1 00:00:00,000 --> 00:00:05,283 So, now, what I want to illustrate is using the risk budgeting stuff for 2 00:00:05,283 --> 00:00:11,146 volatility, we're going to come up with a, a nice summary measure of an asset's 3 00:00:11,146 --> 00:00:16,574 contribution of portfolio volatility. And the summary measure's going to be 4 00:00:16,574 --> 00:00:20,410 called Beta. Beta plays a very key role in, in finance 5 00:00:20,410 --> 00:00:24,246 as, as we'll see. And, and So, how, how does this work? 6 00:00:24,246 --> 00:00:30,181 So, we were looking at a portfolio of assets right, just a weighted average with 7 00:00:30,181 --> 00:00:35,344 our portfolio weights, and we. Derive, you know, the following result. 8 00:00:35,344 --> 00:00:41,970 We use Euler's theorem to additively decompose volatility into its component 9 00:00:41,970 --> 00:00:45,227 pieces. And We showed that the derivative of 10 00:00:45,227 --> 00:00:50,186 portfolio volatility with respect to the portfolio weights is the co-variance 11 00:00:50,186 --> 00:00:53,619 matrix times the vector weights divided by volatility. 12 00:00:53,619 --> 00:00:57,942 And an assets marginal contribution was the eye throw of this thing. 13 00:00:57,942 --> 00:01:00,619 Okay. Now, what we're going to do is, with a 14 00:01:00,619 --> 00:01:05,962 little bit of algebra, we're going to derive an alternative expression for an 15 00:01:05,962 --> 00:01:09,056 asset's marginal contribution to risk. Okay? 16 00:01:09,056 --> 00:01:16,359 So we wanna get a, a, an easier expression for this de-, partial derivative, rather 17 00:01:16,359 --> 00:01:20,284 than saying it's the ith row of this thing. 18 00:01:20,284 --> 00:01:24,420 Okay? And, and so I'm going to show you the 19 00:01:24,420 --> 00:01:30,371 result first and then give the derivation. So the result is going to be that, an, an 20 00:01:30,371 --> 00:01:36,248 asset's, marginal contribution to risk is going to be directly proportional to 21 00:01:36,248 --> 00:01:41,145 something, called its beta. So the beta of asset I, with respect to, 22 00:01:41,145 --> 00:01:48,910 the portfolio is defined as the covariance between asset i, and the return on the 23 00:01:48,910 --> 00:01:54,818 portfolio divided by the variance of the return on the portfolio. 24 00:01:54,818 --> 00:01:58,324 Okay? So it's, And this is key. 25 00:01:58,324 --> 00:02:02,230 So, when we talk about a beta. And. 26 00:02:02,230 --> 00:02:06,206 The beta is always with respect to something else. 27 00:02:06,206 --> 00:02:11,933 So this is beta with respect to portfolio with portfolio weights X, okay? 28 00:02:11,933 --> 00:02:17,660 And in this context asset I is actually a member of this portfolio, okay? 29 00:02:17,660 --> 00:02:21,585 And it's just co variance divided by variance. 30 00:02:21,585 --> 00:02:28,314 Now next week we'll see that beta is in fact a linear regression coefficient. If 31 00:02:28,314 --> 00:02:34,243 you regress the returns of S at I, on the portfolio returns, the regression 32 00:02:34,243 --> 00:02:39,130 co-efficient beta is this covariance divided by variance, so. 33 00:02:39,130 --> 00:02:46,345 And so that's the definition of beta. And the result that we're going to see is 34 00:02:46,345 --> 00:02:51,137 the following. Beta for asset I, measures asset I's 35 00:02:51,137 --> 00:02:57,516 contribution to portfolio volatility. That is, we will show that the marginal 36 00:02:57,516 --> 00:03:03,979 contribution to risk of asset I is equal to its beta multiplied by portfolio 37 00:03:03,979 --> 00:03:08,176 volatility. An asset's contribution to risk is the 38 00:03:08,428 --> 00:03:12,541 portfolio allocation times beta, times volatility. 