WEBVTT 00:00:00.060 --> 00:00:02.500 The following content is provided under a Creative 00:00:02.500 --> 00:00:04.019 Commons license. 00:00:04.019 --> 00:00:06.360 Your support will help MIT OpenCourseWare 00:00:06.360 --> 00:00:10.730 continue to offer high quality educational resources for free. 00:00:10.730 --> 00:00:13.330 To make a donation or view additional materials 00:00:13.330 --> 00:00:17.217 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.217 --> 00:00:17.842 at ocw.mit.edu. 00:00:21.280 --> 00:00:23.640 KENNETH ABBOTT: As I said, my name is Ken Abbott. 00:00:23.640 --> 00:00:26.220 I'm the operating officer for Firm Risk Management 00:00:26.220 --> 00:00:29.850 at Morgan Stanley, which means I'm the everything else guy. 00:00:29.850 --> 00:00:33.480 I'm like the normal stuff with a bar over it. 00:00:33.480 --> 00:00:37.640 The complement of normal-- I get all the odd stuff. 00:00:37.640 --> 00:00:41.065 I consider myself the Harvey Keitel character. 00:00:41.065 --> 00:00:43.954 You know, the fixer? 00:00:43.954 --> 00:00:45.870 And so I get a lot of interesting stuff to do. 00:00:45.870 --> 00:00:47.920 I've covered commodities, I've covered fixed income, 00:00:47.920 --> 00:00:49.900 I've covered equities, I've covered credit derivatives, 00:00:49.900 --> 00:00:51.120 I've covered mortgages. 00:00:51.120 --> 00:00:52.670 Now I'm also the Chief Risk Officer 00:00:52.670 --> 00:00:55.000 for the buy side of Morgan Stanley. 00:00:55.000 --> 00:00:58.170 The investment management business and the private equity 00:00:58.170 --> 00:01:00.550 holdings that we have. 00:01:00.550 --> 00:01:03.160 And I look after lot of that stuff 00:01:03.160 --> 00:01:05.237 and I sit on probably 40 different committees 00:01:05.237 --> 00:01:07.320 because it's become very, very, very bureaucratic. 00:01:07.320 --> 00:01:11.127 But that's the way it goes. 00:01:11.127 --> 00:01:13.710 What I want to talk about today is some of the core approaches 00:01:13.710 --> 00:01:17.250 we use to measure a risk in a market risk setting. 00:01:17.250 --> 00:01:22.380 This is part of a larger course I teach at a couple places. 00:01:22.380 --> 00:01:26.400 I'm a triple alum at NYU-- no I'm a double alum 00:01:26.400 --> 00:01:28.193 and now I'm on their faculty [INAUDIBLE]. 00:01:28.193 --> 00:01:32.370 I have a masters in economics from their arts and sciences 00:01:32.370 --> 00:01:33.115 program. 00:01:33.115 --> 00:01:35.460 I have a masters in statistics from Stern 00:01:35.460 --> 00:01:37.160 when Stern used to have a stat program. 00:01:37.160 --> 00:01:40.260 And now I teach at Courant. 00:01:40.260 --> 00:01:42.915 I also teach at Claremont and I teach at Baruch, 00:01:42.915 --> 00:01:43.790 part of that program. 00:01:43.790 --> 00:01:45.990 So I've been through this material many times. 00:01:45.990 --> 00:01:51.380 So what I want to do is lay the foundation for this notion 00:01:51.380 --> 00:01:54.510 that we call value at risk, this idea of VaR. 00:01:58.785 --> 00:02:01.526 [INAUDIBLE] put this back on. 00:02:01.526 --> 00:02:02.090 Got it. 00:02:02.090 --> 00:02:02.990 I'll make it work. 00:02:05.720 --> 00:02:08.300 I'll talk about it from a mathematical standpoint 00:02:08.300 --> 00:02:10.210 and from a statistical standpoint, 00:02:10.210 --> 00:02:12.960 but also give you some of the intuition behind what 00:02:12.960 --> 00:02:17.190 it is that we're trying to do when we measure this thing. 00:02:17.190 --> 00:02:20.377 First, a couple words about risk management. 00:02:20.377 --> 00:02:21.210 What is the risk do? 00:02:21.210 --> 00:02:25.106 25 years ago, maybe three firms had risk management groups. 00:02:25.106 --> 00:02:26.730 I was part of the first risk management 00:02:26.730 --> 00:02:31.864 group at Bankers Trust in 1986. 00:02:31.864 --> 00:02:33.780 No one else had a risk management group as far 00:02:33.780 --> 00:02:34.330 as I know. 00:02:37.040 --> 00:02:40.590 Market risk management really came to be in the late '80s. 00:02:40.590 --> 00:02:42.090 Credit risk management had obviously 00:02:42.090 --> 00:02:45.700 been around in large financial institutions the whole time. 00:02:45.700 --> 00:02:50.180 So our job is to make sure that management 00:02:50.180 --> 00:02:51.660 knows what's on the books. 00:02:51.660 --> 00:02:54.910 So step one is, what is the risk profile of the firm? 00:02:54.910 --> 00:02:57.582 How do I make sure that management 00:02:57.582 --> 00:02:58.540 is informed about this? 00:02:58.540 --> 00:03:00.160 So it requires two things. 00:03:00.160 --> 00:03:02.520 One, I have to know what the risk profile is 00:03:02.520 --> 00:03:04.000 because I have to know it in order 00:03:04.000 --> 00:03:05.460 to be able to communicate it. 00:03:05.460 --> 00:03:08.210 But the second thing, equally important, particularly 00:03:08.210 --> 00:03:11.880 important for you guys and girls, 00:03:11.880 --> 00:03:16.310 is that you need to be able to express 00:03:16.310 --> 00:03:19.860 relatively complex concepts in simple words 00:03:19.860 --> 00:03:21.190 and pretty pictures. 00:03:21.190 --> 00:03:22.290 All right? 00:03:22.290 --> 00:03:25.900 Chances are if you go to work for big firm, 00:03:25.900 --> 00:03:28.260 your boss won't be a quant. 