1 00:00:00,060 --> 00:00:02,500 The following content is provided under a Creative 2 00:00:02,500 --> 00:00:04,019 Commons license. 3 00:00:04,019 --> 00:00:06,360 Your support will help MIT OpenCourseWare 4 00:00:06,360 --> 00:00:10,730 continue to offer high quality educational resources for free. 5 00:00:10,730 --> 00:00:13,340 To make a donation or view additional materials 6 00:00:13,340 --> 00:00:17,217 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,217 --> 00:00:17,842 at ocw.mit.edu. 8 00:00:22,020 --> 00:00:24,903 PETER KEMPTHORNE: Today's topic is portfolio theory. 9 00:00:24,903 --> 00:00:32,360 An it's really one of the most important topics in finance. 10 00:00:32,360 --> 00:00:37,650 We're going to go through the historical theory of portfolio 11 00:00:37,650 --> 00:00:41,560 optimization, beginning with Markowitz Mean-Variance 12 00:00:41,560 --> 00:00:46,170 Optimization, where we look at portfolios in terms 13 00:00:46,170 --> 00:00:47,670 of their performance characteristics 14 00:00:47,670 --> 00:00:52,490 as determined by the mean return and the volatility returns. 15 00:00:52,490 --> 00:00:56,560 This analysis gets extended by looking at also investing 16 00:00:56,560 --> 00:00:58,590 with a risk-free asset. 17 00:00:58,590 --> 00:01:01,080 The initial theory on portfolio analysis 18 00:01:01,080 --> 00:01:03,370 didn't consider investing in cash, 19 00:01:03,370 --> 00:01:05,510 but just investing in risky assets. 20 00:01:05,510 --> 00:01:07,310 The problem changes quite dramatically when 21 00:01:07,310 --> 00:01:09,950 we add the risk-free asset. 22 00:01:09,950 --> 00:01:13,660 Then the topic of utility theory, von Neumann-Morgenstern 23 00:01:13,660 --> 00:01:19,360 utility theory, in statistical decision theory, 24 00:01:19,360 --> 00:01:22,710 we are trying to make decisions under uncertainty 25 00:01:22,710 --> 00:01:24,450 in a rational way. 26 00:01:24,450 --> 00:01:31,850 And von Neumann and Morgenstern developed a expected utility 27 00:01:31,850 --> 00:01:34,640 hypothesis for rational decision making. 28 00:01:34,640 --> 00:01:36,450 And that's really very important motivator 29 00:01:36,450 --> 00:01:41,700 for decision analysis generally and portfolio selection 30 00:01:41,700 --> 00:01:42,350 in particular. 31 00:01:42,350 --> 00:01:44,500 So we'll go through that. 32 00:01:44,500 --> 00:01:48,240 Then we'll turn to portfolio optimization constraints. 33 00:01:48,240 --> 00:01:51,015 There are realistic issues when you have-- well, 34 00:01:51,015 --> 00:01:53,460 there's how much money you have to invest, 35 00:01:53,460 --> 00:01:55,930 how much capital you have to invest, whether you could 36 00:01:55,930 --> 00:01:59,715 short securities or not, whether there are sort 37 00:01:59,715 --> 00:02:02,190 of limits in capacity of different assets 38 00:02:02,190 --> 00:02:03,200 you might want to trade. 39 00:02:03,200 --> 00:02:07,610 Those can come into play and affect the solutions. 40 00:02:07,610 --> 00:02:11,900 All of these methods are driven by estimates 41 00:02:11,900 --> 00:02:15,160 of how well we expect assets to do, 42 00:02:15,160 --> 00:02:18,910 so estimating returns or estimating alphas, 43 00:02:18,910 --> 00:02:22,470 and also estimating the volatilities and correlations 44 00:02:22,470 --> 00:02:25,130 amongst the assets were trading in. 45 00:02:25,130 --> 00:02:27,120 Most of the theory we'll talk about today deals 46 00:02:27,120 --> 00:02:30,740 with estimates-- well, with the problem where 47 00:02:30,740 --> 00:02:33,480 we're assuming basically certainty in terms 48 00:02:33,480 --> 00:02:36,800 of understanding those underlying return 49 00:02:36,800 --> 00:02:39,330 characteristics, the means, variances, correlations. 50 00:02:39,330 --> 00:02:43,520 But when we have to estimate those that raises problems. 51 00:02:43,520 --> 00:02:45,020 And finally, we'll finish up talking 52 00:02:45,020 --> 00:02:47,630 about some alternative risk measures, which 53 00:02:47,630 --> 00:02:50,240 extend the straightforward mean-- 54 00:02:50,240 --> 00:02:52,195 or simple mean-variance analysis. 55 00:02:59,730 --> 00:03:05,060 Now the mean-variance analysis is a single-period analysis. 56 00:03:05,060 --> 00:03:09,020 What we want to do is consider investing our capital 57 00:03:09,020 --> 00:03:11,160 in some assets. 58 00:03:11,160 --> 00:03:15,300 And we want to do so for just a single period 59 00:03:15,300 --> 00:03:18,760 and do so optimally. 60 00:03:18,760 --> 00:03:22,880 So in terms of notation, we'll consider risky assets 61 00:03:22,880 --> 00:03:26,050 indexed by i, 1 through m. 62 00:03:26,050 --> 00:03:29,970 Single-period returns will be a multivariate vector 63 00:03:29,970 --> 00:03:32,870 of those returns. 64 00:03:32,870 --> 00:03:37,520 The mean and covariance of these assets 65 00:03:37,520 --> 00:03:43,793 are represented in vector of returns and covariance matrix 66 00:03:43,793 --> 00:03:46,410 of squared volatilities, the variances 67 00:03:46,410 --> 00:03:47,930 of the assets along the diagonals, 68 00:03:47,930 --> 00:03:49,810 and the covariances on the off-diagonals. 69 00:03:52,430 --> 00:03:56,840 A portfolio will be represented as basically a weighting 70 00:03:56,840 --> 00:04:00,340 of our investment in these m assets. 71 00:04:00,340 --> 00:04:05,780 So we'll have a vector w, consisting of w_1 up to w_m. 72 00:04:05,780 --> 00:04:08,380 I'm giving the relative weights and absolute weights 73 00:04:08,380 --> 00:04:10,830 of those investments. 74 00:04:10,830 --> 00:04:14,110 We'll assume that we have one unit of capital to invest. 75 00:04:14,110 --> 00:04:16,579 So the sum of these weights equals 1. 76 00:04:16,579 --> 00:04:24,060 And the portfolio return has expected return 77 00:04:24,060 --> 00:04:28,420 given by the expectation of this linear combination 78 00:04:28,420 --> 00:04:30,990 of the returns, which is simply the same linear combination 79 00:04:30,990 --> 00:04:33,040 of the underlying expectations. 80 00:04:33,040 --> 00:04:36,310 And the variance of the portfolio 81 00:04:36,310 --> 00:04:39,950 is given by the variance of the weighted average 82 00:04:39,950 --> 00:04:42,510 of the individual returns, which is 83 00:04:42,510 --> 00:04:46,190 given by this quadratic form in the covariance matrix. 84 00:04:46,190 --> 00:04:51,350 So we've gone through this-- or you understand, I think, 85 00:04:51,350 --> 00:04:54,070 these calculations from our discussion in time series. 86 00:04:54,070 --> 00:04:57,170 And that's quite simple. 87 00:04:57,170 --> 00:05:02,310 Now before going into the theory for this, 88 00:05:02,310 --> 00:05:05,430 let's look at the problem. 89 00:05:05,430 --> 00:05:08,280 Let's look at the portfolio analysis problem 90 00:05:08,280 --> 00:05:10,460 in a very simplified setting. 91 00:05:10,460 --> 00:05:13,880 So we're just going to consider two assets 92 00:05:13,880 --> 00:05:16,770 and talk about optimal portfolios investing 93 00:05:16,770 --> 00:05:18,370 in these two assets. 94 00:05:18,370 --> 00:05:21,410 So m equals 2. 95 00:05:21,410 --> 00:05:25,616 The first asset has return R_1 with mean 0.15. 96 00:05:25,616 --> 00:05:28,725 Now I'm going to think of this as annualized return, so 97 00:05:28,725 --> 00:05:33,820 a 15% annualized return and a volatility of 25%. 98 00:05:33,820 --> 00:05:38,070 The second asset has expected return of 20% 99 00:05:38,070 --> 00:05:43,340 and a volatility of 30%, sigma_2. 100 00:05:43,340 --> 00:05:46,320 These assets are possibly correlated. 101 00:05:46,320 --> 00:05:50,080 We'll let rho denote the correlation between the two. 102 00:05:50,080 --> 00:05:53,660 And a portfolio-- basically all the portfolios 103 00:05:53,660 --> 00:05:57,810 we could invest in, so long as we are limited 104 00:05:57,810 --> 00:06:01,520 to a unit of capital and no shorting, 105 00:06:01,520 --> 00:06:04,360 are given by portfolios indexed by w, 106 00:06:04,360 --> 00:06:07,790 basically the amount of money we invest in the second asset. 107 00:06:07,790 --> 00:06:13,620 So R_w is w R_2 plus 1 minus w R_1. 108 00:06:13,620 --> 00:06:15,780 That's the return on the portfolio 109 00:06:15,780 --> 00:06:19,770 that invest in those proportions in the two assets. 110 00:06:19,770 --> 00:06:22,380 The expected return is this linear combination, 111 00:06:22,380 --> 00:06:25,540 which is given in matrix form in the previous page-- 112 00:06:25,540 --> 00:06:27,310 or in the lecture notes. 113 00:06:27,310 --> 00:06:32,120 And the variance, our squared volatility is what? 114 00:06:32,120 --> 00:06:35,670 It's 1 minus omega squared times the variance 115 00:06:35,670 --> 00:06:41,520 of R_1 plus w squared times the variance of R_2. 116 00:06:41,520 --> 00:06:45,760 Then we have a covariance term that comes into play. 117 00:06:50,130 --> 00:06:54,030 Now the mean-variance analysis that Markowitz addressed 118 00:06:54,030 --> 00:06:57,205 basically looks at the feasible portfolio set. 119 00:07:00,190 --> 00:07:02,890 We simplify things by saying let's focus 120 00:07:02,890 --> 00:07:06,770 just on the volatility or variance and the expected 121 00:07:06,770 --> 00:07:08,810 return of our portfolio. 122 00:07:08,810 --> 00:07:12,420 So we can define the feasible portfolio 123 00:07:12,420 --> 00:07:17,770 set as the collection of volatility return 124 00:07:17,770 --> 00:07:23,820 pairs for all the assets-- or for all the portfolios. 125 00:07:23,820 --> 00:07:28,980 And well, what is pi star, that collection? 126 00:07:28,980 --> 00:07:32,050 Basically what's the universe of possible portfolios 127 00:07:32,050 --> 00:07:33,069 we could construct? 128 00:07:33,069 --> 00:07:35,360 In this two dimensional case it's going to very simple. 129 00:07:35,360 --> 00:07:38,270 In multidimensional cases it's more complicated. 130 00:07:38,270 --> 00:07:40,250 But what portfolios are optimal? 131 00:07:40,250 --> 00:07:42,450 Sub-optimal? 132 00:07:42,450 --> 00:07:45,690 How do we choose or identify a particular portfolio 133 00:07:45,690 --> 00:07:46,700 to invest in? 134 00:07:46,700 --> 00:07:49,270 How's that choice made? 135 00:07:49,270 --> 00:07:53,990 And is there any special structure to these portfolios? 136 00:07:53,990 --> 00:07:55,490 What you'll see in the lecture today 137 00:07:55,490 --> 00:07:59,490 is that the Markowitz theory and extensions of that 138 00:07:59,490 --> 00:08:04,520 provide really elegant answers to these questions. 139 00:08:04,520 --> 00:08:08,190 But let's understand basically what's going on just 140 00:08:08,190 --> 00:08:11,150 with this two-asset case. 141 00:08:11,150 --> 00:08:15,800 And what I'm going to do here is just simulate 500 142 00:08:15,800 --> 00:08:20,200 weekly returns with different values of the correlation 143 00:08:20,200 --> 00:08:23,010 between the two assets. 144 00:08:23,010 --> 00:08:26,320 And we want to examine the cumulative returns 145 00:08:26,320 --> 00:08:26,955 of each asset. 146 00:08:29,490 --> 00:08:32,980 The asset returns in terms of their means, volatilities, 147 00:08:32,980 --> 00:08:35,580 and correlations, a plot of pi star, 148 00:08:35,580 --> 00:08:39,330 and the cumulative returns of each asset 149 00:08:39,330 --> 00:08:40,900 with a minimum variance portfolio. 