's not captured today. 1561 01:10:30,530 --> 01:10:33,790 In other words, this point here for General Dynamics 1562 01:10:33,790 --> 01:10:36,490 doesn't reflect that there's a crisis today. 1563 01:10:36,490 --> 01:10:40,640 What it reflects is, on average, over a long period of time-- 1564 01:10:40,640 --> 01:10:42,760 some of which includes crises-- 1565 01:10:42,760 --> 01:10:45,810 that's what the expected return looks like. 1566 01:10:45,810 --> 01:10:50,580 And so I've already incorporated that into the parameters. 1567 01:10:50,580 --> 01:10:53,310 And if it so happens that today is a bad day for General 1568 01:10:53,310 --> 01:10:55,620 Dynamics, that shouldn't influence 1569 01:10:55,620 --> 01:10:56,850 whether you buy it or not. 1570 01:10:56,850 --> 01:10:58,558 Because what you should be thinking about 1571 01:10:58,558 --> 01:11:03,360 is over the next 15 or 20 years, how will my portfolio do. 1572 01:11:03,360 --> 01:11:06,210 So that's another difference in perspective from us 1573 01:11:06,210 --> 01:11:07,890 versus Warren Buffett. 1574 01:11:07,890 --> 01:11:10,290 Although actually, not as much as you think. 1575 01:11:10,290 --> 01:11:13,290 Warren Buffett typically takes a long term perspective, right? 1576 01:11:13,290 --> 01:11:15,042 His investment in Goldman Sachs? 1577 01:11:15,042 --> 01:11:16,500 We all thought it was a great deal. 1578 01:11:16,500 --> 01:11:18,540 He's lost money in it since he put that money 1579 01:11:18,540 --> 01:11:20,936 in a few weeks ago. 1580 01:11:20,936 --> 01:11:22,060 But is he worried about it? 1581 01:11:22,060 --> 01:11:23,130 I don't think so. 1582 01:11:23,130 --> 01:11:26,040 Because he's invested in it for the next 15 to 20 years. 1583 01:11:26,040 --> 01:11:30,300 And I believe, as he does, that he got a great deal 1584 01:11:30,300 --> 01:11:32,320 over that time period. 1585 01:11:32,320 --> 01:11:35,100 So if you care about what's going to happen tomorrow, 1586 01:11:35,100 --> 01:11:37,380 then all of the things that I'm telling you, 1587 01:11:37,380 --> 01:11:39,450 you should not take that as seriously. 1588 01:11:39,450 --> 01:11:42,510 Because these kind of parameters aren't made for day 1589 01:11:42,510 --> 01:11:44,130 to day forecasting. 1590 01:11:44,130 --> 01:11:45,930 Then you need to get hedge fund models. 1591 01:11:45,930 --> 01:11:48,380 And for that, you need to take not only 433, 1592 01:11:48,380 --> 01:11:50,130 but a number of other courses, and courses 1593 01:11:50,130 --> 01:11:51,210 that aren't even offered yet. 1594 01:11:51,210 --> 01:11:53,070 Because frankly, if I told you how to do it, 1595 01:11:53,070 --> 01:11:53,945 I'd have to kill you. 1596 01:11:53,945 --> 01:11:54,570 Right? 1597 01:11:54,570 --> 01:11:56,850 Highly proprietary. 1598 01:11:56,850 --> 01:11:59,370 But this is the approach where we're not 1599 01:11:59,370 --> 01:12:00,660 trying to forecast markets. 1600 01:12:00,660 --> 01:12:02,460 And so I'm acknowledging that you're 1601 01:12:02,460 --> 01:12:04,470 right that there are instabilities. 1602 01:12:04,470 --> 01:12:06,720 But as long as I'm using a long enough amount of data, 1603 01:12:06,720 --> 01:12:07,590 that it's in there. 1604 01:12:07,590 --> 01:12:10,550 It's captured in there, in that set of parameters. 1605 01:12:10,550 --> 01:12:11,610 OK? 1606 01:12:11,610 --> 01:12:14,310 If it's not, if you know of something that's not in here, 1607 01:12:14,310 --> 01:12:17,250 obviously you can't expect this to give you the right answer. 1608 01:12:17,250 --> 01:12:20,580 And you have to adjust your use of this technology, 1609 01:12:20,580 --> 01:12:23,190 accordingly. 