00:03:28.260 --> 00:03:30.880 My boss happens to have a degree from Carnegie Mellon. 00:03:30.880 --> 00:03:33.770 He can count to 11 with his shoes on. 00:03:33.770 --> 00:03:34.790 His boss is a lawyer. 00:03:34.790 --> 00:03:37.500 His boss is the chairman. 00:03:37.500 --> 00:03:41.220 Commonly, the most senior people are very, very intelligent, 00:03:41.220 --> 00:03:45.900 very, very articulate, very, very learned. 00:03:45.900 --> 00:03:47.432 But not necessarily quants. 00:03:47.432 --> 00:03:49.890 Many of them have had a year or two of calculus, maybe even 00:03:49.890 --> 00:03:51.860 linear algebra. 00:03:51.860 --> 00:03:54.800 You can't show them-- look, when you and I chat and we talk 00:03:54.800 --> 00:03:59.980 about regression analysis, I could say X transpose X inverse 00:03:59.980 --> 00:04:01.310 X transpose y. 00:04:01.310 --> 00:04:03.476 And those of you that have taken a regression course 00:04:03.476 --> 00:04:06.830 think, ah, that's beta hat. 00:04:06.830 --> 00:04:08.080 And we can just stop it there. 00:04:08.080 --> 00:04:11.010 I can just put this form up there and you may recognize it. 00:04:11.010 --> 00:04:13.760 I would have to spend 45 minutes explaining this 00:04:13.760 --> 00:04:15.733 to people on the top floor because this is not 00:04:15.733 --> 00:04:16.649 what they're studying. 00:04:16.649 --> 00:04:20.320 So we can talk the code amongst ourselves, 00:04:20.320 --> 00:04:25.372 but when we go outside our little group-- getting bigger-- 00:04:25.372 --> 00:04:27.830 we have to make sure that we can express ourselves clearly. 00:04:27.830 --> 00:04:30.950 That's done in clear, effective prose, and in graphs. 00:04:30.950 --> 00:04:33.800 And I'll show you some of that stuff as we go on. 00:04:33.800 --> 00:04:36.360 So step one, make sure management 00:04:36.360 --> 00:04:38.130 knows what the risk profile is. 00:04:38.130 --> 00:04:41.450 Step two, protect the firm against unacceptably large 00:04:41.450 --> 00:04:42.290 concentrations. 00:04:42.290 --> 00:04:44.090 This is the subjective part. 00:04:44.090 --> 00:04:46.690 I can know the risk, but how big is big? 00:04:46.690 --> 00:04:48.045 How much is too much? 00:04:48.045 --> 00:04:50.430 How much is too concentrated? 00:04:50.430 --> 00:04:53.550 If I have $1 million of sensitivity per basis point, 00:04:53.550 --> 00:04:56.405 that's a 1/100th of 1% move in a rate. 00:04:56.405 --> 00:04:57.350 Is that big? 00:04:57.350 --> 00:04:58.920 Is that small? 00:04:58.920 --> 00:05:00.197 How do I know how much? 00:05:00.197 --> 00:05:02.280 How much of a particular stock issue should I own? 00:05:02.280 --> 00:05:05.100 How much of a bond issue? 00:05:05.100 --> 00:05:06.610 How much futures open interest? 00:05:06.610 --> 00:05:08.800 How big a limit should I have on this type of risk? 00:05:08.800 --> 00:05:12.121 That's where intuition and experience come into play. 00:05:12.121 --> 00:05:13.620 So that's the second part of our job 00:05:13.620 --> 00:05:17.040 is to protect against unacceptably large losses. 00:05:17.040 --> 00:05:22.150 So the third, no surprises, you can 00:05:22.150 --> 00:05:27.980 liken the trading business-- it's taking calculated risks. 00:05:27.980 --> 00:05:29.560 Sometimes you're going to lose. 00:05:29.560 --> 00:05:31.190 Many times you're going to lose. 00:05:31.190 --> 00:05:35.630 In fact, if you win 51% of the time, life is pretty good. 00:05:35.630 --> 00:05:37.740 So what you want to do is make sure you have 00:05:37.740 --> 00:05:41.090 the right information so you can estimate, if things get bad, 00:05:41.090 --> 00:05:43.240 how bad will they get? 00:05:43.240 --> 00:05:44.750 And to use that, we leverage a lot 00:05:44.750 --> 00:05:49.710 of relatively simple notions that we see in statistics. 00:05:49.710 --> 00:05:54.804 And so I should use a coloring mask here, not a spotlight. 00:05:54.804 --> 00:05:55.720 We do a couple things. 00:05:55.720 --> 00:05:58.750 Just like the way when they talk about the press in your course 00:05:58.750 --> 00:06:01.400 about journalism, we can shine a light 00:06:01.400 --> 00:06:03.430 anywhere we want, and we do all the time. 00:06:03.430 --> 00:06:04.040 You know what? 00:06:04.040 --> 00:06:05.660 I'm going to think about this particular kind of risk. 00:06:05.660 --> 00:06:07.400 I'm going to point out that this is really important. 00:06:07.400 --> 00:06:08.733 You need to pay attention to it. 00:06:08.733 --> 00:06:09.924 And then I could shade it. 00:06:09.924 --> 00:06:12.340 I can make it blue, I can make a red, I can make it green. 00:06:12.340 --> 00:06:14.660 I'd say this is good, this is bad, this is too big, 00:06:14.660 --> 00:06:17.080 this is too small, this is perfectly fine. 00:06:17.080 --> 00:06:21.610 So that's just a little bit of quick background on what we do. 00:06:21.610 --> 00:06:25.035 So I'm going to go through as much of this as I can. 00:06:25.035 --> 00:06:26.760 I'm going to fly through the first part 00:06:26.760 --> 00:06:30.670 and I want to hit these because these are the ways that we 00:06:30.670 --> 00:06:31.990 actually estimate risk. 00:06:31.990 --> 00:06:34.590 Variance, covariance as a quadratic form. 00:06:34.590 --> 00:06:36.610 Monte Carlo simulation, the way I'll show you 00:06:36.610 --> 00:06:38.310 is based on a quadratic form. 00:06:38.310 --> 00:06:40.490 And historical simulation is Monte Carlo simulation 00:06:40.490 --> 00:06:42.130 without the Monte Carlo part. 00:06:42.130 --> 00:06:43.890 It's using historical data. 00:06:43.890 --> 00:06:47.080 And I'll go through that fairly quickly. 