1 00:00:01,010 --> 00:00:03,350 So the seventh week of lectures, is 2 00:00:03,350 --> 00:00:06,440 going to be split into two separate slide decks. 3 00:00:06,440 --> 00:00:10,580 So this one I've labeled 7.1, and it'll be on Lagrange Method. 4 00:00:13,810 --> 00:00:16,546 So I'll start off talking about the idea of an 5 00:00:16,546 --> 00:00:21,570 investment portfolio, and then how you might want to optimize that. 6 00:00:21,570 --> 00:00:24,194 So a portfolio, just, you know, if I buy several 7 00:00:24,194 --> 00:00:28,600 assets, put them all in one folder, that's called a portfolio. 8 00:00:28,600 --> 00:00:32,651 And then the decision I have is, well, it's sort of two parts. 9 00:00:32,651 --> 00:00:34,350 Is A, what am I going to buy? 10 00:00:34,350 --> 00:00:38,892 And I'm going to sort of assume that that's already been answered for us. 11 00:00:38,892 --> 00:00:43,420 And then B is going to be, how much of each thing should I buy? 12 00:00:43,420 --> 00:00:48,814 And, so, if you have investment goals, you can choose exactly 13 00:00:48,814 --> 00:00:53,450 how much you want to buy of each thing, to satisfy them. 14 00:00:53,450 --> 00:00:55,851 So, that's what I'll be talking about, so to setting up 15 00:00:55,851 --> 00:00:58,160 the problem I want to solve in the first group of slides. 16 00:00:59,240 --> 00:01:03,980 Then I'll talk about finding. So it's, labeled Relative Extrema of 17 00:01:03,980 --> 00:01:06,600 Functions of Several Variables. 18 00:01:06,600 --> 00:01:08,742 So we talked about first and second order 19 00:01:08,742 --> 00:01:12,123 conditions, for maximizing a function of a single variable. 20 00:01:12,123 --> 00:01:15,131 And now that we've done some linear algebra, I can tell 21 00:01:15,131 --> 00:01:18,850 you what the conditions are for functions of more than one variable. 22 00:01:20,610 --> 00:01:23,790 And then we'll look at Lagrange Method, which is essentially 23 00:01:23,790 --> 00:01:27,035 trying to do number two, except it also has some constraints. 24 00:01:27,035 --> 00:01:29,255 So it says I want to maximize this function, 25 00:01:29,255 --> 00:01:31,670 subject to another function being satisfied. 26 00:01:33,014 --> 00:01:35,745 in the fourth bit, I'll given example of that. 27 00:01:35,745 --> 00:01:38,626 And then in the fifth bit, I'll go back to one of the 28 00:01:38,626 --> 00:01:40,837 concepts I'm going to introduce in the 29 00:01:40,837 --> 00:01:43,970 first bit, called a Minimum Variance Portfolio. 30 00:01:43,970 --> 00:01:47,550 And just go through the calculation of finding that, using the Lagrange Method. 31 00:01:52,000 --> 00:01:54,810 We'll get started with Optimal Investment Portfolios. 32 00:01:58,190 --> 00:02:03,701 So like I, like I just said, a portfolio is just a collection of n assets. 33 00:02:03,701 --> 00:02:09,404 So n is going to be some positive integer. portfolio just means a folder. 34 00:02:09,404 --> 00:02:13,022 So in the old days I think you, you know, if I went and bought some 35 00:02:13,022 --> 00:02:16,439 stock, I actually got paper certificates that represented 36 00:02:16,439 --> 00:02:18,843 those assets, and I put them in a folder. 37 00:02:18,843 --> 00:02:22,991 And so then my folder full of all of my stocks became known as a portfolio, 38 00:02:22,991 --> 00:02:26,700 and so the folders are gone but the, the word has stuck around. 39 00:02:29,830 --> 00:02:36,389 And I'm going make a vector w where, so i is going to run from one up to n. 40 00:02:36,389 --> 00:02:40,090 So n's the total number of assets. 41 00:02:40,090 --> 00:02:44,798 And then w sub i is going to be the proportion of the portfolio. 42 00:02:44,798 --> 00:02:48,550 So value wise, that's invested in each of the assets. 43 00:02:48,550 --> 00:02:51,366 So if I made it equally weighted portfolio, w i would 44 00:02:51,366 --> 00:02:54,527 be equal to 1 over n for every asset in the portfolio. 