1610 01:12:23,190 --> 01:12:24,650 Other questions? 1611 01:12:24,650 --> 01:12:25,326 Yeah? 1612 01:12:25,326 --> 01:12:29,404 AUDIENCE: [INAUDIBLE] 1613 01:12:29,404 --> 01:12:30,070 ANDREW LO: Yeah. 1614 01:12:30,070 --> 01:12:45,080 AUDIENCE: [INAUDIBLE] 1615 01:12:45,080 --> 01:12:46,430 ANDREW LO: Yeah. 1616 01:12:46,430 --> 01:12:47,230 That's right. 1617 01:12:47,230 --> 01:12:47,760 Yeah. 1618 01:12:47,760 --> 01:12:54,800 AUDIENCE: [INAUDIBLE] 1619 01:12:54,800 --> 01:12:56,660 ANDREW LO: Well, you have to know 1620 01:12:56,660 --> 01:12:58,580 how to build those other variables 1621 01:12:58,580 --> 01:12:59,910 into this kind of a framework. 1622 01:12:59,910 --> 01:13:01,825 That's another way of putting it. 1623 01:13:01,825 --> 01:13:03,200 So in other words, this framework 1624 01:13:03,200 --> 01:13:05,090 is still useful in the case where 1625 01:13:05,090 --> 01:13:06,980 you've got other variables. 1626 01:13:06,980 --> 01:13:09,710 But you just need to know how those other variables will 1627 01:13:09,710 --> 01:13:13,640 impact the parameters for this analysis. 1628 01:13:13,640 --> 01:13:17,690 If you can figure that out, then you're a hedge fund manager. 1629 01:13:17,690 --> 01:13:18,190 Right? 1630 01:13:18,190 --> 01:13:20,000 That's what hedge funds try to do. 1631 01:13:20,000 --> 01:13:22,730 They try to figure out how all of these relations 1632 01:13:22,730 --> 01:13:25,550 impact on how to construct a good portfolio. 1633 01:13:25,550 --> 01:13:26,348 All right? 1634 01:13:26,348 --> 01:13:28,360 AUDIENCE: [INAUDIBLE] 1635 01:13:28,360 --> 01:13:29,360 ANDREW LO: That's right. 1636 01:13:29,360 --> 01:13:30,170 For example. 1637 01:13:30,170 --> 01:13:31,929 That would be one way of looking at it. 1638 01:13:31,929 --> 01:13:32,970 But there are other ways. 1639 01:13:32,970 --> 01:13:34,550 It doesn't have to be quantitative. 1640 01:13:34,550 --> 01:13:36,110 There are a lot of talented hedge fund managers 1641 01:13:36,110 --> 01:13:37,670 that don't know how to use a calculator, 1642 01:13:37,670 --> 01:13:38,503 make a lot of money. 1643 01:13:38,503 --> 01:13:40,100 OK? 1644 01:13:40,100 --> 01:13:42,795 So don't think that it's all about quantitative analysis. 1645 01:13:42,795 --> 01:13:43,670 That's what we think. 1646 01:13:43,670 --> 01:13:45,140 That's how I think. 1647 01:13:45,140 --> 01:13:46,994 That's how MIT may think. 1648 01:13:46,994 --> 01:13:49,160 But that's not the only way to make money out there. 1649 01:13:49,160 --> 01:13:50,870 You know, Warren Buffett, I don't think 1650 01:13:50,870 --> 01:13:53,210 knows how to calculate these covariances. 1651 01:13:53,210 --> 01:13:56,250 But he's done OK. 1652 01:13:56,250 --> 01:13:59,270 So but I think what we can do with this framework is 1653 01:13:59,270 --> 01:14:02,570 to analyze how he does what he does, 1654 01:14:02,570 --> 01:14:06,130 and understand it in the context of this kind of a framework. 1655 01:14:06,130 --> 01:14:06,860 All right. 1656 01:14:06,860 --> 01:14:12,470 So let's now go to the next topic, which is, all right, 1657 01:14:12,470 --> 01:14:15,710 how do we choose a good portfolio, 1658 01:14:15,710 --> 01:14:18,410 given what we now know about this framework? 1659 01:14:18,410 --> 01:14:19,989 Well, we said before-- 1660 01:14:19,989 --> 01:14:21,530 Ryan pointed out-- that we never want 1661 01:14:21,530 --> 01:14:24,410 to be on the lower part of that frontier. 1662 01:14:24,410 --> 01:14:25,830 So that's one thing we know. 1663 01:14:25,830 --> 01:14:26,330 Right? 1664 01:14:26,330 --> 01:14:28,810 We don't want to be on the bottom part. 1665 01:14:28,810 --> 01:14:31,190 So we want to be on the efficient frontier. 