150 00:08:40,900 --> 00:08:47,450 So let me just highlight here that what we want here 151 00:08:47,450 --> 00:08:54,210 is to be plotting sigma_omega verses alpha_omega. 152 00:08:56,940 --> 00:09:00,300 And we have two assets. 153 00:09:00,300 --> 00:09:09,660 We have basically sigma_1 and alpha_1 corresponds 154 00:09:09,660 --> 00:09:11,660 to our first asset. 155 00:09:11,660 --> 00:09:17,910 Sigma_2 and alpha_2 corresponds to our second asset. 156 00:09:17,910 --> 00:09:31,040 So this point here corresponds to w equals 0. 157 00:09:31,040 --> 00:09:39,150 And the other corresponds to w equals 1, so R_1 and R_2. 158 00:09:39,150 --> 00:09:40,810 Now, what's going to happen when we 159 00:09:40,810 --> 00:09:47,060 combine assets in a portfolio? 160 00:09:47,060 --> 00:09:54,705 Well, let's take a look at the simulation. 161 00:10:10,680 --> 00:10:18,210 OK, here is a graph of the simulated asset 162 00:10:18,210 --> 00:10:27,570 returns with basically a mean returns given by 15%, 25%, 163 00:10:27,570 --> 00:10:32,520 volatility is 20 to 30, and with an asset correlation of 0. 164 00:10:32,520 --> 00:10:35,400 So here's just a scatter plot of the weekly returns. 165 00:10:35,400 --> 00:10:38,586 There's basically no apparent correlation there. 166 00:10:38,586 --> 00:10:40,210 There actually is a sample correlation, 167 00:10:40,210 --> 00:10:44,580 because a sample from these distributions 168 00:10:44,580 --> 00:10:48,080 won't have perfectly 0 correlation. 169 00:10:48,080 --> 00:10:52,390 On top, we have graphs of the cumulative returns of the two 170 00:10:52,390 --> 00:10:55,200 individual assets. 171 00:10:55,200 --> 00:10:57,890 Rather obviously, the higher graph 172 00:10:57,890 --> 00:11:02,760 corresponds to the asset with higher return, which 173 00:11:02,760 --> 00:11:03,750 is asset 2. 174 00:11:03,750 --> 00:11:07,350 And the green corresponds to asset 1. 175 00:11:07,350 --> 00:11:09,860 The graph on the right, top right, 176 00:11:09,860 --> 00:11:15,820 is the graph of the feasible set as we allocate 177 00:11:15,820 --> 00:11:20,020 between asset 1 and asset 2. 178 00:11:20,020 --> 00:11:28,380 And so by the simulation, this curve 179 00:11:28,380 --> 00:11:32,430 corresponds to the feasible set of portfolios. 180 00:11:32,430 --> 00:11:41,220 And what's really remarkable is that we can get basically 181 00:11:41,220 --> 00:11:47,180 a reduction in the volatility of the portfolio 182 00:11:47,180 --> 00:11:51,310 without compromising, and in fact improving, 183 00:11:51,310 --> 00:11:54,250 the return the portfolio. 184 00:11:54,250 --> 00:11:59,260 So if we invest fully in asset 1, we're at this point. 185 00:11:59,260 --> 00:12:06,470 As we increase-- as we start allocating towards asset 2, 186 00:12:06,470 --> 00:12:11,120 not only does the return of asset of the portfolio go up, 187 00:12:11,120 --> 00:12:14,360 but volatility goes down. 188 00:12:14,360 --> 00:12:20,650 And let's see, in this simulation, what I've done 189 00:12:20,650 --> 00:12:26,345 is also plotted the return of the portfolio corresponding 190 00:12:26,345 --> 00:12:27,345 to the minimum variance. 191 00:12:30,160 --> 00:12:34,890 So let's just look the minimum variance 192 00:12:34,890 --> 00:12:42,020 portfolio for a minute. 193 00:12:42,020 --> 00:12:46,040 We have sigma squared w is equal to 1 194 00:12:46,040 --> 00:12:52,790 minus w squared plus w squared sigma_2 squared-- 195 00:12:52,790 --> 00:12:55,580 this is sigma 1 squared. 196 00:12:55,580 --> 00:12:58,520 And then plus 0 in the case there's 197 00:12:58,520 --> 00:13:01,160 no correlation between the two assets. 198 00:13:01,160 --> 00:13:06,720 If we want to minimize this portfolio volatility, 199 00:13:06,720 --> 00:13:13,150 we can take the derivative of that with respect to the weight 200 00:13:13,150 --> 00:13:16,770 and set that equal to 0. 201 00:13:16,770 --> 00:13:18,940 So what's that equal to? 202 00:13:18,940 --> 00:13:26,560 Well, it's-- let's see, did I do this right? 203 00:13:26,560 --> 00:13:40,080 Actually just with the previous notes-- sorry, 204 00:13:40,080 --> 00:13:41,600 R_1 is equal to 0. 205 00:13:41,600 --> 00:13:48,340 OK, so we have 2, 1 minus w sigma_1 squared times minus 1, 206 00:13:48,340 --> 00:13:54,260 plus 2 w sigma_2 squared is equal to 0. 207 00:13:54,260 --> 00:14:09,680 And so solving this, we get w is equal to-- well, let's 208 00:14:09,680 --> 00:14:15,670 see, you get 1 over sigma_1 squared-- sorry, 209 00:14:15,670 --> 00:14:18,740 1 over sigma_2 squared divided by 1 over sigma_1 210 00:14:18,740 --> 00:14:21,290 squared plus 1 over sigma_2 squared. 211 00:14:21,290 --> 00:14:24,800 If you solve this out, you basically 212 00:14:24,800 --> 00:14:29,250 get a weighting on the different assets, which weights them 213 00:14:29,250 --> 00:14:34,200 inversely proportional to their squared volatility. 214 00:14:34,200 --> 00:14:38,390 And with this graph here, you can 215 00:14:38,390 --> 00:14:43,760 see that the blue graph is a bit closer to asset 1 than to asset 216 00:14:43,760 --> 00:14:46,050 2's cumulative return. 217 00:14:46,050 --> 00:14:49,470 That corresponds to giving to a slightly higher weight to asset 218 00:14:49,470 --> 00:14:53,970 1, because 1 over sigma_1 squared 219 00:14:53,970 --> 00:14:57,840 is bigger than 1 over sigma_2 squared. 220 00:14:57,840 --> 00:15:02,190 Well, let's look at what happens if we 221 00:15:02,190 --> 00:15:04,630 consider negative correlations between the two assets. 222 00:15:08,000 --> 00:15:11,590 Well, actually, OK, before we do that, 223 00:15:11,590 --> 00:15:14,630 if you were going to choose one of these portfolios 224 00:15:14,630 --> 00:15:19,550 for investing, are there any portfolios 225 00:15:19,550 --> 00:15:20,840 that you wouldn't invest in? 226 00:15:23,928 --> 00:15:27,757 And are any of these portfolios sub-optimal 227 00:15:27,757 --> 00:15:28,840 in terms of mean-variance? 228 00:15:31,684 --> 00:15:33,110 AUDIENCE: w1. 229 00:15:33,110 --> 00:15:37,380 PETER KEMPTHORNE: Well, w1 is certainly infeasible 230 00:15:37,380 --> 00:15:39,915 because we can increase its mean and we 231 00:15:39,915 --> 00:15:42,280 can decrease it's volatility. 232 00:15:45,030 --> 00:15:48,270 So actually all of these points here 233 00:15:48,270 --> 00:15:52,060 are really sub-optimal portfolios. 234 00:15:52,060 --> 00:15:54,690 And from the minimum variance portfolio, 235 00:15:54,690 --> 00:15:59,530 which is basically getting us the vertical tangent, all 236 00:15:59,530 --> 00:16:04,255 the points from here up to the asset 2, fully 237 00:16:04,255 --> 00:16:08,440 investing in asset 2 are feasible. 238 00:16:08,440 --> 00:16:11,650 And none of these portfolios dominates the other. 239 00:16:11,650 --> 00:16:15,175 Basically it's a trade-off between return and volatility. 240 00:16:17,880 --> 00:16:21,890 OK, so that's a really important point that there really 241 00:16:21,890 --> 00:16:28,470 are portfolios that you can just disregard considering 242 00:16:28,470 --> 00:16:32,840 that there are definite-- well, when there's zero correlation, 243 00:16:32,840 --> 00:16:35,430 there is a benefit to diversifying across these two 244 00:16:35,430 --> 00:16:36,700 assets. 245 00:16:36,700 --> 00:16:41,830 So if you have an asset and you are considering 246 00:16:41,830 --> 00:16:45,070 pooling that asset with another in your portfolio, 247 00:16:45,070 --> 00:16:47,900 if those are fully uncorrelated, then 248 00:16:47,900 --> 00:16:51,540 you should be able to improve your portfolio 249 00:16:51,540 --> 00:16:55,720 by adding some allocation to that second asset. 250 00:16:55,720 --> 00:17:06,250 Let's look at the a more negative correlation 251 00:17:06,250 --> 00:17:08,960 between the two assets of minus 0.4. 252 00:17:08,960 --> 00:17:14,160 Here, you'll see that there's basically 253 00:17:14,160 --> 00:17:18,300 a tilt, negative tilt, to the scatter plot of returns. 254 00:17:18,300 --> 00:17:22,310 And with the portfolios of two assets, 255 00:17:22,310 --> 00:17:26,319 basically, the feasible set, that feasible set 256 00:17:26,319 --> 00:17:31,910 is actually stretching further to the left. 257 00:17:31,910 --> 00:17:35,880 So with a negative correlation between the assets, 258 00:17:35,880 --> 00:17:40,960 we're actually able to reduce the volatility even more, 259 00:17:40,960 --> 00:17:44,930 if that's what is our preference. 260 00:17:44,930 --> 00:17:52,830 If we go up here to minus 0.8, it gets even more exaggerated. 261 00:17:52,830 --> 00:17:56,160 Now what's going to happen if we have 262 00:17:56,160 --> 00:18:01,330 a correlation of negative 1 between these two assets? 263 00:18:01,330 --> 00:18:03,315 AUDIENCE: The portfolio can have 0? 264 00:18:03,315 --> 00:18:05,690 PETER KEMPTHORNE: Yes, then a portfolio of the two assets 265 00:18:05,690 --> 00:18:06,880 can have zero variability. 266 00:18:06,880 --> 00:18:14,550 So indeed with the hedging strategies, 267 00:18:14,550 --> 00:18:18,150 one often is considering investing in assets consisting 268 00:18:18,150 --> 00:18:22,910 of perhaps an underlying security and some derivative 269 00:18:22,910 --> 00:18:23,520 on that. 270 00:18:23,520 --> 00:18:26,740 And that derivative, if it's a future say, 271 00:18:26,740 --> 00:18:29,140 could have basically a negative 1 correlation. 272 00:18:29,140 --> 00:18:32,230 So you could essentially hedge out the volatility 273 00:18:32,230 --> 00:18:33,320 almost perfectly. 274 00:18:33,320 --> 00:18:37,900 So that special case actually does exist quite frequently 275 00:18:37,900 --> 00:18:40,120 and is exploited. 276 00:18:40,120 --> 00:18:43,070 But it falls out just from the simple analysis 277 00:18:43,070 --> 00:18:46,570 of this simple simulation. 278 00:18:46,570 --> 00:18:51,530 OK, let's also look at the going to increase in the correlation. 279 00:18:51,530 --> 00:18:52,030 Yes. 280 00:18:52,030 --> 00:18:54,960 AUDIENCE: So if the volatility of the whole portfolio 281 00:18:54,960 --> 00:18:58,630 is zero, does that it mean it will always be constant? 282 00:18:58,630 --> 00:18:59,750 It will be-- 283 00:18:59,750 --> 00:19:01,460 PETER KEMPTHORNE: Does that mean what? 284 00:19:01,460 --> 00:19:03,001 AUDIENCE: So would that mean since it 285 00:19:03,001 --> 00:19:06,100 has zero variance it will actually be constant over time? 286 00:19:06,100 --> 00:19:07,165 The whole portfolio? 287 00:19:07,165 --> 00:19:11,500 PETER KEMPTHORNE: OK, I mean, if it has zero variance, 288 00:19:11,500 --> 00:19:13,970 it should be constant. 289 00:19:13,970 --> 00:19:18,825 Zero variance means that over-- that its value is a constant. 290 00:19:21,570 --> 00:19:25,320 AUDIENCE: Is it a good idea to do that? 291 00:19:25,320 --> 00:19:31,110 PETER KEMPTHORNE: Well, in terms of the markets nothing 292 00:19:31,110 --> 00:19:34,940 has zero-- almost nothing has zero volatility. 293 00:19:37,490 --> 00:19:45,300 And so indeed-- but in the pricing theory that Choongbum 294 00:19:45,300 --> 00:19:48,890 will be discussing in subsequent lectures, 295 00:19:48,890 --> 00:19:53,690 if we have basically a portfolio which has no volatility, 296 00:19:53,690 --> 00:19:57,040 then its return should be equal to whatever 297 00:19:57,040 --> 00:19:59,320 a risk-free rate ought to be. 298 00:19:59,320 --> 00:20:02,000 And so this particular portfolio out 299 00:20:02,000 --> 00:20:04,030 to be structured so that it achieves 300 00:20:04,030 --> 00:20:08,310 a return equal to a risk-free rate, barring transaction costs 301 00:20:08,310 --> 00:20:10,280 and frictions and all that kind of thing. 