39 00:03:12,541 --> 00:03:19,340 And the percent contribution to risk is its allocation weight multiplied by beta. 40 00:03:19,840 --> 00:03:26,242 So the key thing here, when we think about percent contribution to risk of asset i 41 00:03:26,242 --> 00:03:30,302 it's, it's allocation weight multiplied by it's beta. 42 00:03:30,302 --> 00:03:36,080 So beta is really the, the key thing that summarizes an asset contribution. 43 00:03:40,800 --> 00:03:47,410 So I'm going to make some comments. So by construction, the beta of the 44 00:03:47,410 --> 00:03:50,850 portfolio is one. So we talk about. 45 00:03:51,001 --> 00:03:53,726 You know, well, what would be the beta of the portfolio? 46 00:03:53,726 --> 00:03:57,863 The beta of the portfolio would be the co-variance of the return on the portfolio 47 00:03:57,863 --> 00:04:00,537 with itself divided by the variance of the portfolio. 48 00:04:00,537 --> 00:04:03,918 So the co-variance of the, of the portfolio with itself is just the 49 00:04:03,918 --> 00:04:07,400 variance, so we get the variance divided by the variance which is one. 50 00:04:07,400 --> 00:04:11,243 Right? So if we use the definition of beta and we 51 00:04:11,243 --> 00:04:17,126 talk about what is the beta of a portfolio with respect to itself it's one. 52 00:04:17,126 --> 00:04:22,656 And intuitively that should make sense. So you want to think of well, you know, 53 00:04:22,656 --> 00:04:27,166 what is the contribution of the portfolio to its volatility. 54 00:04:27,166 --> 00:04:33,048 Well, it's equal to its allocation which is one, times its beta, which is one. 55 00:04:33,050 --> 00:04:40,408 So, this is important because when we talk the, the value of one for beta is kind of 56 00:04:40,408 --> 00:04:44,942 a benchmark. If an asset has a beta equal to one, then 57 00:04:44,942 --> 00:04:50,590 its risk from a portfolio context is the same as the portfolio. 58 00:04:50,590 --> 00:04:56,220 Because its contribution to the portfolio volatility, you know, is, is propor-, is 59 00:04:56,220 --> 00:04:59,214 its beta times its allocation weight. Okay. 60 00:04:59,214 --> 00:05:03,990 So, when we think about the case when an asset beta is equal to one, 61 00:05:03,990 --> 00:05:09,190 Then it's marginal contribution risk is equal to, you know, beta times portfolio 62 00:05:09,190 --> 00:05:12,481 volatility. And when beta is equal one, so then the 63 00:05:12,481 --> 00:05:16,167 marginal contribution risk is just portfolio volatility. 64 00:05:16,167 --> 00:05:21,038 I mean, so notice that, you know, this is saying how much does the portfolio 65 00:05:21,038 --> 00:05:26,370 volatility change when we incrementally increase its allocation in the portfolio. 66 00:05:26,370 --> 00:05:31,505 Well, the portfolio volatility changes by exactly the, the, the, the portfolio 67 00:05:31,505 --> 00:05:35,343 volatility itself. The contributional risk of an asset when 68 00:05:35,343 --> 00:05:40,132 its beta is one, is its allocation weight times portfolio volatility, right? 69 00:05:40,132 --> 00:05:44,532 So if an asset is you have ten percent of, of, of an asset in a portfolio its 70 00:05:44,532 --> 00:05:48,674 contribution risk would then be ten percent of the portfolio volatility. 71 00:05:48,674 --> 00:05:53,657 And then its percent contribution risk when beta is equal to one, the percent 72 00:05:53,657 --> 00:05:56,440 contribution risk is its allocation weight. 73 00:05:56,440 --> 00:05:58,888 Alright. So if you have beta one, ten percent 74 00:05:58,888 --> 00:06:03,786 portfolio allocation, its percent allocation, its percent contribution to 75 00:06:03,786 --> 00:06:10,609 the risk of the portfolio is, is, is ten%. Now, if a be-, if an asset beta's greater 76 00:06:10,609 --> 00:06:17,347 than one, then essen-, then intuitively you should be thinking that the asset has 77 00:06:17,347 --> 00:06:20,380 greater risk than the portfolio. Now. 78 00:06:20,660 --> 00:06:23,670 How we want to view this is, is the following. 