45 00:02:54,527 --> 00:02:55,031 but there 46 00:02:55,031 --> 00:02:57,190 are a lot of other ways you could do that. 47 00:02:57,190 --> 00:02:59,296 You know, I could say well, I think I'm going to 48 00:02:59,296 --> 00:03:02,025 make more money on a certain, like in technology stocks. 49 00:03:02,025 --> 00:03:04,750 So I might buy technology stocks in a higher proportion. 50 00:03:04,750 --> 00:03:08,782 So those w's would be higher, than the w's for the rest of the assets in the 51 00:03:08,782 --> 00:03:14,634 portfolio. And I'll end up the constraint that, so if 52 00:03:14,634 --> 00:03:21,170 w i is proportions, all of the wi's have to sum up to one, so I have one portfolio. 53 00:03:22,400 --> 00:03:27,380 And so you can think about you know, if you wanted to make this into dollars. 54 00:03:27,380 --> 00:03:29,922 This one isn't a very useful number, but you can 55 00:03:29,922 --> 00:03:32,420 think about it as maybe a thousand or a million. 56 00:03:32,420 --> 00:03:36,030 So I have a million dollars to invest, or I have a thousand dollars to invest. 57 00:03:37,030 --> 00:03:43,289 And then the Wi would be, the amount of money invested in each of the assets. 58 00:03:44,780 --> 00:03:47,444 And just to normalize that, we divide by the 59 00:03:47,444 --> 00:03:49,330 total value of the portfolio. 60 00:03:49,330 --> 00:03:52,870 So you always end up with one on this side, and proportions on this side. 61 00:03:56,460 --> 00:03:59,057 And we're also going to say for, to make the, the problem 62 00:03:59,057 --> 00:04:00,700 mathematically more easy, we can do 63 00:04:00,700 --> 00:04:03,030 something called take long and short positions. 64 00:04:03,030 --> 00:04:07,100 So, a long position in an asset, is something everybody is familiar with. 65 00:04:07,100 --> 00:04:11,960 You go to a market, you buy something, and now you have the something. 66 00:04:11,960 --> 00:04:16,851 A short position is sort of a strange invention, where I can go to a market, 67 00:04:16,851 --> 00:04:21,596 I can sell something I don't have, I can take the money, but I have to promise 68 00:04:21,596 --> 00:04:27,270 that later on I'm going to give the, the person one of these things back. 69 00:04:27,270 --> 00:04:30,503 And the way that's going to manifest itself mathematically, is 70 00:04:30,503 --> 00:04:33,990 it just means there's no constraints on each individual Wi. 71 00:04:33,990 --> 00:04:36,465 So even though they have to sum up to 1, 72 00:04:36,465 --> 00:04:39,315 I could have one Wi greater than 1, and then I 73 00:04:39,315 --> 00:04:42,690 would also have to have some negative Wi's to balance 74 00:04:42,690 --> 00:04:45,713 that out, so that this sum is still equal to 1. 75 00:04:49,280 --> 00:04:51,830 Then I'm going to have mu IB, something 76 00:04:51,830 --> 00:04:54,870 called the expected rate of return on an asset. 77 00:04:56,420 --> 00:04:58,854 And so there will be one of these for each asset. 78 00:04:58,854 --> 00:05:01,040 So that's why it get's a little subscript i. 79 00:05:01,040 --> 00:05:03,112 So different assets in my portfolio. 80 00:05:03,112 --> 00:05:07,830 Some are going to have higher expected rates of return, than others. 81 00:05:07,830 --> 00:05:10,622 and generally, those are going to be the ones that are more risky, too. 82 00:05:10,622 --> 00:05:11,302 [COUGH] 83 00:05:11,302 --> 00:05:20,020 So you can think of sigma i squared then. So ui is the expected rate of return. 84 00:05:20,020 --> 00:05:23,660 Sigma i squared, is going to be the risk of asset i. 85 00:05:23,660 --> 00:05:26,507 And there's also a kind of related quantity, 86 00:05:26,507 --> 00:05:29,750 just sigma sub i, which I'll call the volatility. 87 00:05:29,750 --> 00:05:32,750 So you can think of this as sort of a variance 88 00:05:32,750 --> 00:05:37,330 like quantity, and then sigma is a standard deviation like quantity. 89 00:05:39,210 --> 00:05:41,860 And so again, I get one of these for each asset. 