302 00:20:10,280 --> 00:20:12,940 But yeah. 303 00:20:12,940 --> 00:20:21,800 OK, so if we increase the correlation from 0 to 0.4, 304 00:20:21,800 --> 00:20:26,130 well, we still get a benefit of diversification but less. 305 00:20:26,130 --> 00:20:30,240 We're basically not able to lower the variance as much. 306 00:20:30,240 --> 00:20:39,570 And it's even more exaggerated with a correlation a 0.8. 307 00:20:39,570 --> 00:20:45,402 Now, in looking at the simulation, 308 00:20:45,402 --> 00:20:47,110 one thing I want to highlight to you now, 309 00:20:47,110 --> 00:20:50,500 which will come up later, is here 310 00:20:50,500 --> 00:20:55,000 I've simulated returns according to Gaussian distributions 311 00:20:55,000 --> 00:20:57,610 with these means, volatilities, and correlations. 312 00:20:57,610 --> 00:21:03,006 And in the lower left panel, I give the sample statistics. 313 00:21:03,006 --> 00:21:04,380 Basically, the maximum likelihood 314 00:21:04,380 --> 00:21:08,140 estimate for all the parameters. 315 00:21:08,140 --> 00:21:13,720 And what you'll see is that these sample estimates differ 316 00:21:13,720 --> 00:21:16,730 quite a bit from-- well, they differ 317 00:21:16,730 --> 00:21:19,950 from the theoretical parameters. 318 00:21:19,950 --> 00:21:24,540 In this case, the sample volatility is almost exactly 319 00:21:24,540 --> 00:21:25,950 on point for the first asset. 320 00:21:25,950 --> 00:21:28,760 It's a bit below 0.287 for the second. 321 00:21:28,760 --> 00:21:32,380 The sample mean is 0.21, which is a bit high for the first. 322 00:21:32,380 --> 00:21:36,060 And it's 0.321, which is also a bit high for the second. 323 00:21:36,060 --> 00:21:40,440 Let's just go back and look at a couple of the others, 324 00:21:40,440 --> 00:21:47,440 because it can get-- OK, well here's 325 00:21:47,440 --> 00:21:55,880 one where the sample means are 0.144 and 0.343. 326 00:21:55,880 --> 00:22:00,500 And so those are actually quite different from the population 327 00:22:00,500 --> 00:22:02,450 parameters. 328 00:22:02,450 --> 00:22:07,160 And at least, the second asset has a much higher sample mean. 329 00:22:07,160 --> 00:22:11,410 So it's important to note that sample estimates can 330 00:22:11,410 --> 00:22:14,095 have a certain amount of variability. 331 00:22:14,095 --> 00:22:17,360 It turns out that there's less variability in estimating 332 00:22:17,360 --> 00:22:20,620 covariances and correlations, if the assumptions hold, 333 00:22:20,620 --> 00:22:23,880 and greater variability in the sample means. 334 00:22:23,880 --> 00:22:26,610 So at the end of the day we really need 335 00:22:26,610 --> 00:22:30,260 to be very sensitive to what estimates we use 336 00:22:30,260 --> 00:22:33,110 and how much uncertainty there is those. 337 00:22:36,470 --> 00:22:42,530 OK, let's go back to the lecture notes now. 338 00:22:59,400 --> 00:23:04,100 View, screen, all right. 339 00:23:08,900 --> 00:23:10,740 All right, so we've just gone through how 340 00:23:10,740 --> 00:23:14,100 we are evaluating different portfolios in terms 341 00:23:14,100 --> 00:23:20,130 of the pair of the-- basically the mean of the portfolio 342 00:23:20,130 --> 00:23:23,780 and the squared volatility of the portfolio. 343 00:23:23,780 --> 00:23:26,290 Higher expected returns are obviously desirable. 344 00:23:26,290 --> 00:23:28,690 Low volatility is desirable. 345 00:23:28,690 --> 00:23:33,320 And so what Markowitz did was to pose 346 00:23:33,320 --> 00:23:37,830 this is a quadratic programming problem, where 347 00:23:37,830 --> 00:23:45,250 what we want to do is minimize the squared volatility 348 00:23:45,250 --> 00:23:49,380 of the portfolio subject to a constraint on the mean 349 00:23:49,380 --> 00:23:53,180 of the portfolio and considering that we're fully invested. 350 00:23:53,180 --> 00:24:01,280 So this mathematical problem is a standard convex optimization 351 00:24:01,280 --> 00:24:04,370 problem, a quadratic programming problem, which is very simple. 352 00:24:04,370 --> 00:24:09,500 And we solve it by defining a Lagrangian. 353 00:24:09,500 --> 00:24:14,600 Basically, we take our objective function of the volatility. 354 00:24:14,600 --> 00:24:16,439 And we want to minimize that. 355 00:24:16,439 --> 00:24:17,980 We're going to use a half factor just 356 00:24:17,980 --> 00:24:20,370 to simplify the computations. 357 00:24:20,370 --> 00:24:24,630 And then we add Lagrangians for different constraints 358 00:24:24,630 --> 00:24:25,880 of the problem. 359 00:24:25,880 --> 00:24:29,840 So we have lambda_0 times alpha_0 minus w prime alpha. 360 00:24:29,840 --> 00:24:35,210 We want the mean return, w prime alpha, 361 00:24:35,210 --> 00:24:38,510 to be constrained to equal alpha naught. 362 00:24:38,510 --> 00:24:42,360 And we also want the sum of the weights, w prime 1_m 363 00:24:42,360 --> 00:24:43,140 to equal 1. 364 00:24:45,990 --> 00:24:48,640 And the first-order conditions of the Lagrangian 365 00:24:48,640 --> 00:24:50,340 basically give us those two constraints 366 00:24:50,340 --> 00:24:52,756 for differentiating with respect to lambda_1 and lambda_2. 367 00:24:55,490 --> 00:24:57,510 The initial first-order condition 368 00:24:57,510 --> 00:25:00,410 with respect to the portfolio weights 369 00:25:00,410 --> 00:25:03,380 basically allows us to solve for those weights. 370 00:25:06,500 --> 00:25:08,460 Now, this first-order condition is 371 00:25:08,460 --> 00:25:13,910 going to solve our problem, because if we 372 00:25:13,910 --> 00:25:20,020 take the second-order derivative of this Lagrangian, 373 00:25:20,020 --> 00:25:24,640 it's going to be sigma. 374 00:25:24,640 --> 00:25:34,930 So let me just point out if we take d squared L by dw dw 375 00:25:34,930 --> 00:25:40,540 prime, that's equal to our covariance matrix sigma. 376 00:25:40,540 --> 00:25:44,660 And that is a positive definite or positive semi-definite 377 00:25:44,660 --> 00:25:45,160 matrix. 378 00:25:45,160 --> 00:25:49,000 So we indeed are minimizing the problem. 379 00:25:49,000 --> 00:25:53,965 So this is just a generalization of basically a parabola 380 00:25:53,965 --> 00:25:55,420 in multi-dimensions. 381 00:25:55,420 --> 00:25:58,370 And we're trying to minimize that. 382 00:25:58,370 --> 00:26:01,300 Well, what's the solution to this? 383 00:26:01,300 --> 00:26:05,850 Well, first we can solve for w, the weights, 384 00:26:05,850 --> 00:26:08,230 in terms of lambda_1 and lambda_2. 385 00:26:08,230 --> 00:26:12,840 So we take the first equation of the first-order conditions 386 00:26:12,840 --> 00:26:17,970 and basically pre-multiply by sigma inverse across. 387 00:26:17,970 --> 00:26:21,570 And we get w_0 is equal to lambda_1 sigma inverse alpha 388 00:26:21,570 --> 00:26:26,170 plus lambda_2 sigma inverse the unit vector, 389 00:26:26,170 --> 00:26:27,055 or vector of units. 390 00:26:29,630 --> 00:26:33,440 And then we can just solve for lambda_1 and lambda_2 391 00:26:33,440 --> 00:26:41,340 by plugging this w_0 solution into the second two equations. 392 00:26:41,340 --> 00:26:45,410 And these two equations for the second and third first-order 393 00:26:45,410 --> 00:26:52,035 conditions is just a very simple set of linear equations. 394 00:26:52,035 --> 00:27:01,930 We basically have alpha_0 1 is equal to some matrix times 395 00:27:01,930 --> 00:27:03,960 [lambda_1, lambda_2]. 396 00:27:03,960 --> 00:27:07,460 And that matrix, a, b, c, is given 397 00:27:07,460 --> 00:27:12,390 by alpha prime sigma inverse alpha for the a element. 398 00:27:12,390 --> 00:27:15,200 And the b and c elements are those corresponding elements. 399 00:27:15,200 --> 00:27:17,305 So this solves the problem. 400 00:27:23,500 --> 00:27:27,170 And the variance of the optimal portfolio with a given return 401 00:27:27,170 --> 00:27:32,110 can also be solved by just substituting 402 00:27:32,110 --> 00:27:33,830 in these solutions. 403 00:27:33,830 --> 00:27:37,840 So we get that for a given alpha_0, 404 00:27:37,840 --> 00:27:43,400 our target return, the squared volatility 405 00:27:43,400 --> 00:27:49,530 of that optimal portfolio is this has that form. 406 00:27:49,530 --> 00:27:51,840 Now, what's that form? 407 00:27:51,840 --> 00:27:57,280 It's essentially a parabola in alpha_0. 408 00:27:57,280 --> 00:28:04,860 OK, so when we're looking at-- is there an eraser? 409 00:28:04,860 --> 00:28:09,970 here it is-- when we're looking at this graph 410 00:28:09,970 --> 00:28:12,140 here in the two-dimensional case, 411 00:28:12,140 --> 00:28:14,260 then there's basically a parabola. 412 00:28:16,876 --> 00:28:18,250 I can't draw parabolas very well. 413 00:28:18,250 --> 00:28:20,790 But there's a parabola between those two points 414 00:28:20,790 --> 00:28:23,000 that characterizes the thing. 415 00:28:23,000 --> 00:28:25,590 And in multi-dimensional-- multiple assets, 416 00:28:25,590 --> 00:28:28,746 it's just a multivariate extension of that. 417 00:28:36,600 --> 00:28:45,080 This particular problem can also be looked at in two other ways. 418 00:28:45,080 --> 00:28:47,460 Before we're looking at minimizing the variance, 419 00:28:47,460 --> 00:28:50,570 subject to a constraint on the expected return. 420 00:28:50,570 --> 00:28:52,170 We can also say, well, let's maximize 421 00:28:52,170 --> 00:28:57,000 the return, subject to a constraint on the volatility. 422 00:28:57,000 --> 00:29:05,220 And that problem basically has the same Lagrangian. 423 00:29:05,220 --> 00:29:12,920 And we can also consider just maximizing a weighted average 424 00:29:12,920 --> 00:29:19,840 of the return and have a negative multiple 425 00:29:19,840 --> 00:29:27,500 on the variance and consider maximizing that expression. 426 00:29:27,500 --> 00:29:33,150 This turns out to be the risk aversion optimization, where 427 00:29:33,150 --> 00:29:42,360 we basically are penalizing portfolios w for how 428 00:29:42,360 --> 00:29:45,770 much variance they have. 429 00:29:45,770 --> 00:29:48,490 And the lambda factor tells us how much penalty 430 00:29:48,490 --> 00:29:50,840 to associate per variance unit. 431 00:29:53,970 --> 00:29:59,110 And these are all equivalent problems being 432 00:29:59,110 --> 00:30:02,530 solved by the same Lagrangian. 433 00:30:02,530 --> 00:30:09,570 And from these problems we can define the efficient frontier, 434 00:30:09,570 --> 00:30:14,710 which is the collection of all possible solutions, where 435 00:30:14,710 --> 00:30:20,440 we range the target return amongst values that 436 00:30:20,440 --> 00:30:23,960 are feasible and the volatility amongst values 437 00:30:23,960 --> 00:30:26,550 that are feasible as well. 438 00:30:26,550 --> 00:30:31,560 So the efficient frontier will just trace these. 439 00:30:31,560 --> 00:30:41,620 And in our two-variable case, our two-asset case-- 440 00:30:41,620 --> 00:30:44,120 we have sigma and alpha. 441 00:30:44,120 --> 00:30:45,670 We have two assets. 