00:06:47.080 --> 00:06:50.320 Questions, comments? 00:06:50.320 --> 00:06:51.960 No? 00:06:51.960 --> 00:06:52.780 Excellent. 00:06:52.780 --> 00:06:55.710 Stop me-- look, if any one of you 00:06:55.710 --> 00:06:58.750 doesn't understand something I say, probably many of you 00:06:58.750 --> 00:07:00.084 don't understand it. 00:07:00.084 --> 00:07:02.250 I don't know you guys, so I don't know what you know 00:07:02.250 --> 00:07:02.930 and what you don't know. 00:07:02.930 --> 00:07:05.096 So if there's a term that comes up, you're not sure, 00:07:05.096 --> 00:07:08.930 just say, Ken, I don't have a PhD. 00:07:08.930 --> 00:07:11.420 I work for a living. 00:07:11.420 --> 00:07:13.130 I make fun of academics. 00:07:13.130 --> 00:07:15.010 I know you work for a living too. 00:07:15.010 --> 00:07:16.780 All right. 00:07:16.780 --> 00:07:18.809 There's a guy I tease at Claremont [INAUDIBLE] 00:07:18.809 --> 00:07:21.100 in his class, I say, who is this pointy-headed academic 00:07:21.100 --> 00:07:22.760 [INAUDIBLE]. 00:07:22.760 --> 00:07:24.937 Only kidding. 00:07:24.937 --> 00:07:27.520 All right, so I'm going to talk about one-asset value at risk. 00:07:27.520 --> 00:07:29.200 First I'm going to introduce the notion of value at risk. 00:07:29.200 --> 00:07:30.780 I'm going to talk about one asset. 00:07:30.780 --> 00:07:32.780 I'm going to talk about price-based instruments. 00:07:32.780 --> 00:07:35.052 We're going to go into yield space, 00:07:35.052 --> 00:07:36.510 so we'll talk about the conversions 00:07:36.510 --> 00:07:38.300 we have to do there. 00:07:38.300 --> 00:07:40.481 One thing I'll do after this class is over, 00:07:40.481 --> 00:07:42.980 since I know I'm going to fly through some of the material-- 00:07:42.980 --> 00:07:44.563 and since this is MIT, I'm sure you're 00:07:44.563 --> 00:07:46.276 used to just flying through material. 00:07:46.276 --> 00:07:48.150 And there's a lot of this, the proof of which 00:07:48.150 --> 00:07:50.066 is left to the reader as an exercise. 00:07:50.066 --> 00:07:51.690 I'm sure you get a fair amount of that. 00:07:51.690 --> 00:07:53.120 I will give you papers. 00:07:53.120 --> 00:07:57.070 If you have questions, my email is on the first page. 00:07:57.070 --> 00:07:58.610 I welcome your questions. 00:07:58.610 --> 00:08:00.540 I tell my students that every year. 00:08:00.540 --> 00:08:02.500 I'm OK with you sending me an email asking me 00:08:02.500 --> 00:08:05.450 for a reference, a citation, something. 00:08:05.450 --> 00:08:06.760 I'm perfectly fine with that. 00:08:06.760 --> 00:08:09.040 Don't worry, oh, he's too busy. 00:08:09.040 --> 00:08:10.245 I'm fine. 00:08:10.245 --> 00:08:12.950 If you've got a question, something is not clear, 00:08:12.950 --> 00:08:16.750 I've got access to thousands of papers. 00:08:16.750 --> 00:08:17.790 And I've screened them. 00:08:17.790 --> 00:08:18.860 I've read thousands of papers, I say 00:08:18.860 --> 00:08:20.780 this is a good one, that's a waste of time. 00:08:20.780 --> 00:08:22.475 But I can give you background material 00:08:22.475 --> 00:08:26.110 on regulation, on bond pricing, on derivative algorithms. 00:08:26.110 --> 00:08:26.680 Let me know. 00:08:26.680 --> 00:08:29.000 I'm happy to provide that at any point in time. 00:08:29.000 --> 00:08:30.536 You get that free with your tuition. 00:08:33.789 --> 00:08:35.120 A couple of key metrics. 00:08:35.120 --> 00:08:36.953 I don't want to spend too much time on this. 00:08:36.953 --> 00:08:38.620 Interest rate exposure, how sensitive 00:08:38.620 --> 00:08:40.850 am I to changes in interest rates, equity exposure, 00:08:40.850 --> 00:08:43.200 commodity exposure, credit spread exposure. 00:08:43.200 --> 00:08:47.080 We'll talk about linearity, we won't talk too much 00:08:47.080 --> 00:08:48.512 about regularity of cash flow. 00:08:48.512 --> 00:08:49.970 We won't really get into that here. 00:08:49.970 --> 00:08:53.129 And we need to know correlation across different asset classes. 00:08:53.129 --> 00:08:54.545 And I'll show you what that means. 00:08:57.140 --> 00:09:00.790 At the heart of this notion of value at risk 00:09:00.790 --> 00:09:05.350 is this idea of a statistical order statistic. 00:09:05.350 --> 00:09:08.317 Who here has heard of order statistics? 00:09:08.317 --> 00:09:10.150 All right, I'm going to give you 30 seconds. 00:09:10.150 --> 00:09:15.165 The best simple description of an order statistic. 00:09:15.165 --> 00:09:16.710 PROFESSOR: The maximum or the minimum 00:09:16.710 --> 00:09:19.600 of a set of observations. 00:09:19.600 --> 00:09:21.230 KENNETH ABBOTT: All right? 00:09:21.230 --> 00:09:23.540 When we talk about value at risk, 00:09:23.540 --> 00:09:28.420 I want to know the worst 1% of the outcomes. 00:09:28.420 --> 00:09:32.030 And what's cool about order statistics 00:09:32.030 --> 00:09:34.610 is they're well established in the literature. 