1666 01:14:31,190 --> 01:14:34,310 Now this entire bullet can be viewed 1667 01:14:34,310 --> 01:14:37,550 as what's called the minimum variance boundary. 1668 01:14:37,550 --> 01:14:40,940 Meaning, for any given investment over here, 1669 01:14:40,940 --> 01:14:44,390 we have the absolute minimum variance 1670 01:14:44,390 --> 01:14:46,850 that has the same level of expected return. 1671 01:14:46,850 --> 01:14:49,490 So when I go, when I go horizontally, 1672 01:14:49,490 --> 01:14:52,040 I'm looking at the smallest amount 1673 01:14:52,040 --> 01:14:56,260 of risk I can take for that same level of expected 1674 01:14:56,260 --> 01:14:57,060 rate of return. 1675 01:14:57,060 --> 01:15:00,680 That's what this bullet can be viewed as giving you. 1676 01:15:00,680 --> 01:15:05,610 And it turns out that because we like expected return, 1677 01:15:05,610 --> 01:15:07,790 and we don't like risk, we want to be 1678 01:15:07,790 --> 01:15:09,680 on the upper part of that bullet. 1679 01:15:09,680 --> 01:15:13,500 So this upper part is known as the efficient frontier. 1680 01:15:13,500 --> 01:15:16,680 So that's the one thing we can tell about portfolio theory. 1681 01:15:16,680 --> 01:15:20,060 It's that we want to be on the upper branch. 1682 01:15:20,060 --> 01:15:21,920 But where on the upper branch depends 1683 01:15:21,920 --> 01:15:23,830 upon our risk preferences. 1684 01:15:23,830 --> 01:15:25,370 OK? 1685 01:15:25,370 --> 01:15:27,650 So here's a concrete example. 1686 01:15:27,650 --> 01:15:30,500 Suppose you can invest in any combination of General 1687 01:15:30,500 --> 01:15:32,540 Motors, IBM, and Motorola. 1688 01:15:32,540 --> 01:15:35,010 What portfolio would you choose? 1689 01:15:35,010 --> 01:15:37,260 So these are the data that you would start with. 1690 01:15:37,260 --> 01:15:37,760 OK? 1691 01:15:37,760 --> 01:15:41,030 The means and the standard deviations, and then of course, 1692 01:15:41,030 --> 01:15:42,050 the covariances. 1693 01:15:42,050 --> 01:15:43,820 You've got to have the covariances to be 1694 01:15:43,820 --> 01:15:46,250 able to calculate that bullet. 1695 01:15:46,250 --> 01:15:50,154 So you can invest in any combination. 1696 01:15:50,154 --> 01:15:51,070 What would you choose? 1697 01:15:51,070 --> 01:15:53,890 Let's calculate the expected rate of return 1698 01:15:53,890 --> 01:15:57,730 and the variance of that portfolio. 1699 01:15:57,730 --> 01:16:00,550 And you want to ask the question, what 1700 01:16:00,550 --> 01:16:02,430 looks good to you? 1701 01:16:02,430 --> 01:16:04,200 OK? 1702 01:16:04,200 --> 01:16:06,480 Well, it turns out that when you calculate 1703 01:16:06,480 --> 01:16:09,570 this kind of a bullet, you find out something 1704 01:16:09,570 --> 01:16:13,810 that's, yet again, amazing. 1705 01:16:13,810 --> 01:16:17,560 What you find is that the bullet actually 1706 01:16:17,560 --> 01:16:22,820 is better than any of the three stocks that you started with. 1707 01:16:22,820 --> 01:16:24,640 So look at where the three stocks are-- 1708 01:16:24,640 --> 01:16:27,517 General Motors, IBM, and Motorola-- 1709 01:16:27,517 --> 01:16:29,350 and the bullet, look at where the bullet is. 1710 01:16:29,350 --> 01:16:35,160 The bullet is strictly to the northwest of these stocks. 1711 01:16:35,160 --> 01:16:35,730 Right? 1712 01:16:35,730 --> 01:16:38,850 In other words, you can do better than any one of these, 1713 01:16:38,850 --> 01:16:41,370 either by going here or by going here. 1714 01:16:41,370 --> 01:16:43,830 For the same level of risk, you can get a higher return. 