90 00:05:44,810 --> 00:05:46,471 Then, there'll also be row ij. 91 00:05:46,471 --> 00:05:48,385 So this is going to be the correlation 92 00:05:48,385 --> 00:05:50,959 between the returns on the different assets, 93 00:05:50,959 --> 00:05:54,390 and I'm going to end up with one row ij for every pair of assets. 94 00:05:54,390 --> 00:05:57,685 So that means that row ij is equal to row ji. 95 00:05:57,685 --> 00:05:58,210 [COUGH] 96 00:05:58,210 --> 00:06:01,780 And then once I have all of these inputs, 97 00:06:01,780 --> 00:06:07,580 I can make the expected return for the entire portfolio. 98 00:06:07,580 --> 00:06:11,684 It's just going to be a weigted, so a linear combination with the, 99 00:06:11,684 --> 00:06:15,700 the weights, from w of my expected returns on each of the assets. 100 00:06:17,050 --> 00:06:20,140 And the risk has this more slightly complicated formula. 101 00:06:20,140 --> 00:06:23,704 So you have the weight squared, 102 00:06:23,704 --> 00:06:28,710 times the risk of each asset. But then, you 103 00:06:28,710 --> 00:06:33,980 also have to add in, sort of those extra contributions to the risk that are 104 00:06:33,980 --> 00:06:39,259 coming from, from the correlation between all of the different assets. 105 00:06:46,000 --> 00:06:50,070 So, well I guess if just looking at this thing, it's not a very pretty formula. 106 00:06:50,070 --> 00:06:51,984 So then, first thing I want to do is see, 107 00:06:51,984 --> 00:06:54,020 is there a nicer way I can write this down? 108 00:06:54,020 --> 00:06:57,510 So I'll try doing the same thing using matrix notation. 109 00:06:57,510 --> 00:07:00,114 So now instead of having all the, you know, w1 up 110 00:07:00,114 --> 00:07:03,730 to wn, I'll just think about that as, as a single vector. 111 00:07:03,730 --> 00:07:10,418 And remember when I put a row like this in, in parentheses, I mean column vector. 112 00:07:10,418 --> 00:07:11,041 So I'll 113 00:07:11,041 --> 00:07:14,510 do one vector with the weights in it. 114 00:07:14,510 --> 00:07:17,220 A second vector with the expected returns in it. 115 00:07:17,220 --> 00:07:20,480 So I have a weight for each asset, and an expected return for each asset. 116 00:07:23,110 --> 00:07:26,220 Then I can write the expected return, for the entire portfolio. 117 00:07:26,220 --> 00:07:27,970 So that's what's on the left-hand side 118 00:07:27,970 --> 00:07:30,400 here, expected return here for the entire portfolio. 119 00:07:31,610 --> 00:07:35,250 And the previous slide, that was just the sum of Wi times mu i. 120 00:07:36,280 --> 00:07:40,120 But in matrix vector notation, I can just write this as W transpose mu. 121 00:07:43,170 --> 00:07:49,058 The risk, it turns out, that that sum that I had, that kind of ugly sum on the 122 00:07:49,058 --> 00:07:55,780 previous slide, can be expressed quite simply at just W transpose sigma W. 123 00:07:55,780 --> 00:07:59,470 And here this sigma, so this is not singular values, 124 00:07:59,470 --> 00:08:03,720 this is the covariance matrix for the returns of the assets. 125 00:08:07,170 --> 00:08:09,020 And so that's going to look like this. 126 00:08:09,020 --> 00:08:14,670 So on the diagonal, I have the, the risk of each individual asset. 127 00:08:14,670 --> 00:08:17,574 And then in the off diagonal entries, I 128 00:08:17,574 --> 00:08:21,446 end up with the volatility for here, for instance, 129 00:08:21,446 --> 00:08:25,142 the volatility of asset 1 times the volatility of 130 00:08:25,142 --> 00:08:29,514 asset 2 times the correlation between those two assets. 131 00:08:29,514 --> 00:08:32,234 And that's what's going to give me my 132 00:08:32,234 --> 00:08:34,410 covariance terms. 133 00:08:34,410 --> 00:08:38,776 And if you look back at that sum, I have this sort of strange notation of 134 00:08:38,776 --> 00:08:43,459 1, less than or equal to i, strictly less than j, less than or equal to n. 135 00:08:43,459 --> 00:08:46,219 And essentially what that's doing, I'm just 136 00:08:46,219 --> 00:08:48,691 trying to sum up over the lower triangle. 