79 00:06:23,670 --> 00:06:29,130 So think about, beta's a measure of portfolio risk because it's looking at the 80 00:06:29,130 --> 00:06:33,400 covariance between the return on the asset and the portfolio. 81 00:06:33,400 --> 00:06:39,070 Well the portfolio has N assets in it. So this covariance between asset i and the 82 00:06:39,070 --> 00:06:44,740 portfolio is really the covariance between asset i and asset one, and asset i and 83 00:06:44,740 --> 00:06:50,270 asset two, and asset one and asset N. So beta is really summarizing the variance 84 00:06:50,270 --> 00:06:54,470 N contribution and all the covariance contributions as well. 85 00:06:54,470 --> 00:06:57,827 Okay. Now if a, if a Beta's greater than one, 86 00:06:57,827 --> 00:07:03,593 then that's saying, like, asset i has a large covariance with all of the acids in 87 00:07:03,593 --> 00:07:06,878 the portfolio. So, that when you increase it's 88 00:07:06,878 --> 00:07:12,279 allocation, you kind of increase the average covariance in the portfolio and 89 00:07:12,279 --> 00:07:17,794 that's gonna increase it's riskiness. So when an asset beta is greater than one, 90 00:07:17,796 --> 00:07:22,743 its marginal contribution to risk is greater than portfolio volatility. 91 00:07:22,743 --> 00:07:28,102 Its contribution risk is greater than its allocation weight times portfolio 92 00:07:28,102 --> 00:07:33,805 volatility and it's percent contribution risk is going to be bigger than its, its 93 00:07:33,805 --> 00:07:37,721 asset allocation. And similarly when an asset beta's less 94 00:07:37,721 --> 00:07:40,749 than one. Its marginal contribution is, is less than 95 00:07:40,749 --> 00:07:44,363 portfolio volatility. Its contribution to risk is less than the 96 00:07:44,363 --> 00:07:49,469 asset allocation times volatility. And its percentage contribution to risk is 97 00:07:49,469 --> 00:07:54,711 less than its allocation rate. So the key thing about beta is it, it's a 98 00:07:54,711 --> 00:08:01,839 measure of an asset contribution portfolio risk and a, and a, the benchmark value of 99 00:08:01,839 --> 00:08:07,765 one you know, kind of tells you, you know, is the asset riskier or less risky than 100 00:08:07,765 --> 00:08:10,600 the portfolio? So when you have a, 101 00:08:10,600 --> 00:08:15,057 You know, assets with beta less than one are less risky to the portfolio, and in 102 00:08:15,057 --> 00:08:19,401 summary sense or like diversifiers. Right, so if you have an asset with a beta 103 00:08:19,401 --> 00:08:23,858 less then one and, and, and you increase your allocation to the portfolio, you're 104 00:08:23,858 --> 00:08:26,284 actually going to decrease the portfolio risk. 105 00:08:26,284 --> 00:08:30,628 Where if you have an asset with a beta greater than one and you increase the 106 00:08:30,628 --> 00:08:34,577 allocation then you're going to increase the volatility of the portfolio. 107 00:08:34,577 --> 00:08:37,060 So this really emphasizes the, the notion of, 108 00:08:37,060 --> 00:08:42,010 Of riskiness in a portfolio context. Because, I mean, when you form a portfolio 109 00:08:42,010 --> 00:08:46,638 you're, you're diversifying, right. You're trying to spread your risks out, 110 00:08:46,638 --> 00:08:51,588 but not all assets are diversifiers. If an asset is very high correlated with 111 00:08:51,588 --> 00:08:54,674 everything else it's not a diversifier, you know. 112 00:08:54,674 --> 00:09:00,010 It, it is actually risk increaser, and so beta is kind of measuring when an asset is 113 00:09:00,010 --> 00:09:03,933 a diversifier versus when it isn't. Okay. 114 00:09:03,933 --> 00:09:09,873 So that's the intuition about beta and so now we'll do the derivation of the result, 115 00:09:09,873 --> 00:09:14,257 which is, it's, it's a little tedious but it's not difficult. 