442 00:30:45,670 --> 00:30:48,930 It's basically a parabola like that. 443 00:30:48,930 --> 00:30:55,870 If we have another two assets that we're investing with, 444 00:30:55,870 --> 00:30:59,600 then if we consider the two-asset portfolios of just 445 00:30:59,600 --> 00:31:01,540 these two assets, there's basically 446 00:31:01,540 --> 00:31:04,870 another parabola there. 447 00:31:04,870 --> 00:31:09,520 And as we consider different other assets in the mix, 448 00:31:09,520 --> 00:31:11,250 we basically get all these parabolas 449 00:31:11,250 --> 00:31:12,390 of two-asset portfolios. 450 00:31:12,390 --> 00:31:16,230 And then combinations of those two-asset portfolio 451 00:31:16,230 --> 00:31:20,650 gives us our feasible set. 452 00:31:20,650 --> 00:31:23,190 And so at the end of the day, we basically 453 00:31:23,190 --> 00:31:30,450 have a convex set of all feasible assets, which 454 00:31:30,450 --> 00:31:33,970 define the efficient frontier. 455 00:31:33,970 --> 00:31:36,090 And the efficient frontier is always 456 00:31:36,090 --> 00:31:43,620 going to be basically the top side of that curve. 457 00:31:43,620 --> 00:31:55,600 Well, let's see, the next topic considers 458 00:31:55,600 --> 00:31:59,117 adding basically a risk-free asset to invest in. 459 00:31:59,117 --> 00:32:00,700 The problem as it's been stated so far 460 00:32:00,700 --> 00:32:04,770 says we want to be fully invested across our m assets. 461 00:32:04,770 --> 00:32:07,900 And what are the optimal portfolios? 462 00:32:07,900 --> 00:32:15,700 Well, what if we want to invest some of our money, our capital, 463 00:32:15,700 --> 00:32:16,810 just in cash? 464 00:32:16,810 --> 00:32:19,790 Or not invest our full capital and invest 465 00:32:19,790 --> 00:32:22,800 the rest in the portfolio? 466 00:32:22,800 --> 00:32:31,660 So if we consider adding a risk-free asset, 467 00:32:31,660 --> 00:32:32,740 then this is an asset. 468 00:32:32,740 --> 00:32:35,430 We'll call it the 0th asset. 469 00:32:35,430 --> 00:32:39,320 It has, say, some expected return or not. 470 00:32:39,320 --> 00:32:40,050 It's risk free. 471 00:32:40,050 --> 00:32:41,520 So that's a constant. 472 00:32:41,520 --> 00:32:44,630 It has zero variance. 473 00:32:44,630 --> 00:32:49,160 And if we're investing as well in that possible risky asset, 474 00:32:49,160 --> 00:32:53,270 then we can basically consider investing w weights 475 00:32:53,270 --> 00:32:56,520 in the risky asset and 1 minus w times 476 00:32:56,520 --> 00:33:01,590 the unit vector in the risk-free asset. 477 00:33:01,590 --> 00:33:05,545 And let's see, I want to draw this graph here. 478 00:33:09,350 --> 00:33:10,550 Let's see. 479 00:33:16,570 --> 00:33:21,420 OK, suppose we have two assets where 480 00:33:21,420 --> 00:33:24,660 this is the efficient set. 481 00:33:24,660 --> 00:33:29,870 And we consider now allowing for a risk-free asset 482 00:33:29,870 --> 00:33:32,330 to be invested in as well. 483 00:33:32,330 --> 00:33:37,690 Well, that risk-free asset is basically this point here. 484 00:33:40,530 --> 00:33:46,540 And it has mean r_0 and variance 0. 485 00:33:46,540 --> 00:33:53,840 Now, if we combine this asset with any of these portfolios, 486 00:33:53,840 --> 00:33:55,530 what's the feasible set going to be? 487 00:33:58,150 --> 00:34:04,010 Well, we can basically invest some money 488 00:34:04,010 --> 00:34:07,220 in the risk-free asset and some in asset 2. 489 00:34:10,790 --> 00:34:14,179 So we can get any point along this line. 490 00:34:14,179 --> 00:34:18,780 Basically, if we invest in the risk-free asset-- 491 00:34:18,780 --> 00:34:23,760 if we invest some fraction in asset 2 492 00:34:23,760 --> 00:34:27,100 and the remainder in the risk-free asset, 493 00:34:27,100 --> 00:34:33,150 then our expected return is the linear line 494 00:34:33,150 --> 00:34:37,760 between r_0 and alpha_2. 495 00:34:37,760 --> 00:34:42,500 And the volatility of a weighted average-- 496 00:34:42,500 --> 00:34:47,530 of a fraction of asset 2 has a volatility 497 00:34:47,530 --> 00:34:52,050 given by whatever that weight is times the volatility. 498 00:34:52,050 --> 00:35:05,430 So this point here corresponds to, say, w times sigma_2 499 00:35:05,430 --> 00:35:15,860 for its volatility and r_0 plus w times alpha_2 minus r_0. 500 00:35:20,500 --> 00:35:23,750 Now the dramatic thing is that we actually 501 00:35:23,750 --> 00:35:31,310 are able to achieve an improvement over points 502 00:35:31,310 --> 00:35:35,630 on the efficient frontier, such as, say, the minimum variance 503 00:35:35,630 --> 00:35:38,258 portfolio. 504 00:35:38,258 --> 00:35:42,030 And if we consider the fraction here, 505 00:35:42,030 --> 00:35:46,400 we have a higher return and a lower variance 506 00:35:46,400 --> 00:35:48,590 than the minimum variance portfolio 507 00:35:48,590 --> 00:35:51,760 that we considered before. 508 00:35:51,760 --> 00:35:53,890 And so investing in the risk-free asset 509 00:35:53,890 --> 00:35:56,980 does sort of enlarge our opportunity 510 00:35:56,980 --> 00:35:59,190 space quite dramatically. 511 00:35:59,190 --> 00:36:03,130 And there are some very special results that come from this. 512 00:36:08,530 --> 00:36:10,580 So let's go through the mathematics 513 00:36:10,580 --> 00:36:12,420 for solving this problem. 514 00:36:12,420 --> 00:36:14,000 And in the lecture notes, I'm going 515 00:36:14,000 --> 00:36:18,020 to go over this pretty quickly-- really, 516 00:36:18,020 --> 00:36:19,730 when you go through it slowly you'll say, 517 00:36:19,730 --> 00:36:22,070 oh, this is very straightforward and logical. 518 00:36:24,740 --> 00:36:27,240 So we're basically going to minimize the volatility subject 519 00:36:27,240 --> 00:36:31,510 to the constraint that the return is equal to alpha_0. 520 00:36:31,510 --> 00:36:36,780 We define the Lagrangian and do the first-order conditions, 521 00:36:36,780 --> 00:36:38,210 solve those. 522 00:36:38,210 --> 00:36:44,010 And if we solve those, we get-- basically lambda_1 523 00:36:44,010 --> 00:36:45,520 has this nice form here. 524 00:36:45,520 --> 00:36:48,890 And this is obtained by just very simple equation 525 00:36:48,890 --> 00:36:51,120 solving of those two first-order conditions. 526 00:36:55,390 --> 00:37:01,860 And if we have m assets available 527 00:37:01,860 --> 00:37:08,710 when we look at the solution, then we 528 00:37:08,710 --> 00:37:13,480 get an optimal portfolio that has 529 00:37:13,480 --> 00:37:18,420 a w_0 vector given by this solution and a lambda_1 530 00:37:18,420 --> 00:37:19,440 given by this solution. 531 00:37:19,440 --> 00:37:23,610 Now w_0 is the proportion or is the allocation 532 00:37:23,610 --> 00:37:24,620 to the risky assets. 533 00:37:27,360 --> 00:37:35,120 What varies depending on our target return alpha_naught, 534 00:37:35,120 --> 00:37:43,090 is simply this lambda_1 Lagrangian multiplier. 535 00:37:43,090 --> 00:37:48,010 Basically, alpha_0, our target return, 536 00:37:48,010 --> 00:37:50,900 only affects the value of lambda_1. 537 00:37:50,900 --> 00:37:55,520 And our weights across the risky asset 538 00:37:55,520 --> 00:38:03,080 is a simple multiple of a fixed vector of relative weights. 539 00:38:03,080 --> 00:38:06,740 And this fixed vector of relative weights 540 00:38:06,740 --> 00:38:11,450 is the inverse of a covariance matrix across the assets 541 00:38:11,450 --> 00:38:16,590 times alpha minus 1_m R_0. 542 00:38:16,590 --> 00:38:23,200 And so what we have is a portfolio 543 00:38:23,200 --> 00:38:29,950 that basically invests in the risky assets in the same way. 544 00:38:29,950 --> 00:38:32,860 The only thing that differs is how much weight 545 00:38:32,860 --> 00:38:35,520 we give to that particular portfolio. 546 00:38:35,520 --> 00:38:40,129 As we increase lambda_1, then we give more allocation 547 00:38:40,129 --> 00:38:41,420 to this fixed vector of weight. 548 00:38:41,420 --> 00:38:43,580 So we invest proportionally in the assets. 549 00:38:43,580 --> 00:38:47,460 And we just scale how much that overall factor 550 00:38:47,460 --> 00:38:49,815 is to achieve different levels of return. 551 00:38:55,130 --> 00:38:59,530 OK, we can get nice, closed-form expressions for the portfolio 552 00:38:59,530 --> 00:39:08,590 variance just as an extension of the two-asset case. 553 00:39:08,590 --> 00:39:12,160 Basically the portfolio variance is a parabola 554 00:39:12,160 --> 00:39:15,520 in the target return. 555 00:39:15,520 --> 00:39:20,660 And so as we increase our target return, 556 00:39:20,660 --> 00:39:24,220 we have to increase our portfolio variability. 557 00:39:24,220 --> 00:39:27,135 There is that trade-off when we use optimal portfolios. 558 00:39:34,750 --> 00:39:40,890 And if we consider the fully invested optimal portfolio, 559 00:39:40,890 --> 00:39:48,000 well, the fully invested portfolio-- and that's 560 00:39:48,000 --> 00:39:49,560 nothing in cash, invest everything 561 00:39:49,560 --> 00:39:52,960 in the risky assets-- we actually 562 00:39:52,960 --> 00:39:55,350 are going to call that the market portfolio. 563 00:39:58,030 --> 00:40:01,650 And the expressions here give us the weights 564 00:40:01,650 --> 00:40:06,030 for that market portfolio. 565 00:40:06,030 --> 00:40:13,054 And this may look a little complicated 566 00:40:13,054 --> 00:40:14,220 in terms of the expressions. 567 00:40:14,220 --> 00:40:16,730 But it's actually quite simple closed-form expressions 568 00:40:16,730 --> 00:40:21,165 for the expected return and the variance 569 00:40:21,165 --> 00:40:22,165 of the market portfolio. 570 00:40:25,070 --> 00:40:37,540 And what's happening with this is we have-- basically, 571 00:40:37,540 --> 00:40:43,300 for the risky assets we have some efficient or set 572 00:40:43,300 --> 00:40:47,770 of feasible portfolios, which is all in this range. 573 00:40:47,770 --> 00:40:55,430 And then with our risky asset, we could be investing here. 574 00:40:55,430 --> 00:41:03,720 And so the optimal portfolio, in the case 575 00:41:03,720 --> 00:41:09,120 where we can invest in the risk-free asset, 576 00:41:09,120 --> 00:41:12,930 will basically correspond to sort 577 00:41:12,930 --> 00:41:18,420 of maximizing the mean return across all portfolios. 578 00:41:18,420 --> 00:41:20,880 So it's actually going to correspond 579 00:41:20,880 --> 00:41:30,840 to the simple tangent line, which basically crosses 580 00:41:30,840 --> 00:41:35,780 the market portfolio, which is the fully invested portfolio. 581 00:41:44,670 --> 00:41:50,860 Now, this structure of the problem 582 00:41:50,860 --> 00:41:54,760 is actually incredibly powerful. 583 00:41:54,760 --> 00:41:57,830 And there's an important paper by Tobin, 584 00:41:57,830 --> 00:42:02,540 which states that basically every portfolio is going 585 00:42:02,540 --> 00:42:07,870 to-- under these assumptions for the investment problem-- 586 00:42:07,870 --> 00:42:10,080 every optimal portfolio invests in a combination 587 00:42:10,080 --> 00:42:14,840 of the risk-free asset and the market portfolio. 588 00:42:14,840 --> 00:42:25,600 So regardless how much risk you want to take, 589 00:42:25,600 --> 00:42:28,970 the optimal portfolio is essentially the same. 590 00:42:28,970 --> 00:42:31,240 It just depends on how much capital 591 00:42:31,240 --> 00:42:34,880 you're going to put into that optimal portfolio. 