00:09:34.610 --> 00:09:36.840 Pretty well understood. 00:09:36.840 --> 00:09:39.430 And so people are familiar with it. 00:09:39.430 --> 00:09:42.180 Once we put our toe into the academic water 00:09:42.180 --> 00:09:44.160 and we start talking about this notion, 00:09:44.160 --> 00:09:46.071 there's a vast body of literature 00:09:46.071 --> 00:09:47.570 that says this is how this thing is. 00:09:47.570 --> 00:09:48.460 This is how it pays. 00:09:48.460 --> 00:09:50.680 This is what the distribution looks like. 00:09:50.680 --> 00:09:55.200 And so we can estimate these things. 00:09:55.200 --> 00:09:57.830 And so what we're looking at in value at risk, 00:09:57.830 --> 00:10:03.867 if my distribution of returns, how much I make. 00:10:03.867 --> 00:10:05.450 In particular, if I look historically, 00:10:05.450 --> 00:10:06.860 I have a position. 00:10:06.860 --> 00:10:10.630 How much would this position have earned me over the last n 00:10:10.630 --> 00:10:13.140 days, n weeks, n months. 00:10:13.140 --> 00:10:16.430 If I look at a frequency distribution of that, 00:10:16.430 --> 00:10:19.690 I'm likely-- don't have to-- I'm likely to get something that's 00:10:19.690 --> 00:10:20.417 symmetric. 00:10:20.417 --> 00:10:22.250 I'm likely to get something that's unimodal. 00:10:24.860 --> 00:10:26.610 It may or may not have fat tails. 00:10:26.610 --> 00:10:29.290 We'll talk about that a little later. 00:10:29.290 --> 00:10:32.040 If my return distribution were beautifully symmetric 00:10:32.040 --> 00:10:38.490 and beautifully normal and independent, then the risk-- 00:10:38.490 --> 00:10:40.790 I could measure this 1% order statistic. 00:10:40.790 --> 00:10:46.570 What's the 1% likely worst case outcome tomorrow? 00:10:46.570 --> 00:10:49.960 I might do that by integrating the normal function 00:10:49.960 --> 00:10:54.150 from negative infinity-- for all intents and purposes 00:10:54.150 --> 00:10:55.640 five or six standard deviations. 00:10:55.640 --> 00:10:59.970 Anyway, from negative infinity to negative 2.33 00:10:59.970 --> 00:11:01.080 standard deviations. 00:11:01.080 --> 00:11:01.710 Why? 00:11:01.710 --> 00:11:05.250 Because the area under the curve, that's 0.01. 00:11:05.250 --> 00:11:07.440 Now this is a one-sided confidence interval 00:11:07.440 --> 00:11:09.540 as opposed to a two-sided confidence integral. 00:11:09.540 --> 00:11:12.060 And this is one of these things that as an undergrad 00:11:12.060 --> 00:11:14.370 you learn two-sided, and then the first time someone 00:11:14.370 --> 00:11:15.670 shows you one sided you're like, wait a minute. 00:11:15.670 --> 00:11:16.212 What is this? 00:11:16.212 --> 00:11:17.336 Than you say, oh, I get it. 00:11:17.336 --> 00:11:18.690 You're just looking at the area. 00:11:18.690 --> 00:11:21.210 I could build a gazillion two-sided confidence intervals. 00:11:21.210 --> 00:11:23.620 One sided, it's got to stop at one place. 00:11:23.620 --> 00:11:25.870 All right so this set of outcomes-- and this 00:11:25.870 --> 00:11:31.480 is standardized-- this is in standard deviation space-- 00:11:31.480 --> 00:11:34.150 negative infinity to 2.33. 00:11:34.150 --> 00:11:37.900 If I want 95%, or 5% likely loss, so I could say, 00:11:37.900 --> 00:11:39.910 tomorrow there's a 5% chance my loss is 00:11:39.910 --> 00:11:42.230 going to be x or greater, I would go 00:11:42.230 --> 00:11:45.950 to 1.645 standard deviations. 00:11:45.950 --> 00:11:48.120 Because the integral from negative infinity to 1.645 00:11:48.120 --> 00:11:52.970 standard deviations is about 0.05. 00:11:52.970 --> 00:11:55.650 It's not just a good idea, it's the law. 00:11:55.650 --> 00:11:58.560 Does that make sense? 00:11:58.560 --> 00:12:00.602 And again, I'm going to say assuming the normal. 00:12:00.602 --> 00:12:02.060 That's like the old economist joke, 00:12:02.060 --> 00:12:04.630 assume a can opener when he's on a desert island. 00:12:04.630 --> 00:12:05.900 You guys don't know that one. 00:12:05.900 --> 00:12:07.410 I got lots of economics jokes. 00:12:07.410 --> 00:12:10.970 I'll tell them later on maybe-- or after class. 00:12:10.970 --> 00:12:13.950 If I'm assuming normal distribution, and that's 00:12:13.950 --> 00:12:16.084 what I'm going to do, what I want to do 00:12:16.084 --> 00:12:19.600 is I'm going to set this thing up in a normal distribution 00:12:19.600 --> 00:12:20.140 framework. 00:12:20.140 --> 00:12:24.200 Now doing this approach and assuming normal distributions, 00:12:24.200 --> 00:12:28.720 I liken it to using Latin. 00:12:28.720 --> 00:12:32.240 Nobody really uses it anymore but everything we do 00:12:32.240 --> 00:12:33.872 is based upon it. 00:12:33.872 --> 00:12:35.080 So that's our starting point. 00:12:35.080 --> 00:12:37.710 And it's really easy to teach it this way 00:12:37.710 --> 00:12:39.560 and then we relax the assumptions 00:12:39.560 --> 00:12:41.210 like so many things in life. 00:12:41.210 --> 00:12:42.880 I teach you the strict case then we 00:12:42.880 --> 00:12:45.630 relax the assumptions to get to the way it's done now. 00:12:45.630 --> 00:12:48.360 So this makes sense? 00:12:48.360 --> 00:12:48.870 All right. 00:12:48.870 --> 00:12:51.290 So let's get there. 