137 00:08:48,691 --> 00:08:53,430 So all of the values that are strictly below the main diagonal. 138 00:08:53,430 --> 00:08:54,930 And then I have a 2 in front of that, 139 00:08:54,930 --> 00:08:57,660 and that's just because this matrix is going to be symmetric. 140 00:08:57,660 --> 00:09:04,730 So if I, I have sigma12 row 12, it's the same as sigma 21 row 21. 141 00:09:04,730 --> 00:09:08,378 And so I'm basically, I'm summing the main 142 00:09:08,378 --> 00:09:12,470 diagonal, and then 2 times the lower triangle. 143 00:09:13,510 --> 00:09:15,490 And so that, that's what my formula is giving me. 144 00:09:16,500 --> 00:09:19,046 And so I, when I'd have this as a, a vector of 145 00:09:19,046 --> 00:09:22,798 weight times a variance, covariance matrix it, it gets a lot more 146 00:09:22,798 --> 00:09:24,000 compact notation. 147 00:09:27,280 --> 00:09:27,780 Okay. 148 00:09:30,690 --> 00:09:34,366 So if I want to use these two formulas I've come up with now. 149 00:09:34,366 --> 00:09:36,886 The, the mu and the Sigma, these are going to 150 00:09:36,886 --> 00:09:39,850 be properties of the assets that I'm looking at. 151 00:09:39,850 --> 00:09:43,555 There, there's something that generally, the, you know, 152 00:09:43,555 --> 00:09:46,790 generally the investor doesn't have any control over. 153 00:09:46,790 --> 00:09:50,710 You just have to look at historical data and say, you know, Boeing 154 00:09:50,710 --> 00:09:54,590 has been producing a return of so much for the last two years. 155 00:09:54,590 --> 00:09:56,098 You know, I think it'll probably 156 00:09:56,098 --> 00:09:57,879 stay about that for the next six months. 157 00:09:59,670 --> 00:10:03,180 but it's not something, I can't go to Boeing, for instance, 158 00:10:03,180 --> 00:10:06,300 and say be less risky to get this number to go down. 159 00:10:06,300 --> 00:10:10,670 you know, you're stuck with whatever you can, you can find out from the market. 160 00:10:10,670 --> 00:10:16,150 The thing you can select, are the weights in your portfolio. 161 00:10:16,150 --> 00:10:19,440 So the investor can select, you know, how much of Boeing am I going to buy? 162 00:10:19,440 --> 00:10:21,180 How much of Starbucks am I going to buy? 163 00:10:21,180 --> 00:10:23,080 How much of Microsoft am I going to buy? 164 00:10:24,500 --> 00:10:28,922 And, once they've decided on a particular, vector of weights w, then you can 165 00:10:28,922 --> 00:10:33,340 figure out what the portfolio return, and the portfolio risk are going to be. 166 00:10:36,280 --> 00:10:39,410 And so, how should you go about picking w. 167 00:10:39,410 --> 00:10:41,484 So we have two notions of optimality that 168 00:10:41,484 --> 00:10:43,924 I'm going to talk about, now, and later I'll 169 00:10:43,924 --> 00:10:47,927 show you, and so in the fifth session I'll show you how to solve one of these. 170 00:10:47,927 --> 00:10:52,160 Then hopefully at the end of week eight, we'll get around to solving the other one. 171 00:10:56,650 --> 00:11:01,049 So, I can try and design a portfolio to have a specific expected 172 00:11:01,049 --> 00:11:06,638 return. And then, once I have, once I've decided 173 00:11:06,638 --> 00:11:12,550 I want to have this specific expected return, then I, I choose w. 174 00:11:12,550 --> 00:11:14,290 So there going to be a whole bunch of 175 00:11:14,290 --> 00:11:17,610 choices, that will give me that one expected return. 176 00:11:17,610 --> 00:11:21,888 And among all of those, I would want to choose the portfolio that's going 177 00:11:21,888 --> 00:11:24,020 to have the minimum amount of risk. 178 00:11:26,800 --> 00:11:29,638 And so that'll, that'll give me an optimization 179 00:11:29,638 --> 00:11:31,616 problem, that I can use to find w. 180 00:11:31,616 --> 00:11:33,786 So I want to choose w so that the risk of 181 00:11:33,786 --> 00:11:37,960 the portfolio is minimum, given a certain level of expected return. 