1715 01:16:43,830 --> 01:16:47,980 For the same level of expected return, you can get lower risk. 1716 01:16:47,980 --> 01:16:49,450 OK? 1717 01:16:49,450 --> 01:16:52,450 So the first point is that portfolio theory, 1718 01:16:52,450 --> 01:16:55,600 for multiple stocks, is even more compelling. 1719 01:16:55,600 --> 01:16:57,530 Because now, with those two stocks, 1720 01:16:57,530 --> 01:17:02,290 at least the two stocks were on that minimum variance boundary. 1721 01:17:02,290 --> 01:17:05,050 Now, with three stocks and more, it's 1722 01:17:05,050 --> 01:17:07,360 possible that none of the stocks are 1723 01:17:07,360 --> 01:17:10,080 going to be on the minimum variance boundary. 1724 01:17:10,080 --> 01:17:10,898 Yeah? 1725 01:17:10,898 --> 01:17:12,890 AUDIENCE: What I don't understand is this. 1726 01:17:12,890 --> 01:17:15,880 S This is backward looking. 1727 01:17:15,880 --> 01:17:20,420 So if people between companies and so on all make this graph 1728 01:17:20,420 --> 01:17:23,980 and they all decide to strategize based on this graph, 1729 01:17:23,980 --> 01:17:26,046 the price of these stocks next day 1730 01:17:26,046 --> 01:17:27,920 are going to be correlated in a different way 1731 01:17:27,920 --> 01:17:30,640 than you expect because it's going to effect the-- 1732 01:17:30,640 --> 01:17:31,810 ANDREW LO: Very good point. 1733 01:17:31,810 --> 01:17:34,360 But it turns out your conclusion is false. 1734 01:17:34,360 --> 01:17:36,040 But that's a good question to ask. 1735 01:17:36,040 --> 01:17:38,890 And I'm going to answer that, not today, but on Monday. 1736 01:17:38,890 --> 01:17:42,040 When we go over the question, what 1737 01:17:42,040 --> 01:17:44,820 happens when everybody does it, I'm 1738 01:17:44,820 --> 01:17:46,660 going to deal with that question head on. 1739 01:17:46,660 --> 01:17:47,380 OK? 1740 01:17:47,380 --> 01:17:49,270 That's going to be a very important point. 1741 01:17:49,270 --> 01:17:53,380 And this is going to be one of the very few instances where 1742 01:17:53,380 --> 01:17:57,360 when everybody does what I tell you they're going to do, 1743 01:17:57,360 --> 01:17:59,760 that's actually going to give you an equilibrium. 1744 01:17:59,760 --> 01:18:03,240 And everything will work out in just the right way. 1745 01:18:03,240 --> 01:18:05,820 That's the magic of the CAPM. 1746 01:18:05,820 --> 01:18:08,100 But you're right that Warren Buffett could not 1747 01:18:08,100 --> 01:18:10,055 answer that criticism. 1748 01:18:10,055 --> 01:18:11,680 If you talk to Warren Buffett and said, 1749 01:18:11,680 --> 01:18:16,947 Warren, if everybody did what you did, then it wouldn't work. 1750 01:18:16,947 --> 01:18:18,280 And he would say, you know what? 1751 01:18:18,280 --> 01:18:18,821 That's right. 1752 01:18:18,821 --> 01:18:21,700 That's why nobody can ever do what I do. 1753 01:18:21,700 --> 01:18:23,260 I'm just smarter than everybody else. 1754 01:18:23,260 --> 01:18:24,010 And so, I'm sorry. 1755 01:18:24,010 --> 01:18:25,110 That's the way it goes. 1756 01:18:25,110 --> 01:18:28,180 Here, I'm not appealing to everybody being 1757 01:18:28,180 --> 01:18:29,440 smarter than everybody else. 1758 01:18:29,440 --> 01:18:32,450 Because frankly, we're not. 1759 01:18:32,450 --> 01:18:34,610 What I'm appealing to, what I will appeal to, 1760 01:18:34,610 --> 01:18:37,710 is if everybody does the right thing-- by the right thing, 1761 01:18:37,710 --> 01:18:40,490 I'm about to tell you what that right thing is-- everybody 1762 01:18:40,490 --> 01:18:43,040 is on that upper branch and maximizes 1763 01:18:43,040 --> 01:18:48,080 their risk/reward trade off, a very special thing happens. 