00:12:51.290 --> 00:12:53.670 This is way oversimplified-- but let's say 00:12:53.670 --> 00:12:55.117 I have something like this. 00:12:55.117 --> 00:12:56.700 Who has taken intermediate statistics? 00:13:01.530 --> 00:13:05.290 We have the notion of stationarity 00:13:05.290 --> 00:13:06.720 that we talk about all the time. 00:13:06.720 --> 00:13:10.022 The mean and variance constant is one simplistic way 00:13:10.022 --> 00:13:10.980 of thinking about this. 00:13:10.980 --> 00:13:13.147 Do you have a better way for me to put that to them? 00:13:13.147 --> 00:13:15.146 Because you know what their background would be. 00:13:15.146 --> 00:13:15.870 PROFESSOR: No. 00:13:15.870 --> 00:13:16.953 KENNETH ABBOTT: All right. 00:13:16.953 --> 00:13:20.400 Just, mean and variance are constant. 00:13:20.400 --> 00:13:22.930 When I look at the time series itself, 00:13:22.930 --> 00:13:26.110 the time series mean and the time series variance 00:13:26.110 --> 00:13:28.520 are not constant. 00:13:28.520 --> 00:13:30.920 And there also could be other time series stuff going on. 00:13:30.920 --> 00:13:34.900 There could be seasonality, there could be autocorrelation. 00:13:34.900 --> 00:13:38.430 This looks something like a random walk 00:13:38.430 --> 00:13:40.560 but it's not stationary. 00:13:40.560 --> 00:13:43.371 It's hard for me to draw inference by looking at that 00:13:43.371 --> 00:13:43.870 alone. 00:13:43.870 --> 00:13:45.328 So we want to try to predict what's 00:13:45.328 --> 00:13:47.460 going to happen in the future, it's kind of hard. 00:13:47.460 --> 00:13:51.180 And the game, here, that we're playing, is we want to know 00:13:51.180 --> 00:13:55.510 how much money do I need to hold to support that position? 00:13:55.510 --> 00:13:58.146 Now, who here has taken an accounting course? 00:13:58.146 --> 00:14:00.780 All right, word to the wise-- there's 00:14:00.780 --> 00:14:03.020 two things I tell students in quant finance programs. 00:14:03.020 --> 00:14:05.686 First of all, I know you have to take a time series course-- I'm 00:14:05.686 --> 00:14:07.447 sure-- this is MIT. 00:14:07.447 --> 00:14:09.030 If you don't get a time series course, 00:14:09.030 --> 00:14:11.910 get your money back because you've got to take time series. 00:14:11.910 --> 00:14:13.540 Accounting is important. 00:14:13.540 --> 00:14:15.500 Accounting is important because so much 00:14:15.500 --> 00:14:17.350 of what we do, the way we think about things 00:14:17.350 --> 00:14:18.760 is predicated on the dollars. 00:14:18.760 --> 00:14:22.320 And you need to know how the dollars are recorded. 00:14:22.320 --> 00:14:22.850 Quick aside. 00:14:28.650 --> 00:14:29.530 Balance sheet. 00:14:29.530 --> 00:14:31.405 I'll give you a 30 second accounting lecture. 00:14:34.800 --> 00:14:38.230 Assets, what we own. 00:14:38.230 --> 00:14:41.410 Everything we own-- we have stuff, it's assets. 00:14:41.410 --> 00:14:43.560 We came to that stuff one of two ways. 00:14:43.560 --> 00:14:46.750 We either pay for it out of our pocket, or we borrowed money. 00:14:46.750 --> 00:14:48.150 There's no third way. 00:14:48.150 --> 00:14:50.624 So everything we own, we either paid 00:14:50.624 --> 00:14:52.290 for out of our pocket or borrowed money. 00:14:52.290 --> 00:14:57.950 The amount we paid for out of our pocket is the equity. 00:14:57.950 --> 00:15:01.000 The ratio of this to this is called leverage 00:15:01.000 --> 00:15:02.510 among other things. 00:15:02.510 --> 00:15:04.040 All right? 00:15:04.040 --> 00:15:05.400 If I'm this company. 00:15:05.400 --> 00:15:08.230 I have this much stuff and I bought it 00:15:08.230 --> 00:15:10.630 with this much debt, and this much equity. 00:15:10.630 --> 00:15:13.760 Again, that's a gross oversimplification. 00:15:13.760 --> 00:15:16.710 When this gets down to zero, it's game over. 00:15:16.710 --> 00:15:17.870 Belly up. 00:15:17.870 --> 00:15:19.160 All right? 00:15:19.160 --> 00:15:20.520 Does that make sense? 00:15:20.520 --> 00:15:23.590 Now you've taken a semester of accounting. 00:15:23.590 --> 00:15:26.050 No, only kidding. 00:15:26.050 --> 00:15:30.970 But it's actually important to have a grip on how that works. 00:15:30.970 --> 00:15:33.270 Because what we need to make sure of 00:15:33.270 --> 00:15:36.560 is that if we're going to take this position and hold it, 00:15:36.560 --> 00:15:40.690 we need to make sure that with some level of certainty-- 00:15:40.690 --> 00:15:43.310 every time we lose money this gets reduced. 00:15:43.310 --> 00:15:47.040 When this goes down to zero, I go bankrupt. 00:15:47.040 --> 00:15:48.650 So that's what we're trying to do. 00:15:48.650 --> 00:15:51.340 We need to protect this, and we do it 00:15:51.340 --> 00:15:55.220 by knowing how much of this could move against us. 00:15:55.220 --> 00:15:56.247 Everybody with me? 00:15:56.247 --> 00:15:57.080 Anybody not with me? 00:15:57.080 --> 00:16:00.624 It's OK to have questions, it really is. 00:16:00.624 --> 00:16:03.540 Excellent. 00:16:03.540 --> 00:16:05.680 All right, so if I do a frequency distribution 00:16:05.680 --> 00:16:10.980 of this time series, I just say, show me the frequency 00:16:10.980 --> 00:16:13.270 with which this thing shows. 00:16:13.270 --> 00:16:16.891 I get this thing, it's kind of trimodal. 