182 00:11:37,960 --> 00:11:43,875 And then I can solve sort of a related problem of, I could instead of targeting 183 00:11:43,875 --> 00:11:49,680 the expected return level, I could say, I want to have this much risk. 184 00:11:49,680 --> 00:11:52,048 So if, for some reason, I, you know, 185 00:11:52,048 --> 00:11:54,032 said, I, I want a very conservative 186 00:11:54,032 --> 00:11:56,778 portfolio, or I want a high-growth portfolio. 187 00:11:56,778 --> 00:11:59,286 I can communicate, you know, some number of 188 00:11:59,286 --> 00:12:02,790 some level of risk I'm willing to tolerate. 189 00:12:02,790 --> 00:12:07,206 And so if I can tolerate a certain level of risk, then I want to choose w so that 190 00:12:07,206 --> 00:12:10,173 my portfolio gives me the highest level of expected 191 00:12:10,173 --> 00:12:12,900 return, that I can get at that level of risk. 192 00:12:16,810 --> 00:12:18,811 And so both of these turn out 193 00:12:18,811 --> 00:12:23,389 to be something called constrained optimization problems. 194 00:12:23,389 --> 00:12:28,607 So the constraint, is I'm not trying to maximize the expected return. 195 00:12:28,607 --> 00:12:32,790 So that would be an unconstrained optimization problem. 196 00:12:32,790 --> 00:12:36,705 But I want to maximize my expected return, subject to 197 00:12:36,705 --> 00:12:41,840 a constraint oops, sorry I want to minimize my portfolio risk. 198 00:12:41,840 --> 00:12:46,940 So in the first one I want to minimize my risk, subject to the constraint that I'm 199 00:12:46,940 --> 00:12:52,780 hitting my target expected return. And in the second, the second task. 200 00:12:53,820 --> 00:12:55,516 So I have a target level of risk. 201 00:12:55,516 --> 00:12:59,510 So my constraint is, I want a certain level of risk in my portfolio. 202 00:12:59,510 --> 00:13:02,030 And now I want to maximize the expected return, 203 00:13:02,030 --> 00:13:04,590 that I can get at that level of risk. 204 00:13:04,590 --> 00:13:06,280 So they're optimization problems. 205 00:13:06,280 --> 00:13:06,940 I'm trying 206 00:13:06,940 --> 00:13:11,220 to find a minimum or a maximum, but I'm subject to a certain constraint. 207 00:13:11,220 --> 00:13:16,030 And then there's actually one more constraint, that's hanging over from a 208 00:13:16,030 --> 00:13:19,520 previous slide, is that the w vector has to sum up to 1. 209 00:13:19,520 --> 00:13:22,200 So, there's two constraints for each of these. 210 00:13:22,200 --> 00:13:25,896 So, the first one has expected return, has to be equal to 211 00:13:25,896 --> 00:13:29,730 some value, and my portfolio weights have to sum up to 1. 212 00:13:29,730 --> 00:13:32,370 And, satisfy those two constraints. 213 00:13:32,370 --> 00:13:36,350 And then find the one that has the minimum risk. 214 00:13:36,350 --> 00:13:40,270 And, and then sort of, related problem for the second goal. 215 00:13:42,080 --> 00:13:45,030 And see, generally the way I like to write these things down. 216 00:13:46,590 --> 00:13:50,050 So I have a, a minimum variance optimization. 217 00:13:50,050 --> 00:13:51,500 So, what I'm trying to do. 218 00:13:53,080 --> 00:13:56,540 This, this turns out to be something called the portfolio variance. 219 00:13:56,540 --> 00:13:57,661 I'm calling it risk, 220 00:13:57,661 --> 00:14:01,142 because that's a, it's a mathematical concept, that can be used to 221 00:14:01,142 --> 00:14:05,510 describe sort of, how spread out the return from my portfolio will be. 222 00:14:05,510 --> 00:14:09,310 So how, how far away from the expected return, do I expect it to be? 223 00:14:10,580 --> 00:14:12,809 So I want to minimize this quantity. 224 00:14:12,809 --> 00:14:16,400 So this is minimizing the variance or minimizing the risk. 225 00:14:16,400 --> 00:14:18,832 And then I write below that, subject to, and 226 00:14:18,832 --> 00:14:21,230 the I just list the constraints that I have. 227 00:14:22,640 --> 00:14:22,776 So 228 00:14:22,776 --> 00:14:25,680 I've been using e to be a vector of all one's. 229 00:14:25,680 --> 00:14:27,760 So e transposed w. 230 00:14:27,760 --> 00:14:31,870 That just means that the sum of w, has to be equal to 1. 