116 00:09:14,257 --> 00:09:20,056 But it's informative to go through the derivation because it enforces some of the 117 00:09:20,268 --> 00:09:26,562 I don't know, some of the rules of working with random variables that we've done so 118 00:09:26,562 --> 00:09:29,107 far throughout the course. All right. 119 00:09:29,107 --> 00:09:33,043 So. The, what we're, again, what we're trying 120 00:09:33,043 --> 00:09:37,540 to derive here is. Is this result. 121 00:09:37,540 --> 00:09:42,734 That the marginal contra risk to asset I is equal to its beta multiplied by 122 00:09:42,734 --> 00:09:46,835 portfolio volatility. So this is, this is the key result that 123 00:09:46,835 --> 00:09:50,047 was stated that we now want to actually derive. 124 00:09:50,047 --> 00:09:55,378 So when you take the vector of partial derivatives, we did this, the derivative 125 00:09:55,378 --> 00:10:00,368 of sigma P with respect to X is the covariance matrix times X divided by 126 00:10:00,368 --> 00:10:03,748 volatility. Now if we write this thing out, you know, 127 00:10:03,748 --> 00:10:06,771 longhand. Let's actually look at the numerator. 128 00:10:06,771 --> 00:10:12,161 If we look at Sigma times X what is that? Well, we have the co-variance matrix that 129 00:10:12,161 --> 00:10:16,301 looks like this, and X is just the vector the portfolio weights. 130 00:10:16,301 --> 00:10:19,221 Okay. Now what I'm going to do is I'm going to 131 00:10:19,221 --> 00:10:23,885 concentrate on the first row of this, of this of this vector. 132 00:10:24,107 --> 00:10:29,957 You know, so sigma one times X one, sigma one that should be a sigma one squared. 133 00:10:29,957 --> 00:10:37,940 What the heck did I do? So the first row is going to be. 134 00:10:38,600 --> 00:10:44,177 Sigma one squared times x1, sigma twelve times x2 plus sigma 1n times xn. 135 00:10:44,177 --> 00:10:50,296 So, that's, if we want to know what is the marginal contribution of asset one to 136 00:10:50,296 --> 00:10:56,183 portfolio volatility, then it's going to be the first row of this divided by 137 00:10:56,183 --> 00:11:01,864 portfolio volatility. Consider the first row of sigma time X is 138 00:11:01,864 --> 00:11:06,189 X1 sigma1 squared X2 sigma 1,2 plus XN sigma 1N. 139 00:11:06,189 --> 00:11:10,424 Okay. So turns out that this here is, is in fact 140 00:11:10,424 --> 00:11:15,361 the co-varince between asset one and the portfolio Okay. 141 00:11:15,361 --> 00:11:19,180 So why is that? So, if we look at the co-variance between 142 00:11:19,180 --> 00:11:24,978 asset one and the portfolio, the port, the port, it's equal to the co-variance 143 00:11:24,978 --> 00:11:29,344 between asset one, and X one R one plus X two R two plus X N R N. 144 00:11:29,344 --> 00:11:32,550 Right? Because the portfolio has N pieces to it. 145 00:11:32,550 --> 00:11:38,856 Now you had a homework assignment, I don't know if you remember, that covariance is 146 00:11:38,856 --> 00:11:42,702 additive. So this covariance can be written as the 147 00:11:42,702 --> 00:11:49,932 covariance between R1 and X1 R1 plus the covariance between R1 and X2 plus the 148 00:11:49,932 --> 00:11:56,315 covariance between R1 and X N R N. And the covariance between R1 r! and X1 R1 149 00:11:56,546 --> 00:12:01,873 is X1 times its variance. Right? 150 00:12:01,873 --> 00:12:10,176 Cuz the co-variance between R1 and itself is its variance, and the co-variance of AX 151 00:12:10,176 --> 00:12:15,020 and BY is AB times a co-variance between X and Y. 152 00:12:15,900 --> 00:12:23,290 And then the covariance between R1 and X2 R2 is X2 times sigma one two, and then the 153 00:12:23,290 --> 00:12:28,217 covariance between R1 and XN RN is XN times sigma one N. 154 00:12:28,217 --> 00:12:35,696 So we see that the first row of sigma X is the same as the covariance between R1 and 155 00:12:35,696 --> 00:12:37,087 RP. Alright. 