00:16:16.891 --> 00:16:17.890 It's all over the place. 00:16:17.890 --> 00:16:19.480 It doesn't tell me anything. 00:16:19.480 --> 00:16:22.200 If I look at the levels-- the frequency distribution, 00:16:22.200 --> 00:16:24.290 the relative frequency distribution of the levels 00:16:24.290 --> 00:16:26.800 themselves, I don't get a whole lot of intuition. 00:16:26.800 --> 00:16:29.080 If I go into return space, which is either 00:16:29.080 --> 00:16:31.440 looking at the log differences from day to day, 00:16:31.440 --> 00:16:33.340 or the percentage changes from day to day, 00:16:33.340 --> 00:16:36.030 or perhaps the absolute changes from day to day-- 00:16:36.030 --> 00:16:38.810 it varies from market to market. 00:16:38.810 --> 00:16:41.910 Oh, look, now we're in familiar territory. 00:16:41.910 --> 00:16:44.260 So what I'm doing here-- and this 00:16:44.260 --> 00:16:46.350 is why I started out with a normal distribution 00:16:46.350 --> 00:16:48.640 because this thing is unimodal. 00:16:48.640 --> 00:16:50.310 It's more or less symmetric. 00:16:50.310 --> 00:16:50.960 Right? 00:16:50.960 --> 00:16:52.940 Now is it a perfect measure? 00:16:52.940 --> 00:16:55.146 No, because it's probably got fat tails. 00:16:55.146 --> 00:16:56.520 So it's a little bit like looking 00:16:56.520 --> 00:16:59.500 for the glasses you lost up on 67th Street down on 59th street 00:16:59.500 --> 00:17:01.140 because there's more light there. 00:17:01.140 --> 00:17:03.410 But it's a starting point. 00:17:03.410 --> 00:17:09.369 So what I'm saying to you is once I difference it-- no, 00:17:09.369 --> 00:17:10.660 I won't talk about [INAUDIBLE]. 00:17:10.660 --> 00:17:12.099 Once I difference the timeshares, 00:17:12.099 --> 00:17:14.682 once I take the timeshares and look at the percentage changes, 00:17:14.682 --> 00:17:17.140 and I look at the frequency distribution of those changes, 00:17:17.140 --> 00:17:20.109 I get this which is far more amenable. 00:17:20.109 --> 00:17:22.140 And I can draw inference from that. 00:17:22.140 --> 00:17:24.950 I can say, ah, now if this thing is normal, 00:17:24.950 --> 00:17:28.630 then I know that x% of my observations 00:17:28.630 --> 00:17:30.410 will take place over here. 00:17:30.410 --> 00:17:33.280 Now I can start drawing inferences. 00:17:33.280 --> 00:17:35.310 And a thing to keep in mind here, 00:17:35.310 --> 00:17:39.180 one thing we do constantly in statistics 00:17:39.180 --> 00:17:43.900 is we do parameter estimates. 00:17:43.900 --> 00:17:46.410 And remember, every time you estimate something 00:17:46.410 --> 00:17:48.480 you estimate it with error. 00:17:48.480 --> 00:17:50.610 I think that maybe the single most important thing 00:17:50.610 --> 00:17:52.951 I learned when I got my statistics degree. 00:17:52.951 --> 00:17:54.950 Everything you estimate you estimate with error. 00:17:54.950 --> 00:17:57.060 People do means, they say, oh, it's x. 00:17:57.060 --> 00:17:59.712 No, that's the average and that's an unbiased estimator, 00:17:59.712 --> 00:18:01.670 but guess what, there's a huge amount of noise. 00:18:01.670 --> 00:18:03.045 And there's a certain probability 00:18:03.045 --> 00:18:04.400 that you're wrong by x%. 00:18:04.400 --> 00:18:08.110 So every time we come up with a number, when somebody tells me 00:18:08.110 --> 00:18:11.300 the risk is 10, that means it's probably not 10,000, 00:18:11.300 --> 00:18:13.532 it's probably not zero. 00:18:13.532 --> 00:18:14.490 Just keep that in mind. 00:18:14.490 --> 00:18:17.200 Just sort of throw that in on the side for nothing. 00:18:17.200 --> 00:18:20.250 All right, so when I take the returns of this same time 00:18:20.250 --> 00:18:22.980 series, I get something that's unimodal, symmetric, 00:18:22.980 --> 00:18:24.560 may or may not have fat tails. 00:18:24.560 --> 00:18:26.750 That has important implications for whether or not 00:18:26.750 --> 00:18:28.550 my normal distribution underestimates 00:18:28.550 --> 00:18:30.300 the amount of risk I'm taking. 00:18:30.300 --> 00:18:32.990 Everybody with me on that more or less? 00:18:32.990 --> 00:18:33.744 Questions? 00:18:33.744 --> 00:18:34.660 Now would be the time. 00:18:38.000 --> 00:18:41.160 Good enough? 00:18:41.160 --> 00:18:42.660 He's lived this. 00:18:42.660 --> 00:18:44.170 All right. 00:18:44.170 --> 00:18:47.610 So once I have my time series of returns, which I just 00:18:47.610 --> 00:18:50.170 plotted there, I can gauge their dispersion 00:18:50.170 --> 00:18:52.382 with this measure called variance. 00:18:52.382 --> 00:18:53.715 And you guys probably know this. 00:19:01.390 --> 00:19:07.360 Variance, the expected value of x_i minus x bar-- I 00:19:07.360 --> 00:19:10.960 love these thick chalks-- squared. 00:19:10.960 --> 00:19:19.640 And it's the sum of x_i minus x bar squared over n minus 1. 00:19:19.640 --> 00:19:22.680 It's a measure of dispersion. 00:19:22.680 --> 00:19:25.940 Variance has its-- Now, I should say 00:19:25.940 --> 00:19:31.840 that this is sigma squared hat. 00:19:31.840 --> 00:19:32.340 Right? 00:19:32.340 --> 00:19:33.660 Estimate-- parameter estimate. 00:19:33.660 --> 00:19:35.190 Parameter. 00:19:35.190 --> 00:19:37.600 Parameter estimate. 00:19:37.600 --> 00:19:41.990 This is measured with error. 