231 00:14:31,870 --> 00:14:35,240 So my first constraint is that the weights sum up to 1. 232 00:14:35,240 --> 00:14:38,780 So I, I spend exactly the amount of money I have to invest. 233 00:14:41,770 --> 00:14:43,250 And then the second constraint. 234 00:14:44,320 --> 00:14:48,290 So this, I argued before was the expected return on my portfolio. 235 00:14:49,770 --> 00:14:54,189 And I'm trying to minimize the variance, at at given level of expected return. 236 00:14:54,189 --> 00:14:57,569 So I have to choose an expected level of, of a level of expected 237 00:14:57,569 --> 00:15:02,290 return, that I'm calling mu p, so that's the expected return for the portfolio. 238 00:15:03,530 --> 00:15:06,698 And I'm limiting myself to vectors w, 239 00:15:06,698 --> 00:15:12,055 that satisfy this linear equation here. So this is the n asset case. 240 00:15:12,055 --> 00:15:16,578 So, you can write it all out just, using matrices and vectors. 241 00:15:16,578 --> 00:15:21,170 And we know this doesn't get any more complicated, as n gets bigger. 242 00:15:21,170 --> 00:15:23,294 It's just the matrices and the vectors are 243 00:15:23,294 --> 00:15:26,818 getting bigger, but writing it down stays pretty compact. 244 00:15:26,818 --> 00:15:31,080 you can already see what this looks like just for the two asset case. 245 00:15:31,080 --> 00:15:31,728 And this is 246 00:15:31,728 --> 00:15:36,192 actually trivial because, if I look at the, the two constraints, I have 247 00:15:36,192 --> 00:15:40,693 two equations and two unknowns. So there's only one point that's going to 248 00:15:40,693 --> 00:15:46,852 satisfy the, the constraints here. So there's actually no minimization to do. 249 00:15:46,852 --> 00:15:49,085 There's just one one portfolio that I would have here. 250 00:15:49,085 --> 00:15:52,652 So, I just wanted to show what it looks like written out. 251 00:15:52,652 --> 00:15:56,981 So this a quadratic form, and it's just equal 252 00:15:56,981 --> 00:16:00,422 to what w transpose sigma w would look 253 00:16:00,422 --> 00:16:05,670 like, if you did the, did the matrix multiplication. 254 00:16:07,020 --> 00:16:11,430 And then my few constraints, are just the sum of the w's has to be 1. 255 00:16:11,430 --> 00:16:14,277 And the expected return of the portfolio, has 256 00:16:14,277 --> 00:16:17,700 to equal my pre-specified level of expected return. 257 00:16:19,990 --> 00:16:22,518 So then the other possibility, was to 258 00:16:22,518 --> 00:16:25,790 maximize the expected return of the portfolio. 259 00:16:26,800 --> 00:16:30,040 So, this was my expression for the expected return. 260 00:16:30,040 --> 00:16:33,172 So now I'm going to say maximize, and put the 261 00:16:33,172 --> 00:16:36,942 quantity that I want to maximize on the first line. 262 00:16:36,942 --> 00:16:40,680 And my constraints are going to be, well, I still 263 00:16:40,680 --> 00:16:43,770 have to have my vector w summing up to 1. 264 00:16:45,280 --> 00:16:48,230 But now I have to choose a target level of risk. 265 00:16:48,230 --> 00:16:50,933 So I have to, have to say who ever is going to do this portfolio 266 00:16:50,933 --> 00:16:54,650 optimization problem, you know, this is the risk level that I would like to have. 267 00:16:55,800 --> 00:16:59,560 And that's then going to give this quadratic form, as the constraint now. 268 00:16:59,560 --> 00:17:04,022 So I'm, I'm limited to choosing from vectors, that 269 00:17:04,022 --> 00:17:07,280 are going to give me this level of risk. 270 00:17:09,710 --> 00:17:15,260 And so if I write that out pretty much all that happens is the, the constraint here, 271 00:17:15,260 --> 00:17:20,900 or the, the objective here, the thing I'm trying to minimize, becomes a constraint. 272 00:17:22,740 --> 00:17:26,120 I still have the constraint w1 plus w2 equals 1. 273 00:17:27,510 --> 00:17:30,480 And now, it's just my expected return that I'm trying to optimize.