156 00:12:37,087 --> 00:12:40,173 Now, let's remember the definition of beta. 157 00:12:40,173 --> 00:12:45,537 Beta for asset one is the covariance between asset one and the portfolio, 158 00:12:45,537 --> 00:12:50,460 divided by the variance of the portfolio. This probably should be a. 159 00:12:54,880 --> 00:13:00,263 Okay. And so we can write the numerator here. 160 00:13:00,263 --> 00:13:09,440 The co-variance between R1 and the portfolio is the beta multiplied by the 161 00:13:09,440 --> 00:13:11,520 variance, okay? So. 162 00:13:11,520 --> 00:13:17,329 Again, by algebra, the co-variance between R1 and RP is beta one times the variance. 163 00:13:17,329 --> 00:13:20,589 But the co-variance between R1 and RP is this, 164 00:13:20,589 --> 00:13:25,973 Which is the first row of sigma X. So we can get that the first row of sigma 165 00:13:25,973 --> 00:13:29,800 times X is equal to beta one times portfolio variance. 166 00:13:32,600 --> 00:13:39,063 And so the marginal contribution to risk of asset one which is the first row of 167 00:13:39,063 --> 00:13:45,365 sigma x divided by portfolio volatility is beta one times variance divided by 168 00:13:45,365 --> 00:13:49,889 volatility. And so that's beta one times volatility, 169 00:13:49,889 --> 00:13:56,110 Okay. So this is the, the result that the marginal contributional risk of asset one 170 00:13:56,110 --> 00:14:02,780 is directly proportional to its beta. And then using a similar analysis you can 171 00:14:02,780 --> 00:14:14,452 get the marginal contribution whereas with asset I, is just it's the Ith row of sigma 172 00:14:14,452 --> 00:14:20,594 X, sigma times X divided by volatility, and then we see that's beta I time times 173 00:14:20,594 --> 00:14:24,625 volatility. So again, what, what's key, what's nice 174 00:14:24,625 --> 00:14:31,232 about this example where we do risk budgeting with volatility, is that we can 175 00:14:31,232 --> 00:14:37,153 re-express an assets marginal contribution risk in terms of its beta. 176 00:14:37,153 --> 00:14:44,190 And beta is a measure that summarizes an asset's variance plus all the co-variance 177 00:14:44,190 --> 00:14:50,712 contributions to, to the portfolio. So it's a summary measure of how an asset 178 00:14:50,712 --> 00:14:54,660 contributes to the variability of a portfolio. 179 00:14:56,120 --> 00:15:03,338 Alright. Alright so some key points to take away, 180 00:15:03,799 --> 00:15:13,295 about beta as a measure of portfolio risk. So, the first point is that asset specific 181 00:15:13,295 --> 00:15:16,825 risk can be diversified away by forming portfolios. 182 00:15:16,825 --> 00:15:22,155 So, the reason why we create a portfolio is to try to spread our risks around. 183 00:15:22,155 --> 00:15:27,554 And when we diversify a portfolio diversification eliminates certain types 184 00:15:27,554 --> 00:15:33,022 of risk but you don't eliminate all risk when you form a diversified portfolio. 185 00:15:33,022 --> 00:15:38,005 Some risks still remains, right? So , you can have a portfolio of, of 5,000 186 00:15:38,005 --> 00:15:40,982 assets but that portfolio is not risk-free. 187 00:15:41,189 --> 00:15:46,829 You know, the portfolio still has risk. Alright and so what is the risk that 188 00:15:46,829 --> 00:15:49,420 remains. It turns out that. 189 00:15:49,420 --> 00:15:54,920 Beta measures the risk that remains after you do diversification. 190 00:15:55,680 --> 00:16:04,234 So, what remains after you diversify a portfolio, is, is called portfolio risk, 191 00:16:04,234 --> 00:16:05,750 right. [laugh]. 