00:19:41.990 --> 00:19:45.616 Anybody here know what the distribution of this is? 00:19:45.616 --> 00:19:46.750 Anyone? 00:19:46.750 --> 00:19:49.270 $5. 00:19:49.270 --> 00:19:50.530 Close. 00:19:50.530 --> 00:19:51.810 n chi-squared. 00:19:51.810 --> 00:19:52.550 Worth $2. 00:19:52.550 --> 00:19:56.010 Talk to me after class. 00:19:56.010 --> 00:19:57.487 It's a chi-squared distribution. 00:19:57.487 --> 00:19:58.320 What does that mean? 00:19:58.320 --> 00:20:04.177 That means that we know it can't be 0 or less than 0. 00:20:04.177 --> 00:20:06.510 If you figure out a way to get variances less than zero, 00:20:06.510 --> 00:20:08.750 let's talk. 00:20:08.750 --> 00:20:10.810 And it's got a long right tail, but that's 00:20:10.810 --> 00:20:12.020 because this is squared. 00:20:12.020 --> 00:20:14.660 [INAUDIBLE] one point can move it up. 00:20:14.660 --> 00:20:18.100 Anyway, once I have my returns, I 00:20:18.100 --> 00:20:20.960 have a measure of the dispersion of these returns called 00:20:20.960 --> 00:20:22.290 variance. 00:20:22.290 --> 00:20:24.270 I take the square root of the variance, 00:20:24.270 --> 00:20:28.390 which is the standard deviation, or the volatility. 00:20:28.390 --> 00:20:29.890 When I'm doing it with a data set, 00:20:29.890 --> 00:20:32.220 I usually refer to it as the standard deviation. 00:20:32.220 --> 00:20:36.120 When I'm referring to the standard deviation 00:20:36.120 --> 00:20:39.500 of the distribution, I usually call it the standard error. 00:20:39.500 --> 00:20:42.072 Is that a law or is that just common parlance? 00:20:42.072 --> 00:20:43.052 PROFESSOR: Both. 00:20:43.052 --> 00:20:45.218 The standard error is typically for something that's 00:20:45.218 --> 00:20:47.264 random, like an estimate. 00:20:47.264 --> 00:20:50.087 Whereas the standard deviation is more like for sample-- 00:20:50.087 --> 00:20:51.170 KENNETH ABBOTT: Empirical. 00:20:51.170 --> 00:20:54.909 See, it's important because when you first learn this, 00:20:54.909 --> 00:20:55.950 they don't tell you that. 00:20:55.950 --> 00:20:57.420 And they flip them back and forth. 00:20:57.420 --> 00:20:59.420 And then when you take the intermediate courses, 00:20:59.420 --> 00:21:01.360 they say, no, don't use standard deviation 00:21:01.360 --> 00:21:03.460 when you mean standard error. 00:21:03.460 --> 00:21:06.350 And you'll get points off on your exam for that, right? 00:21:06.350 --> 00:21:09.710 All right, so, the standard deviation 00:21:09.710 --> 00:21:11.790 is the square root of the variance, 00:21:11.790 --> 00:21:13.790 also called the volatility. 00:21:13.790 --> 00:21:16.470 In a normal distribution, 1% of the observations 00:21:16.470 --> 00:21:18.775 is outside of 2.33 standard deviations. 00:21:21.330 --> 00:21:27.380 For 95%, it's out past 1.64, 1.645 standard deviations. 00:21:27.380 --> 00:21:28.880 Now you're saying, wait a minute, 00:21:28.880 --> 00:21:32.000 where did my 1.96 go that I learned as an undergrad. 00:21:32.000 --> 00:21:33.740 Two-sided. 00:21:33.740 --> 00:21:37.740 So if I go from the mean to 1.96 standard deviations 00:21:37.740 --> 00:21:40.570 on either side, that encompasses 95% 00:21:40.570 --> 00:21:43.000 of the total area of the integral from negative infinity 00:21:43.000 --> 00:21:43.880 to positive infinity. 00:21:43.880 --> 00:21:44.963 Everybody with me on that? 00:21:44.963 --> 00:21:46.560 Does that make sense? 00:21:46.560 --> 00:21:48.020 The two-sided versus one-sided. 00:21:48.020 --> 00:21:49.270 That's confused me. 00:21:49.270 --> 00:21:51.310 When I was your age, it confused me a lot. 00:21:51.310 --> 00:21:53.950 But I got there. 00:21:53.950 --> 00:21:56.960 All right so this is how we do it. 00:21:56.960 --> 00:22:00.092 Excel functions are VAR and-- you don't need to know that. 00:22:00.092 --> 00:22:02.300 All right, so in this case, I estimating the variance 00:22:02.300 --> 00:22:04.340 of this particular time series. 00:22:04.340 --> 00:22:06.920 I took the standard deviation by taking the square root 00:22:06.920 --> 00:22:09.040 of the variance. 00:22:09.040 --> 00:22:10.610 It's in percentages. 00:22:10.610 --> 00:22:12.990 When you do this, I tell you, it's like physics, 00:22:12.990 --> 00:22:16.737 your units will screw you up every time. 00:22:16.737 --> 00:22:17.570 What am I measuring? 00:22:17.570 --> 00:22:18.740 What are my units? 00:22:18.740 --> 00:22:20.130 I still make units mistakes. 00:22:20.130 --> 00:22:20.890 I want you to know that. 00:22:20.890 --> 00:22:21.880 And I'm in this business 30 years. 00:22:21.880 --> 00:22:23.400 I still make units mistakes. 00:22:23.400 --> 00:22:24.950 Just like physics. 00:22:24.950 --> 00:22:26.740 I'm in percentage change space, so I 00:22:26.740 --> 00:22:28.573 want to talk in terms of percentage changes. 00:22:28.573 --> 00:22:31.820 The standard deviation is 1.8% of that time series 00:22:31.820 --> 00:22:33.090 I showed you. 00:22:33.090 --> 00:22:36.410 So 2.33 standard deviations times the standard deviation 00:22:36.410 --> 00:22:38.720 is about 4.2%. 00:22:38.720 --> 00:22:46.280 What that says, given this data set-- one time series-- I'm 00:22:46.280 --> 00:22:50.430 saying, I expect to lose, on any given day, 00:22:50.430 --> 00:22:54.540 if I have that position, 99% of the time I'm going 00:22:54.540 --> 00:22:58.240 to lose 4.2% of it or less. 00:23:00.840 --> 00:23:01.472 Very important. 