1764 01:18:48,080 --> 01:18:48,740 All right? 1765 01:18:48,740 --> 01:18:51,470 I'm going to keep that as a surprise for Monday. 1766 01:18:51,470 --> 01:18:52,400 OK, question? 1767 01:18:52,400 --> 01:18:53,140 A question? 1768 01:18:53,140 --> 01:18:53,835 Yeah. 1769 01:18:53,835 --> 01:18:56,210 AUDIENCE: [INAUDIBLE] 1770 01:18:56,210 --> 01:18:57,314 ANDREW LO: Yeah. 1771 01:18:57,314 --> 01:18:58,730 Well, first of all, it's not clear 1772 01:18:58,730 --> 01:19:00,271 that Warren Buffett is number one. 1773 01:19:00,271 --> 01:19:00,770 All right? 1774 01:19:00,770 --> 01:19:03,650 He's got a long track record and he's got the biggest pie. 1775 01:19:03,650 --> 01:19:05,300 But in terms of actual track record, 1776 01:19:05,300 --> 01:19:06,550 he doesn't have the best track record. 1777 01:19:06,550 --> 01:19:07,340 There are people that you've never 1778 01:19:07,340 --> 01:19:08,946 heard of that have a better track 1779 01:19:08,946 --> 01:19:10,070 record than Warren Buffett. 1780 01:19:10,070 --> 01:19:13,250 For example, there's a fellow by the name of James Simons-- 1781 01:19:13,250 --> 01:19:16,280 who I may have mentioned earlier on as a hedge fund manager-- 1782 01:19:16,280 --> 01:19:19,190 who also happens to be a first rate mathematician, 1783 01:19:19,190 --> 01:19:20,780 who started up a hedge fund called 1784 01:19:20,780 --> 01:19:22,400 Renaissance Technologies. 1785 01:19:22,400 --> 01:19:24,710 That is probably the single best track 1786 01:19:24,710 --> 01:19:28,190 record of any manager in the history of investments. 1787 01:19:28,190 --> 01:19:30,290 And he does it completely quantitatively, 1788 01:19:30,290 --> 01:19:33,200 completely automated. 1789 01:19:33,200 --> 01:19:39,410 He hires something like 100 PhDs that work on nothing else 1790 01:19:39,410 --> 01:19:42,350 but how to forecast the next minute, as well as 1791 01:19:42,350 --> 01:19:44,732 the next hour, the next year. 1792 01:19:44,732 --> 01:19:46,190 It's an extraordinary track record. 1793 01:19:46,190 --> 01:19:47,840 So there are a number of folks like that. 1794 01:19:47,840 --> 01:19:49,256 Let me not dwell on that, but I'll 1795 01:19:49,256 --> 01:19:51,680 come back to that in a few lectures 1796 01:19:51,680 --> 01:19:54,390 when we talk about performance attribution. 1797 01:19:54,390 --> 01:19:55,370 OK. 1798 01:19:55,370 --> 01:19:57,980 So I'm going to leave you with one final thought 1799 01:19:57,980 --> 01:19:59,880 since we're out of time. 1800 01:19:59,880 --> 01:20:03,530 There is going to be a very special role played 1801 01:20:03,530 --> 01:20:07,490 by a portfolio called the Tangency Portfolio. 1802 01:20:07,490 --> 01:20:11,790 And I want you to think about how your risk/reward trade off 1803 01:20:11,790 --> 01:20:15,830 would look when you mix your T-bill 1804 01:20:15,830 --> 01:20:20,620 risk-free asset with arbitrary portfolios on this bullet. 1805 01:20:20,620 --> 01:20:22,620 Think about what that trade off would look like. 1806 01:20:22,620 --> 01:20:24,200 And ask yourself the question, does 1807 01:20:24,200 --> 01:20:29,150 there exist a special portfolio on that efficient frontier 1808 01:20:29,150 --> 01:20:32,581 that everybody in this classroom is going to want to have? 1809 01:20:32,581 --> 01:20:33,080 All right? 1810 01:20:33,080 --> 01:20:34,110 I want you to identify them. 1811 01:20:34,110 --> 01:20:36,151 And I'm going to ask you that question on Monday, 1812 01:20:36,151 --> 01:20:37,631 and I expect an answer. 1813 01:20:37,631 --> 01:20:38,130 All right? 1814 01:20:38,130 --> 01:20:40,280 I'll see you on Monday.