592 00:42:34,880 --> 00:42:39,510 And so all of the optimal portfolios 593 00:42:39,510 --> 00:42:41,320 will invest in the same risky assets 594 00:42:41,320 --> 00:42:44,510 as the market portfolio in same proportions. 595 00:42:44,510 --> 00:42:48,570 And the only difference is their total weight. 596 00:42:48,570 --> 00:42:53,350 Now, plugging in the expressions for the different Lagrange 597 00:42:53,350 --> 00:43:00,510 multipliers for a given portfolio, P, with return-- 598 00:43:00,510 --> 00:43:01,870 and we get that expression. 599 00:43:01,870 --> 00:43:05,060 So let me just summarize that. 600 00:43:05,060 --> 00:43:09,140 So suppose we want the portfolio P such 601 00:43:09,140 --> 00:43:17,660 that the expectation of our P is equal to alpha_0, 602 00:43:17,660 --> 00:43:42,950 then this portfolio R_P is going to equal-- basically, it's 603 00:43:42,950 --> 00:43:46,210 going to invest in the risk-free asset with 1 604 00:43:46,210 --> 00:43:59,240 minus 1 transpose w_m plus w_m transpose-- 605 00:43:59,240 --> 00:44:06,250 I'm sorry, w_m times the return of the market. 606 00:44:06,250 --> 00:44:10,840 And the actual weights are given by this expression 607 00:44:10,840 --> 00:44:19,162 here, which gives us the expression that I was just 608 00:44:19,162 --> 00:44:20,162 writing up on the board. 609 00:44:25,250 --> 00:44:30,280 We have the expected return is r_0 plus w_m times market 610 00:44:30,280 --> 00:44:31,840 return minus r_0. 611 00:44:31,840 --> 00:44:35,510 And the volatility, sigma squared P, 612 00:44:35,510 --> 00:44:38,740 is simply the square of the weight in the market 613 00:44:38,740 --> 00:44:39,960 times the market volatility. 614 00:44:47,430 --> 00:44:54,260 This leads to the definition of basically the capital market 615 00:44:54,260 --> 00:45:05,100 line, which is essentially this line here. 616 00:45:05,100 --> 00:45:11,860 This is our capital market line for the portfolio optimization. 617 00:45:11,860 --> 00:45:15,510 And the structure of this line is such 618 00:45:15,510 --> 00:45:19,880 that this point for the market portfolio that 619 00:45:19,880 --> 00:45:24,070 has volatility given by the market portfolio's volatility 620 00:45:24,070 --> 00:45:29,860 sigma_m and the return on the market alpha_m, 621 00:45:29,860 --> 00:45:34,120 or expected value of R_m. 622 00:45:34,120 --> 00:45:48,060 And the slope of this line is given by alpha_m minus r_0 623 00:45:48,060 --> 00:45:51,480 over sigma_m. 624 00:45:51,480 --> 00:45:59,730 And points along the line are given by r_0 plus sigma_p 625 00:45:59,730 --> 00:46:00,810 times this factor. 626 00:46:00,810 --> 00:46:06,930 So this is the expected value of R_p 627 00:46:06,930 --> 00:46:11,740 is any optimal portfolio has return 628 00:46:11,740 --> 00:46:18,260 equal to the risk-free rate plus a multiple 629 00:46:18,260 --> 00:46:22,090 of the return per risk of the market portfolio. 630 00:46:24,860 --> 00:46:33,820 This term here is called the price of market risk. 631 00:46:33,820 --> 00:46:34,402 Yes? 632 00:46:34,402 --> 00:46:36,360 AUDIENCE: Is that the same as the Sharpe ratio? 633 00:46:36,360 --> 00:46:41,070 PETER KEMPTHORNE: It's close to the Sharpe ratio. 634 00:46:41,070 --> 00:46:46,550 The Sharpe ratio is-- yes, this is the Sharpe 635 00:46:46,550 --> 00:46:49,425 ratio for the market portfolio. 636 00:46:52,800 --> 00:46:59,540 And so, if we want to invest in the market, 637 00:46:59,540 --> 00:47:01,660 our decision is reduced to how much 638 00:47:01,660 --> 00:47:03,770 risk are we willing to take. 639 00:47:03,770 --> 00:47:07,970 And the compensation for taking extra risk is extra return. 640 00:47:07,970 --> 00:47:10,350 And we achieve that by essentially investing 641 00:47:10,350 --> 00:47:14,490 in the same portfolio, only we change the scale at which we're 642 00:47:14,490 --> 00:47:15,700 investing in that portfolio. 643 00:47:24,760 --> 00:47:27,515 So far we've been considering sort 644 00:47:27,515 --> 00:47:33,500 of points between being fully invested in the market 645 00:47:33,500 --> 00:47:35,570 or fully invested in cash. 646 00:47:35,570 --> 00:47:41,960 If we are able to borrow money at the risk-free rate, 647 00:47:41,960 --> 00:47:45,175 then we can basically allocate additional weights 648 00:47:45,175 --> 00:47:49,360 to the market portfolio and achieve 649 00:47:49,360 --> 00:47:54,920 points that are beyond-- have higher return, 650 00:47:54,920 --> 00:47:56,850 higher volatility than the market portfolio. 651 00:47:56,850 --> 00:47:58,900 We can basically lever the strategy 652 00:47:58,900 --> 00:48:05,030 by borrowing money and investing that in this market portfolio. 653 00:48:05,030 --> 00:48:14,880 So the efficient frontier, if we can borrow risklessly, 654 00:48:14,880 --> 00:48:17,430 would just be the capital market line extended. 655 00:48:21,940 --> 00:48:24,050 Here's a listing of the papers that 656 00:48:24,050 --> 00:48:27,450 go through sort of the classical foundations for this. 657 00:48:27,450 --> 00:48:29,712 These are all accessible on the web. 658 00:48:29,712 --> 00:48:31,920 And I encourage you to actually look at these papers, 659 00:48:31,920 --> 00:48:36,110 because the arguments are very straightforward. 660 00:48:36,110 --> 00:48:41,660 The motivation and background is interesting to read. 661 00:48:41,660 --> 00:48:45,310 Virtually everyone on this page actually has a Nobel Prize, 662 00:48:45,310 --> 00:48:48,900 except Lintner, I think he died before they gave the Nobel 663 00:48:48,900 --> 00:48:53,060 Prize to Markowitz and Sharpe, but he certainly 664 00:48:53,060 --> 00:48:55,035 would've been included in this case. 665 00:48:55,035 --> 00:48:57,194 AUDIENCE: And the latest one. 666 00:48:57,194 --> 00:48:58,360 PETER KEMPTHORNE: Pardon me? 667 00:48:58,360 --> 00:48:59,310 AUDIENCE: The latest. 668 00:48:59,310 --> 00:49:01,476 PETER KEMPTHORNE: And the latest one, yes, yes, Fama 669 00:49:01,476 --> 00:49:04,110 was just awarded, what, two weeks ago. 670 00:49:07,830 --> 00:49:17,410 All right, let's move on to von Neumann-Morgenstern Utility 671 00:49:17,410 --> 00:49:17,910 Theory. 672 00:49:23,130 --> 00:49:27,070 Now, in the Markowitz mean-variance analysis 673 00:49:27,070 --> 00:49:29,110 we sort of reduced all portfolios 674 00:49:29,110 --> 00:49:31,850 to the properties of what's their expected return 675 00:49:31,850 --> 00:49:35,805 and what's their variance or volatility of the returns. 676 00:49:39,350 --> 00:49:43,450 Under what circumstances would that be a really good decision 677 00:49:43,450 --> 00:49:47,930 to be making for how to do portfolio optimization, 678 00:49:47,930 --> 00:49:49,950 portfolio allocations? 679 00:49:49,950 --> 00:49:54,180 Well, the von Neumann-Morgenstern theory 680 00:49:54,180 --> 00:50:00,480 is a theory which motivates making decisions 681 00:50:00,480 --> 00:50:06,120 under uncertainty, where you should specify a utility 682 00:50:06,120 --> 00:50:12,090 function for your wealth, and you 683 00:50:12,090 --> 00:50:15,430 should engage in decisions that maximize the expected 684 00:50:15,430 --> 00:50:18,540 utility of your wealth. 685 00:50:18,540 --> 00:50:26,120 And the theory is really very powerful 686 00:50:26,120 --> 00:50:33,870 in that when you're making decisions under uncertainty, 687 00:50:33,870 --> 00:50:37,210 there are sort of rational things you should do. 688 00:50:40,960 --> 00:50:45,440 If you like higher return, the decision 689 00:50:45,440 --> 00:50:48,880 should be consistent with preferring outcomes 690 00:50:48,880 --> 00:50:49,930 with higher returns. 691 00:50:49,930 --> 00:50:52,200 If you don't like variability, then you 692 00:50:52,200 --> 00:50:54,270 should be preferring returns that 693 00:50:54,270 --> 00:50:58,990 have the same expected return, that have lower volatility. 694 00:50:58,990 --> 00:51:01,700 But depending on how your utility function is defined, 695 00:51:01,700 --> 00:51:04,460 you may get different outcomes. 696 00:51:04,460 --> 00:51:10,209 And so to set up this problem, let's just consider the problem 697 00:51:10,209 --> 00:51:11,250 that Markowitz addressed. 698 00:51:11,250 --> 00:51:13,160 You have a one-period investment. 699 00:51:13,160 --> 00:51:15,790 You start with initial wealth W_0. 700 00:51:15,790 --> 00:51:18,360 You're going to choose a portfolio P. 701 00:51:18,360 --> 00:51:21,490 And the wealth after one period is 702 00:51:21,490 --> 00:51:24,650 simply going to be 1 plus the return on that portfolio, 703 00:51:24,650 --> 00:51:29,280 giving terminal wealth W. And our utility function 704 00:51:29,280 --> 00:51:32,930 is going to be some quantitative measure of the outcome, what's 705 00:51:32,930 --> 00:51:34,100 the value to the investor. 706 00:51:34,100 --> 00:51:38,490 And we'd simply want to compute the expected volatility 707 00:51:38,490 --> 00:51:40,800 of different strategies and find the strategy that 708 00:51:40,800 --> 00:51:42,515 maximizes the expected utility. 709 00:51:45,380 --> 00:51:51,120 What are the basic properties of utility functions? 710 00:51:51,120 --> 00:52:01,420 Well, if we graph over wealth, our utility of wealth, 711 00:52:01,420 --> 00:52:06,160 starting at W_0, our initial wealth, 712 00:52:06,160 --> 00:52:14,790 if we have greater wealth, presumably we'll 713 00:52:14,790 --> 00:52:17,220 have greater utility. 714 00:52:17,220 --> 00:52:23,230 So the slope of the utility function should be increasing. 715 00:52:23,230 --> 00:52:26,300 Perhaps as we get more and more wealthy, 716 00:52:26,300 --> 00:52:28,870 the marginal benefit of additional wealth 717 00:52:28,870 --> 00:52:33,530 isn't quite as much as it was when we were poor. 718 00:52:33,530 --> 00:52:38,100 And so perhaps the curve for the utility function 719 00:52:38,100 --> 00:52:40,970 should taper off a bit. 720 00:52:40,970 --> 00:52:44,320 OK, these conditions would correspond 721 00:52:44,320 --> 00:52:47,090 to the first derivative being increasing always 722 00:52:47,090 --> 00:52:49,875 and the second derivative being less than 0 perhaps. 723 00:52:53,370 --> 00:52:55,260 There are definitions in the literature 724 00:52:55,260 --> 00:53:00,380 of risk aversion, absolute risk aversion, and relative risk 725 00:53:00,380 --> 00:53:02,760 aversion. 726 00:53:02,760 --> 00:53:04,475 And these are simple functions of 727 00:53:04,475 --> 00:53:06,850 the first and second derivatives of the utility function. 728 00:53:06,850 --> 00:53:08,860 To see where these come into play, 729 00:53:08,860 --> 00:53:13,850 let's assume the utility function is a smooth function 730 00:53:13,850 --> 00:53:18,550 and consider a Taylor series approximation of that. 731 00:53:18,550 --> 00:53:21,800 So if we consider expanding the utility function 732 00:53:21,800 --> 00:53:25,690 about some base wealth W star, then it's 733 00:53:25,690 --> 00:53:30,230 simply equal to that value of W star 734 00:53:30,230 --> 00:53:32,940 plus first derivative times W minus W star 735 00:53:32,940 --> 00:53:37,270 plus 1/2 the second derivative times the squared deviation 736 00:53:37,270 --> 00:53:39,375 of the wealth from W star. 737 00:53:39,375 --> 00:53:44,500 And if we take expectations of this, 738 00:53:44,500 --> 00:53:47,430 and if W star is the actual expected 739 00:53:47,430 --> 00:53:54,230 wealth of the random variable, then this expected utility 740 00:53:54,230 --> 00:53:57,900 is actually proportional to the expected 741 00:53:57,900 --> 00:54:03,090 wealth minus 1/2 lambda times the variance of the wealth. 