192 00:16:05,750 --> 00:16:10,830 And it's kind of like the, you know, in any a sense, so, so, this portfolio risk 193 00:16:10,830 --> 00:16:15,308 is actually measured by beta. And beta again, is a measure , you know, 194 00:16:15,308 --> 00:16:19,519 it has asset, it has a variance contribution and the covariance 195 00:16:19,519 --> 00:16:22,995 contributions. So, that's that, that's what remains. 196 00:16:22,995 --> 00:16:26,589 Now, when we think about. The risk of a single asset. 197 00:16:26,589 --> 00:16:31,967 Portfolio theory tells us the following. If certain types of risks can be 198 00:16:31,967 --> 00:16:35,791 diversified away. Then when we view the riskiness of an 199 00:16:35,791 --> 00:16:38,633 asset. You know, if we think of a portfolio 200 00:16:38,633 --> 00:16:42,084 context. We should really think about how the asset 201 00:16:42,084 --> 00:16:47,430 influenced the risk of the portfolio. Not necessarily the it's stand alone risk. 202 00:16:47,430 --> 00:16:50,947 Alright? Because when you know, certain assets can 203 00:16:50,947 --> 00:16:56,087 be you know, good in a portfolio context, that is they could be a diversifier. 204 00:16:56,087 --> 00:17:01,363 And some portfolios, some assets can be bad in a portfolio context, that is you 205 00:17:01,363 --> 00:17:04,272 know, they don't diversify very well at all. 206 00:17:04,272 --> 00:17:09,345 So when you think of you know, the riskiness of an asset you can also think 207 00:17:09,345 --> 00:17:14,080 of risk from a portfolio context. And this is what beta does for us. 208 00:17:15,060 --> 00:17:20,100 And the last one is beta measures the portfolio risk of an asset. 209 00:17:21,820 --> 00:17:26,767 Now. This was recognized early on. 210 00:17:26,767 --> 00:17:32,759 You know, I guess you know, people developing finance theory in the 1960's. 211 00:17:32,759 --> 00:17:38,979 And in particular William Sharpe who is one of the Nobel Prize winners for his 212 00:17:38,979 --> 00:17:44,895 work in the Capital Asset Pricing Model. You know, again recognized that in the 213 00:17:44,895 --> 00:17:50,964 context of forming portfolios you can summarize an asset's portfolio risk using 214 00:17:50,964 --> 00:17:54,529 data. And this lead to the following conjecture. 215 00:17:54,529 --> 00:18:02,112 And If Beta is, the appropriate measure of risk, you know from the context of a 216 00:18:02,112 --> 00:18:07,750 portfolio, and if we believe there is a risk return trade off, that is assets with 217 00:18:07,750 --> 00:18:12,623 higher risks should generate higher expected returns, then perhaps the 218 00:18:12,623 --> 00:18:17,843 expected return on assets should be related to its beta, right'cause assets 219 00:18:17,843 --> 00:18:23,621 with high beta are risky from a portfolio context, so they should generate a higher 220 00:18:23,621 --> 00:18:29,189 risk premium, and assets with low betas are less risky in a portfolio context, so, 221 00:18:29,189 --> 00:18:34,814 you know they should not. You know, generate a high-risk premium. 222 00:18:34,814 --> 00:18:37,930 And so. Willam Sahrpe and others essentially 223 00:18:37,930 --> 00:18:42,370 conjectured if, if the data of an asset with respect to a portfolio is the 224 00:18:42,370 --> 00:18:46,990 appropriate measure of the risk of an asset then the asset's expected return 225 00:18:46,990 --> 00:18:51,250 should, should depend upon its data. So, the expected return on the asset 226 00:18:51,250 --> 00:18:55,750 should be some function of data. And this is what essentially formalized in 227 00:18:55,750 --> 00:19:01,287 the capital asset pricing model Capital asset pricing model says the risk return 228 00:19:01,287 --> 00:19:04,904 trade-off between assets, is dictated by its beta. 229 00:19:04,904 --> 00:19:10,539 Assets with high beta should, should give, give you high risk premiums, assets with 230 00:19:10,539 --> 00:19:13,600 low betas should give you low risk premiums.