742 00:54:03,090 --> 00:54:08,800 So sort of to a second order of approximation, this expected 743 00:54:08,800 --> 00:54:14,520 utility is a function which is looking 744 00:54:14,520 --> 00:54:18,305 at expected return minus a multiple of the volatility 745 00:54:18,305 --> 00:54:20,990 of that return, or the wealth if we're 746 00:54:20,990 --> 00:54:22,220 considering it on that scale. 747 00:54:25,250 --> 00:54:30,740 So there are various different utility functions 748 00:54:30,740 --> 00:54:34,020 that economists have worked with. 749 00:54:34,020 --> 00:54:38,474 And basically the kinds of functions they work with 750 00:54:38,474 --> 00:54:40,515 are all the simple functions we as mathematicians 751 00:54:40,515 --> 00:54:43,300 know about, linear functions, quadratic functions, 752 00:54:43,300 --> 00:54:46,780 exponential functions, power functions, and log functions. 753 00:54:46,780 --> 00:54:50,820 So these are just the first ones to come 754 00:54:50,820 --> 00:54:52,990 to mind for economists perhaps. 755 00:54:52,990 --> 00:54:56,100 But there's actually some rich theory 756 00:54:56,100 --> 00:55:02,130 in terms of sort of investment choice with different utility 757 00:55:02,130 --> 00:55:05,480 functions of these types. 758 00:55:05,480 --> 00:55:09,870 And there is some interesting work there in the economics 759 00:55:09,870 --> 00:55:11,590 literature. 760 00:55:11,590 --> 00:55:16,200 One thing that is to be pointed out 761 00:55:16,200 --> 00:55:19,760 is that with quadratic utility, then 762 00:55:19,760 --> 00:55:22,320 if we consider the expected utility 763 00:55:22,320 --> 00:55:25,780 under quadratic utility, that is the expected utility 764 00:55:25,780 --> 00:55:28,150 function that only depends on expected 765 00:55:28,150 --> 00:55:31,310 wealth and the variance of the wealth, which depends only 766 00:55:31,310 --> 00:55:34,160 on the expected return and the variance of the return 767 00:55:34,160 --> 00:55:36,290 of the portfolio. 768 00:55:36,290 --> 00:55:41,870 So if we are working with a quadratic utility function, 769 00:55:41,870 --> 00:55:45,210 then this mean-variance analysis is the right thing 770 00:55:45,210 --> 00:55:48,390 to be doing under the von Neumann-Morgenstern 771 00:55:48,390 --> 00:55:52,010 expected utility theory. 772 00:55:52,010 --> 00:55:58,080 So doing that is a good decision under that 773 00:55:58,080 --> 00:55:59,810 if that's the utility function. 774 00:55:59,810 --> 00:56:04,220 Now when is that solution not to be preferred? 775 00:56:04,220 --> 00:56:05,950 Well, really it's not be preferred 776 00:56:05,950 --> 00:56:11,160 if you have a different utility function possibly. 777 00:56:11,160 --> 00:56:14,300 And maybe the utility function should 778 00:56:14,300 --> 00:56:18,580 be adding some penalties for skewness or kurtosis. 779 00:56:18,580 --> 00:56:21,160 And that's being ignored here. 780 00:56:21,160 --> 00:56:25,620 So extensions can be looked at. 781 00:56:25,620 --> 00:56:32,920 An interesting mathematical fact is that under the assumption, 782 00:56:32,920 --> 00:56:35,730 or should I say the imagination, that returns 783 00:56:35,730 --> 00:56:42,180 are Gaussian distributed, then the sort of means and variances 784 00:56:42,180 --> 00:56:45,100 of portfolios completely characterize 785 00:56:45,100 --> 00:56:48,510 all the distributions of portfolios of assets, 786 00:56:48,510 --> 00:56:51,500 if the underlying assets are Gaussian distributions. 787 00:56:51,500 --> 00:56:54,430 So a mean-variance analysis is actually 788 00:56:54,430 --> 00:57:00,140 optimal under non-quadratic utility functions, 789 00:57:00,140 --> 00:57:04,680 if the underlying assets are Gaussian distributions, 790 00:57:04,680 --> 00:57:07,570 because only the means and variances are 791 00:57:07,570 --> 00:57:11,090 going to characterize the optimal portfolios 792 00:57:11,090 --> 00:57:12,500 and properties of those. 793 00:57:12,500 --> 00:57:16,360 And I guess, the stochastic dominance 794 00:57:16,360 --> 00:57:18,820 of functions of these variables will generally 795 00:57:18,820 --> 00:57:22,800 apply when there's that corresponding stochastic 796 00:57:22,800 --> 00:57:26,510 dominance in terms of their means and variances. 797 00:57:26,510 --> 00:57:30,160 Anyway, this kind of theory can get involved. 798 00:57:30,160 --> 00:57:35,760 It allows us to extend the basic model that we consider here. 799 00:57:35,760 --> 00:57:40,090 Let's turn then to the topic of the portfolio constraints. 800 00:57:40,090 --> 00:57:42,800 So far, in looking at this problem, 801 00:57:42,800 --> 00:57:45,660 we haven't made any constraints on the problem. 802 00:57:45,660 --> 00:57:52,000 We just want to maximize return, minimize variance, and consider 803 00:57:52,000 --> 00:57:55,360 trade-offs between those. 804 00:57:55,360 --> 00:58:00,174 With practical portfolio optimization problems, 805 00:58:00,174 --> 00:58:02,340 there are different constraints that come into play. 806 00:58:05,470 --> 00:58:08,010 Portfolios that are long only constrain the weights 807 00:58:08,010 --> 00:58:10,710 to only be positive. 808 00:58:10,710 --> 00:58:14,250 There can be holding constraints, 809 00:58:14,250 --> 00:58:20,210 which it may be that we don't want the amount of a given 810 00:58:20,210 --> 00:58:21,800 asset to be too big. 811 00:58:24,570 --> 00:58:26,400 There's the equity strategies, you 812 00:58:26,400 --> 00:58:28,770 don't want to be holding-- if you 813 00:58:28,770 --> 00:58:32,210 have a sort of medium-frequency trading strategy in equities, 814 00:58:32,210 --> 00:58:35,180 you don't want to be holding much more than the trading 815 00:58:35,180 --> 00:58:39,240 volume of a few hours or maybe a day in the asset, 816 00:58:39,240 --> 00:58:42,380 because if you happen to have to sell it, 817 00:58:42,380 --> 00:58:47,100 there really won't be liquidity to trade that. 818 00:58:47,100 --> 00:58:49,600 But we can add-- 819 00:58:49,600 --> 00:58:51,800 There's a simple linear constraints on the holdings. 820 00:58:51,800 --> 00:58:53,300 There could be turnover constraints. 821 00:58:57,060 --> 00:59:05,270 If we're considering basically adjusting our portfolio 822 00:59:05,270 --> 00:59:09,962 from one period to the next, there 823 00:59:09,962 --> 00:59:11,920 are limits in terms of how much we can actually 824 00:59:11,920 --> 00:59:15,180 trade of the different assets. 825 00:59:15,180 --> 00:59:18,720 There can be benchmark exposure constraints. 826 00:59:18,720 --> 00:59:22,280 Suppose we want to invest in a portfolio that's 827 00:59:22,280 --> 00:59:30,040 very much like there's a market index, say, the S&P 500. 828 00:59:30,040 --> 00:59:33,660 But we want to try and do better than the S&P 500. 829 00:59:33,660 --> 00:59:42,120 But we want to protect ourselves from basically not being 830 00:59:42,120 --> 00:59:46,540 very close to the benchmark if our bets, basically, 831 00:59:46,540 --> 00:59:48,670 an allocation, are wrong. 832 00:59:48,670 --> 00:59:55,210 Then one basically can control how different 833 00:59:55,210 --> 00:59:59,790 from the benchmark allocation we are limit, 834 00:59:59,790 --> 01:00:04,910 then limit that departure so we can basically 835 01:00:04,910 --> 01:00:08,260 limit how far from the benchmark weights we want our portfolio 836 01:00:08,260 --> 01:00:09,140 to be. 837 01:00:09,140 --> 01:00:13,690 And this is useful basically for considering strategies 838 01:00:13,690 --> 01:00:19,430 that do as well or better than particular benchmark indices. 839 01:00:19,430 --> 01:00:21,650 Related to this is our tracking error constraints. 840 01:00:25,220 --> 01:00:31,180 In addition to basically not having allocation weights that 841 01:00:31,180 --> 01:00:35,540 depart much from the benchmark weights, 842 01:00:35,540 --> 01:00:37,980 we can consider how much variation is there 843 01:00:37,980 --> 01:00:44,790 in our portfolio compared with an underlying benchmark. 844 01:00:44,790 --> 01:00:48,770 And we can consider tracking error between our portfolio 845 01:00:48,770 --> 01:00:52,660 P and the underlying benchmark and measure that tracking error 846 01:00:52,660 --> 01:00:55,000 in terms of variability, that difference, 847 01:00:55,000 --> 01:00:56,149 and want to control that. 848 01:01:00,550 --> 01:01:04,970 There are also risk factor constraints. 849 01:01:04,970 --> 01:01:08,080 In equity markets there are many different factors 850 01:01:08,080 --> 01:01:12,610 that affect returns. 851 01:01:12,610 --> 01:01:19,040 And these factors can be identified empirically 852 01:01:19,040 --> 01:01:24,690 and controlled for by limiting the exposure 853 01:01:24,690 --> 01:01:28,030 to different factors in the portfolio. 854 01:01:28,030 --> 01:01:33,570 We'll see this in next lecture. 855 01:01:33,570 --> 01:01:36,980 But basically if there are underlying factors 856 01:01:36,980 --> 01:01:42,890 f, which-- basically a return on the asset i 857 01:01:42,890 --> 01:01:49,470 can be explained by sort of an idiosyncratic alpha for that 858 01:01:49,470 --> 01:01:54,360 asset, plus certain correlations with market factors 859 01:01:54,360 --> 01:02:00,550 f, and then a residual innovation epsilon. 860 01:02:00,550 --> 01:02:08,550 And so the returns on these assets can be explained perhaps 861 01:02:08,550 --> 01:02:15,480 50% by underlying market forces, and given by the underlying 862 01:02:15,480 --> 01:02:19,160 factors f_(j, t), which are constant across i. 863 01:02:19,160 --> 01:02:21,440 So these are affecting all the stocks with different 864 01:02:21,440 --> 01:02:24,430 coefficients beta_(i,k). 865 01:02:24,430 --> 01:02:27,060 And in our portfolio, we may want 866 01:02:27,060 --> 01:02:31,580 to limit the exposure to a given factor. 867 01:02:31,580 --> 01:02:33,780 And with many strategies, you actually 868 01:02:33,780 --> 01:02:39,410 want to neutralize the portfolio to those market risk factors. 869 01:02:39,410 --> 01:02:41,390 So we can actually constrain our weights 870 01:02:41,390 --> 01:02:46,575 to have zero exposure across these different market factors. 871 01:02:49,550 --> 01:02:56,450 There are other constraints, minimum transaction size-- 872 01:02:56,450 --> 01:03:00,120 generally it's the case that trades in equities 873 01:03:00,120 --> 01:03:03,680 are in 100 share units, although that's changing. 874 01:03:03,680 --> 01:03:06,880 There can be minimum holding sizes. 875 01:03:06,880 --> 01:03:09,990 And there's also integer constraints 876 01:03:09,990 --> 01:03:13,260 that can be applied. 877 01:03:13,260 --> 01:03:18,750 If you're trading assets that have sort of large values, 878 01:03:18,750 --> 01:03:22,857 then these integer constraints actually come into play. 879 01:03:22,857 --> 01:03:24,690 If you're trading Google stock, and how much 880 01:03:24,690 --> 01:03:26,120 is Google stock worth now? 881 01:03:26,120 --> 01:03:27,237 AUDIENCE: $800. 882 01:03:27,237 --> 01:03:28,820 PETER KEMPTHORNE: Something like that. 883 01:03:28,820 --> 01:03:33,997 So compare that with Ford, which is like $50, or I don't know, 884 01:03:33,997 --> 01:03:36,760 but there's orders of magnitude difference. 885 01:03:36,760 --> 01:03:39,036 So integer constraints can come into play. 886 01:03:39,036 --> 01:03:40,660 If you're dealing in a very large size, 887 01:03:40,660 --> 01:03:42,660 then these things don't really have much impact. 888 01:03:42,660 --> 01:03:48,650 But they can with smaller portfolios. 889 01:03:48,650 --> 01:03:51,620 Now, all of these different constraints 890 01:03:51,620 --> 01:03:54,950 can be expressed as linear and quadratic constraints 891 01:03:54,950 --> 01:03:55,600 on the weights. 892 01:03:58,300 --> 01:04:02,030 So the set-up for the portfolio optimization problem 893 01:04:02,030 --> 01:04:05,040 can be the same as before, except we 894 01:04:05,040 --> 01:04:07,040 add in these additional constraints. 895 01:04:07,040 --> 01:04:10,134 So we basically add in additional Lagrange multipliers 896 01:04:10,134 --> 01:04:11,550 times these particular constraints 897 01:04:11,550 --> 01:04:13,300 with their linear quadratic. 898 01:04:13,300 --> 01:04:16,060 And we can implement the portfolio optimization problem 899 01:04:16,060 --> 01:04:18,450 that way. 900 01:04:18,450 --> 01:04:24,050 Let me turn to an example. 901 01:04:24,050 --> 01:04:54,040 Let's go-- OK, I want to go through an example. 902 01:04:54,040 --> 01:05:16,400 of-- OK, I want to consider US sector exchange-traded funds 903 01:05:16,400 --> 01:05:22,130 between 2009 and last week. 904 01:05:22,130 --> 01:05:26,100 Basically, exchange-traded funds allow 905 01:05:26,100 --> 01:05:31,580 you to invest in equity markets with sort of single assets 906 01:05:31,580 --> 01:05:33,615 that represent different sectors. 907 01:05:33,615 --> 01:05:34,990 The ones that are considered here 908 01:05:34,990 --> 01:05:38,770 are basically spider-traded funds, 909 01:05:38,770 --> 01:05:43,220 which invest in the different major industrial sectors 910 01:05:43,220 --> 01:05:47,230 of the US market, ranging from materials, 911 01:05:47,230 --> 01:05:49,400 health care, consumer staples, down 912 01:05:49,400 --> 01:05:50,650 to technology and utilities. 913 01:05:54,900 --> 01:06:02,490 Here is a graph of the cumulative returns 914 01:06:02,490 --> 01:06:09,690 of these nine different exchange-traded funds 915 01:06:09,690 --> 01:06:17,040 and over the period from 2009 up through last week. 916 01:06:17,040 --> 01:06:20,210 And what one can see is basically 917 01:06:20,210 --> 01:06:22,700 they perform differently. 918 01:06:22,700 --> 01:06:26,060 And what I'd like to do is just examines what 919 01:06:26,060 --> 01:06:30,110 would have been an optimal allocation 920 01:06:30,110 --> 01:06:33,980 across these exchange-traded funds over this period. 921 01:06:33,980 --> 01:06:36,910 So basically looking back at the data 922 01:06:36,910 --> 01:06:43,690 and seeing how our portfolio analysis 923 01:06:43,690 --> 01:06:49,630 tools would result in particular application allocations. 924 01:06:49,630 --> 01:06:53,370 So we can look at the risk versus return 925 01:06:53,370 --> 01:06:54,890 annualized of these. 926 01:06:54,890 --> 01:07:09,610 Let's see-- this is-- I guess that good enough. 927 01:07:09,610 --> 01:07:12,300 Anyway, what you can see is a volatility range 928 01:07:12,300 --> 01:07:15,510 between 0 and 30. 929 01:07:15,510 --> 01:07:19,630 Annualized want return between 0 and 25. 930 01:07:19,630 --> 01:07:23,880 We have the different sectors in this plot. 931 01:07:23,880 --> 01:07:27,970 So this is our feasible set-- this 932 01:07:27,970 --> 01:07:32,060 is the plot of return versus volatility for these nine 933 01:07:32,060 --> 01:07:33,840 exchange-traded funds. 934 01:07:33,840 --> 01:07:35,820 This happens to be utilities. 935 01:07:35,820 --> 01:07:37,680 There's financials over there. 936 01:07:37,680 --> 01:07:39,450 Here's consumer staples. 937 01:07:39,450 --> 01:07:41,970 Very top is a consumer discretionary. 938 01:07:50,780 --> 01:07:54,880 Let's see, I applied a mean-variance optimization 939 01:07:54,880 --> 01:07:59,740 to this problem, assuming that there's 940 01:07:59,740 --> 01:08:06,240 a constraint of 30% of the capital per asset. 941 01:08:06,240 --> 01:08:07,970 So we don't want to invest more than 30% 942 01:08:07,970 --> 01:08:12,080 of the capital in any single exchange-traded fund. 943 01:08:12,080 --> 01:08:23,160 And this graph here shows how, as we vary our target return 944 01:08:23,160 --> 01:08:25,340 from sort of the minimum value up 945 01:08:25,340 --> 01:08:28,899 to the maximum value for being fully invested, 946 01:08:28,899 --> 01:08:31,390 how much of the capital we're going to invest 947 01:08:31,390 --> 01:08:34,005 in the different sectors. 948 01:08:34,005 --> 01:08:35,880 So what you can see is this yellow one, which 949 01:08:35,880 --> 01:08:40,910 is consumer staples, is coming in with a really high weight. 950 01:08:40,910 --> 01:08:45,910 And the green is energy-- let's see, 951 01:08:45,910 --> 01:08:49,229 I think it's energy-- that is the next one. 952 01:08:49,229 --> 01:08:52,529 And orange is health. 953 01:08:52,529 --> 01:09:02,170 And what's important to see is that when 954 01:09:02,170 --> 01:09:07,130 we're looking at sort of the lower range of the returns, 955 01:09:07,130 --> 01:09:10,790 basically expecting returns above the risk-free rate that's 956 01:09:10,790 --> 01:09:15,810 very low, then the relative proportion 957 01:09:15,810 --> 01:09:18,029 investment in the different assets is the same. 958 01:09:18,029 --> 01:09:20,660 Basically, these allocations are slowly scaling up, 959 01:09:20,660 --> 01:09:21,714 just linearly. 960 01:09:21,714 --> 01:09:24,529 And that corresponds to investing 961 01:09:24,529 --> 01:09:30,240 in the optimal portfolio without constraints with a fixed value. 962 01:09:30,240 --> 01:09:34,760 But once we hit the 30% then that constraint 963 01:09:34,760 --> 01:09:36,189 starts to be active. 964 01:09:36,189 --> 01:09:37,689 And we can't invest anymore in that. 965 01:09:37,689 --> 01:09:41,160 So we have to add more to other securities. 966 01:09:41,160 --> 01:09:49,300 And so basically more is given to the consumers discretionary 967 01:09:49,300 --> 01:09:50,200 and the other. 968 01:09:50,200 --> 01:09:53,760 Now actually here's another graph of the same data, 969 01:09:53,760 --> 01:09:57,400 just stacking the allocation. 970 01:09:57,400 --> 01:10:02,220 So we consider ranging from how are we investing our capital. 971 01:10:02,220 --> 01:10:05,860 This dark red is how much money we invest in cash. 972 01:10:05,860 --> 01:10:08,240 And then these colored lines indicate 973 01:10:08,240 --> 01:10:10,920 by their vertical length how much 974 01:10:10,920 --> 01:10:15,120 we're investing in the different exchange traded funds. 975 01:10:15,120 --> 01:10:20,190 So basically as the constraints get 976 01:10:20,190 --> 01:10:24,110 hit, the portfolio sort of change 977 01:10:24,110 --> 01:10:26,230 in terms of their overall structure. 978 01:10:26,230 --> 01:10:32,160 And what can happen is-- actually 979 01:10:32,160 --> 01:10:35,720 as we're trying to achieve more and more return, 980 01:10:35,720 --> 01:10:38,010 well, there may not be assets that 981 01:10:38,010 --> 01:10:40,390 provide that additional return, or there 982 01:10:40,390 --> 01:10:43,610 may be a very few assets that provide that return. 983 01:10:43,610 --> 01:10:47,910 And so in this case, technology stocks 984 01:10:47,910 --> 01:10:49,820 are coming in as we're trying to get 985 01:10:49,820 --> 01:10:54,050 really high levels of return in our optimal portfolios. 986 01:10:54,050 --> 01:10:58,880 And so if we really want that higher return, 987 01:10:58,880 --> 01:11:01,840 that's going to come at higher risk. 988 01:11:01,840 --> 01:11:04,490 And we actually are de-allocating 989 01:11:04,490 --> 01:11:11,560 some from the consumer staples and consumer discretionary 990 01:11:11,560 --> 01:11:12,970 at that point. 991 01:11:19,540 --> 01:11:27,000 OK, here's a graph of the efficient frontier as estimated 992 01:11:27,000 --> 01:11:29,520 with the data. 993 01:11:29,520 --> 01:11:33,900 And what you can see is that these portfolios, 994 01:11:33,900 --> 01:11:37,090 optimal portfolios, yield improvements 995 01:11:37,090 --> 01:11:39,820 over each of the individual exchange-traded funds, 996 01:11:39,820 --> 01:11:42,930 in terms of having higher return with lower risk. 997 01:11:49,050 --> 01:11:52,990 And if we consider just a target return of 10%-- OK, 998 01:11:52,990 --> 01:11:54,960 there's the portfolio. 999 01:11:54,960 --> 01:11:57,350 I've graphed here sort of in solid blue 1000 01:11:57,350 --> 01:12:02,620 what the optimal portfolio with a 10% volatility. 1001 01:12:02,620 --> 01:12:06,450 That's the solid blue line there. 1002 01:12:06,450 --> 01:12:14,370 So these are the results with this 30% capital constraint. 1003 01:12:14,370 --> 01:12:16,540 Now how is this problem going to change 1004 01:12:16,540 --> 01:12:20,811 if we reduce the capital constraint from 30% per asset 1005 01:12:20,811 --> 01:12:21,310 to 15%? 1006 01:12:27,050 --> 01:12:31,582 Who can comment on what will change? 1007 01:12:31,582 --> 01:12:33,037 AUDIENCE: I'm a little confused. 1008 01:12:33,037 --> 01:12:35,120 Could you put a constraint on the risk-free asset. 1009 01:12:35,120 --> 01:12:36,203 Could you do that as well? 1010 01:12:41,360 --> 01:12:46,570 PETER KEMPTHORNE: Well, the risk-free asset has no risk. 1011 01:12:46,570 --> 01:12:50,122 And so putting a constraint-- well, 1012 01:12:50,122 --> 01:12:52,080 you could put a constraint on it in terms of it 1013 01:12:52,080 --> 01:12:55,010 not going negative and it not being greater than one. 1014 01:12:55,010 --> 01:12:56,430 So in fact that is a constraint. 1015 01:12:56,430 --> 01:12:58,138 Those are constraints being imposed here. 1016 01:13:03,770 --> 01:13:09,150 But, I guess, in some-- well, there are, like I guess, 1017 01:13:09,150 --> 01:13:11,960 endowments, say, that have investment policies that 1018 01:13:11,960 --> 01:13:15,480 say that they want a certain fraction assets invested 1019 01:13:15,480 --> 01:13:17,210 in risky assets and not in cash. 1020 01:13:17,210 --> 01:13:23,190 So that is a realistic constraint 1021 01:13:23,190 --> 01:13:25,510 in certain circumstances. 1022 01:13:25,510 --> 01:13:30,890 And this is just highlighting how the allocations vary 1023 01:13:30,890 --> 01:13:32,950 across the risky asset if we only 1024 01:13:32,950 --> 01:13:37,000 have very simple constraints on the cash investment. 1025 01:13:37,000 --> 01:13:41,030 Well, the question I want to focus on 1026 01:13:41,030 --> 01:13:44,542 is what happens if we make the maximum allocation 1027 01:13:44,542 --> 01:13:45,500 constraint more severe. 1028 01:13:45,500 --> 01:13:47,580 So it's 15% instead of 30%. 1029 01:13:47,580 --> 01:13:55,030 Then what's going to happen is these capital constraints are 1030 01:13:55,030 --> 01:13:57,820 going to start hitting sooner. 1031 01:13:57,820 --> 01:14:03,320 So we have to allocate to the other exchange traded funds. 1032 01:14:03,320 --> 01:14:06,937 So let me show you how that works out with the same graph. 1033 01:14:10,520 --> 01:14:24,500 Let's see-- OK, let me just close that and then we go here. 1034 01:14:28,010 --> 01:14:31,120 OK, this shows with a 15% allocation, 1035 01:14:31,120 --> 01:14:34,270 how the allocations vary. 1036 01:14:34,270 --> 01:14:39,890 And I'll be posting this on the course website. 1037 01:14:39,890 --> 01:14:44,950 But it turns out the efficient frontier is actually lower. 1038 01:14:44,950 --> 01:14:48,390 Basically, for higher returns, we actually 1039 01:14:48,390 --> 01:14:51,571 can't get those higher returns by getting 1040 01:14:51,571 --> 01:14:52,820 the biggest bang for the buck. 1041 01:14:52,820 --> 01:14:55,480 We don't want to allocate too much to those higher 1042 01:14:55,480 --> 01:14:57,620 return, higher risk securities. 1043 01:14:57,620 --> 01:15:03,070 And so the optimal, the efficient frontier as estimated 1044 01:15:03,070 --> 01:15:05,520 basically slopes down. 1045 01:15:05,520 --> 01:15:15,840 Now, with this particular example 1046 01:15:15,840 --> 01:15:19,230 of these exchange-traded funds and how 1047 01:15:19,230 --> 01:15:23,210 you allocate across those, this provides some insight 1048 01:15:23,210 --> 01:15:26,990 into portfolio optimization. 1049 01:15:26,990 --> 01:15:29,700 It's not really a realistic setting, 1050 01:15:29,700 --> 01:15:31,310 because we're looking over the past 1051 01:15:31,310 --> 01:15:33,860 and using the actual sample performance 1052 01:15:33,860 --> 01:15:36,820 to define these portfolios. 1053 01:15:36,820 --> 01:15:42,680 Let me just highlight an example of applying these kinds 1054 01:15:42,680 --> 01:15:51,440 of methods with a-- is it here? 1055 01:15:51,440 --> 01:15:51,980 Yep, OK. 1056 01:16:04,642 --> 01:16:07,077 Is that the right graph? 1057 01:16:10,490 --> 01:16:16,170 OK, suppose we consider not investing 1058 01:16:16,170 --> 01:16:17,860 in exchange-traded funds. 1059 01:16:17,860 --> 01:16:21,310 But we are a hedge fund. 1060 01:16:21,310 --> 01:16:24,390 And we have sector pricing models 1061 01:16:24,390 --> 01:16:26,910 across all these different sectors. 1062 01:16:26,910 --> 01:16:31,320 And what we can do is consider going long/short 1063 01:16:31,320 --> 01:16:33,470 in these sectors. 1064 01:16:33,470 --> 01:16:38,330 And in fact, we consider sort of a market-neutral strategy, 1065 01:16:38,330 --> 01:16:44,920 sector by sector, and consider investing sector 1066 01:16:44,920 --> 01:16:47,700 by sector in these different sector-based models. 1067 01:16:47,700 --> 01:16:51,460 OK, here's a graph of different multi-factor pricing 1068 01:16:51,460 --> 01:16:54,940 models for trading market-neutral programs 1069 01:16:54,940 --> 01:16:59,790 within each of the nine industrial sectors, 1070 01:16:59,790 --> 01:17:04,840 the same ones corresponding to the exchange-traded funds. 1071 01:17:04,840 --> 01:17:08,720 And because these strategies are market-neutral, 1072 01:17:08,720 --> 01:17:11,080 the sort of total returns over-- this 1073 01:17:11,080 --> 01:17:18,340 is a five-year period-- are rather modest, namely 1074 01:17:18,340 --> 01:17:26,530 sort of 60% for some of the models, 20% for other models 1075 01:17:26,530 --> 01:17:30,660 What's particularly relevant with these models 1076 01:17:30,660 --> 01:17:35,400 though is that they tend to be less correlated. 1077 01:17:35,400 --> 01:17:41,900 And the diversification benefits can be rather dramatic. 1078 01:17:41,900 --> 01:17:47,010 And so here is a graph of the optimal allocations 1079 01:17:47,010 --> 01:17:51,720 across these different sector market-neutral models. 1080 01:17:51,720 --> 01:17:54,040 And we can see that this red model 1081 01:17:54,040 --> 01:17:55,860 is getting a lot of weight. 1082 01:17:55,860 --> 01:18:00,140 That actually is the utility sector. 1083 01:18:00,140 --> 01:18:13,210 And then these other models are industrials and energy. 1084 01:18:13,210 --> 01:18:29,070 And so if we consider investing in these, 1085 01:18:29,070 --> 01:18:33,580 we can actually achieve a target volatility of 10%-- 1086 01:18:33,580 --> 01:18:36,370 we're going to achieve this solid blue line-- 1087 01:18:36,370 --> 01:18:39,090 as the portfolio strategy. 1088 01:18:39,090 --> 01:18:43,250 And one can see how by combining these assets together-- 1089 01:18:43,250 --> 01:18:46,190 these assets are different market-neutral trading 1090 01:18:46,190 --> 01:18:49,660 models-- we have to get quite a bit of benefit 1091 01:18:49,660 --> 01:18:52,830 from the portfolio optimization. 1092 01:18:52,830 --> 01:18:56,000 And the greater benefit in the portfolio optimization 1093 01:18:56,000 --> 01:19:02,114 is because these strategies tend to have much lower correlations 1094 01:19:02,114 --> 01:19:02,780 with each other. 1095 01:19:07,240 --> 01:19:22,821 Let's go back now, finish up with-- how do we go back here? 1096 01:19:26,300 --> 01:19:28,288 Where's my red dot? 1097 01:20:06,640 --> 01:20:15,650 OK, just to finish up the discussion, with these methods 1098 01:20:15,650 --> 01:20:18,300 it's important to highlight that what we've been using 1099 01:20:18,300 --> 01:20:20,690 were estimated returns, estimated volatilities, 1100 01:20:20,690 --> 01:20:21,830 and correlations. 1101 01:20:21,830 --> 01:20:27,720 And these estimates can have a huge impact on the results. 1102 01:20:27,720 --> 01:20:30,310 There's basically choices of estimation period. 1103 01:20:30,310 --> 01:20:32,540 There's estimation error. 1104 01:20:32,540 --> 01:20:37,790 Different techniques can modulate these issues. 1105 01:20:37,790 --> 01:20:40,090 Exponential moving average are often applied. 1106 01:20:40,090 --> 01:20:44,590 Having dynamic factor models is used. 1107 01:20:44,590 --> 01:20:46,494 Basically, rather than using sample estimates 1108 01:20:46,494 --> 01:20:47,910 of the variance/covariance matrix, 1109 01:20:47,910 --> 01:20:52,120 using factor models to estimate the variance-covariance matrix 1110 01:20:52,120 --> 01:20:59,450 results in more precise inputs to the optimization. 1111 01:20:59,450 --> 01:21:03,834 And finally, just with different risk measures-- 1112 01:21:03,834 --> 01:21:05,625 we've been focusing on portfolio volatility 1113 01:21:05,625 --> 01:21:10,180 and minimizing that-- the methodologies can be extended 1114 01:21:10,180 --> 01:21:13,010 to have different measures of risk-- 1115 01:21:13,010 --> 01:21:17,330 mean absolute deviation, for example, or semi-variance. 1116 01:21:19,860 --> 01:21:24,230 In terms of the motivation for squared volatility 1117 01:21:24,230 --> 01:21:29,100 and minimizing that, you'll recall that in likelihood 1118 01:21:29,100 --> 01:21:31,746 analysis of Gaussian distributions, 1119 01:21:31,746 --> 01:21:33,620 the sort of squared deviations from the means 1120 01:21:33,620 --> 01:21:36,390 are characterizing variability. 1121 01:21:36,390 --> 01:21:40,870 Well, if we focus on mean absolute deviation, then 1122 01:21:40,870 --> 01:21:44,440 probability distributions where the distributions relate 1123 01:21:44,440 --> 01:21:47,072 to the absolute deviation from the mean characterizing 1124 01:21:47,072 --> 01:21:49,720 the distribution might be appropriate. 1125 01:21:49,720 --> 01:21:54,460 And indeed, distributions with heavier tails, 1126 01:21:54,460 --> 01:21:58,000 like exponential distributions, are 1127 01:21:58,000 --> 01:22:00,410 appropriate when we consider mean absolute deviation. 1128 01:22:00,410 --> 01:22:02,840 So some alternate risk measures are 1129 01:22:02,840 --> 01:22:05,260 appropriate when the volatility measure 1130 01:22:05,260 --> 01:22:09,710 itself isn't necessarily appropriate because 1131 01:22:09,710 --> 01:22:12,400 of the distributional assumptions. 1132 01:22:12,400 --> 01:22:15,860 Semi-variance was introduced actually by Markowitz, 1133 01:22:15,860 --> 01:22:18,590 where you're interested in controlling the downside risk 1134 01:22:18,590 --> 01:22:20,140 as opposed to the upside risk. 1135 01:22:22,990 --> 01:22:25,540 There's value-at-risk measures which 1136 01:22:25,540 --> 01:22:30,280 are now completely standard in portfolio management 1137 01:22:30,280 --> 01:22:38,700 and management of risky assets, where one is actually-- well, 1138 01:22:38,700 --> 01:22:40,335 this is introduced by Ken Abbott in one 1139 01:22:40,335 --> 01:22:42,270 of the first few lectures. 1140 01:22:42,270 --> 01:22:46,600 Value at risk is a very simple idea 1141 01:22:46,600 --> 01:22:51,850 of characterizing what outcome is 1142 01:22:51,850 --> 01:22:54,940 sort of the threshold of extreme, at say 1143 01:22:54,940 --> 01:22:58,840 the 5% level or the 1% level, and to basically keep 1144 01:22:58,840 --> 01:23:00,520 and monitor the value at risk. 1145 01:23:00,520 --> 01:23:03,390 So if we had a risk that was basically 1146 01:23:03,390 --> 01:23:06,610 in the worst 5% or 1% of the time. 1147 01:23:06,610 --> 01:23:08,130 What is that level? 1148 01:23:08,130 --> 01:23:10,600 That's the value at risk. 1149 01:23:10,600 --> 01:23:17,130 That as a nice risk measure is simple to define and reasonably 1150 01:23:17,130 --> 01:23:19,400 simple to estimate. 1151 01:23:19,400 --> 01:23:22,950 But it doesn't characterize sort of what the potential exposure 1152 01:23:22,950 --> 01:23:24,680 is if you exceed that. 1153 01:23:24,680 --> 01:23:26,060 And there are extensions of that. 1154 01:23:26,060 --> 01:23:28,730 The conditional value at risk which 1155 01:23:28,730 --> 01:23:32,480 is looking at the expected loss given 1156 01:23:32,480 --> 01:23:35,230 that you exceed the value at risk threshold. 1157 01:23:35,230 --> 01:23:40,040 And this method is now-- I think it's 1158 01:23:40,040 --> 01:23:47,950 been going to be incorporated into regulatory requirements 1159 01:23:47,950 --> 01:23:50,180 for banks in terms of how they measure risk. 1160 01:23:50,180 --> 01:23:53,950 Right now, I believe it's almost all value at risk. 1161 01:23:53,950 --> 01:23:56,670 Extensions like this conditional value at risk 1162 01:23:56,670 --> 01:24:00,900 is definitely going to be applied. 1163 01:24:00,900 --> 01:24:04,220 And in the discussion of different risk measures, 1164 01:24:04,220 --> 01:24:05,840 there's literature talking about what 1165 01:24:05,840 --> 01:24:08,630 are appropriate risk measures and how do we define those. 1166 01:24:08,630 --> 01:24:13,010 Well, it really depends on what your assets are 1167 01:24:13,010 --> 01:24:20,190 and if your assets are simply simple investments in stocks 1168 01:24:20,190 --> 01:24:25,720 or bonds, sort of cash assets or cash instruments, 1169 01:24:25,720 --> 01:24:28,530 then value at risk is quite reasonable. 1170 01:24:28,530 --> 01:24:30,890 If you're investing in derivatives, 1171 01:24:30,890 --> 01:24:36,135 which have non-linear payoffs, then things get complicated. 1172 01:24:36,135 --> 01:24:42,900 And so things need to be handled on a case by case basis there. 1173 01:24:42,900 --> 01:24:45,174 But if you're interested in risk analysis, 1174 01:24:45,174 --> 01:24:47,340 there's a whole discussion on coherent risk measures 1175 01:24:47,340 --> 01:24:50,479 that you can look into. 1176 01:24:50,479 --> 01:24:51,520 That's quite interesting. 1177 01:24:51,520 --> 01:24:52,720 OK, let's stop there. 1178 01:24:52,720 --> 01:24:54,570 Thank you.