00:23:01.472 --> 00:23:02.180 Think about that. 00:23:02.180 --> 00:23:04.750 Is that clear? 00:23:04.750 --> 00:23:06.960 That's how I get there. 00:23:06.960 --> 00:23:09.880 I'm making a statement about the probability of loss. 00:23:09.880 --> 00:23:12.490 I'm saying there's a 1% probability, 00:23:12.490 --> 00:23:19.910 for that particular time series-- which is-- all right? 00:23:19.910 --> 00:23:21.530 If this is my historical data set 00:23:21.530 --> 00:23:24.640 and it's my only historical data set, and I own this, 00:23:24.640 --> 00:23:28.812 tomorrow I may be 4.2% lighter than I was today 00:23:28.812 --> 00:23:30.520 because the market could move against me. 00:23:30.520 --> 00:23:34.930 And I'm 99% sure, if the future's like the past, 00:23:34.930 --> 00:23:40.240 that my loss tomorrow is going to be 4.2% or less. 00:23:40.240 --> 00:23:42.670 That's VaR. 00:23:42.670 --> 00:23:45.860 Simplest case, assuming normal distribution, 00:23:45.860 --> 00:23:50.500 single asset, not fixed income. 00:23:50.500 --> 00:23:51.290 Yes, no? 00:23:51.290 --> 00:23:52.180 Questions, comments? 00:23:52.180 --> 00:23:58.779 AUDIENCE: Yes, [INAUDIBLE] positive and [INAUDIBLE]. 00:23:58.779 --> 00:23:59.820 KENNETH ABBOTT: Yes, yes. 00:23:59.820 --> 00:24:02.960 Assuming my distribution is symmetric. 00:24:02.960 --> 00:24:08.210 Now that's the right assumption to point out. 00:24:08.210 --> 00:24:11.110 Because in the real world, it may not be symmetric. 00:24:11.110 --> 00:24:13.580 And when we go into historical simulation, 00:24:13.580 --> 00:24:15.081 we use empirical distributions where 00:24:15.081 --> 00:24:17.163 we don't care if it's symmetric because we're only 00:24:17.163 --> 00:24:18.310 looking at the downside. 00:24:18.310 --> 00:24:21.390 And whether I'm long or short, I might 00:24:21.390 --> 00:24:24.337 care about the downside or the putative upside. 00:24:24.337 --> 00:24:26.170 Because I'm short, and I care about how much 00:24:26.170 --> 00:24:27.220 is going to move up. 00:24:27.220 --> 00:24:27.750 Make sense? 00:24:27.750 --> 00:24:29.720 That's the right question to ask. 00:24:29.720 --> 00:24:30.350 Yes? 00:24:30.350 --> 00:24:34.172 AUDIENCE: [INAUDIBLE] if you're doing it for upside as well? 00:24:34.172 --> 00:24:35.005 KENNETH ABBOTT: Yes. 00:24:35.005 --> 00:24:36.000 AUDIENCE: Could it just be the same thing? 00:24:36.000 --> 00:24:36.833 KENNETH ABBOTT: Yes. 00:24:36.833 --> 00:24:38.970 In fact, in this case, in what we're doing here 00:24:38.970 --> 00:24:41.380 of variance/covariance or closed form VaR, 00:24:41.380 --> 00:24:43.954 it's for long or short. 00:24:43.954 --> 00:24:45.870 But getting your signs right, I'm telling you, 00:24:45.870 --> 00:24:47.380 it's like physics. 00:24:47.380 --> 00:24:49.940 I still make that mistake. 00:24:49.940 --> 00:24:50.688 Yes? 00:24:50.688 --> 00:24:53.178 AUDIENCE: [INAUDIBLE] symmetric. 00:24:53.178 --> 00:24:55.600 Do you guys still use this process to say, OK-- 00:24:55.600 --> 00:24:59.300 KENNETH ABBOTT: I use it all the time as a heuristic. 00:24:59.300 --> 00:25:00.290 All right? 00:25:00.290 --> 00:25:05.530 Because let's say I've got-- and that's a very good question-- 00:25:05.530 --> 00:25:09.260 let's say I've got five years worth of data 00:25:09.260 --> 00:25:13.330 and I don't have time to do an empirical estimate. 00:25:13.330 --> 00:25:17.090 It could be lopsided. 00:25:17.090 --> 00:25:19.910 If you tell me a two standard deviation move 00:25:19.910 --> 00:25:22.880 is x, that means something to me. 00:25:22.880 --> 00:25:25.230 Now, there's a problem with that. 00:25:25.230 --> 00:25:28.432 And the problem is that people extrapolate that. 00:25:28.432 --> 00:25:30.890 Sometimes people talk to me and, oh, it's an eight standard 00:25:30.890 --> 00:25:32.610 deviation move. 00:25:32.610 --> 00:25:34.769 Eight standard deviation moves don't happen. 00:25:34.769 --> 00:25:36.935 I don't think we've seen an eight standard deviation 00:25:36.935 --> 00:25:40.120 move in the Cenozoic era. 00:25:40.120 --> 00:25:43.020 It just doesn't happen. 00:25:43.020 --> 00:25:45.500 Three standard deviation-- you will see a three standard 00:25:45.500 --> 00:25:50.160 deviation move once every 10,000 observations. 00:25:50.160 --> 00:25:53.874 Now, I learned this the hard way by just, 00:25:53.874 --> 00:25:55.540 see how many times do I have to do this? 00:25:55.540 --> 00:25:57.789 And then I looked it up in the table, oh, I was right. 00:26:00.400 --> 00:26:02.950 When we oversimplify, and start to talk about everything 00:26:02.950 --> 00:26:04.491 in terms of that normal distribution, 00:26:04.491 --> 00:26:07.100 we really just lose our grip on reality. 00:26:07.100 --> 00:26:10.020 But I use it as a heuristic all the time. 00:26:10.020 --> 00:26:12.457 I'll do it even now, and I know better. 00:26:12.457 --> 00:26:14.290 But I'll go, what's two standard deviations? 00:26:14.290 --> 00:26:15.664 What's three standard deviations? 00:26:15.664 --> 00:26:22.270 Because by and large-- and I still do this, I get my data 00:26:22.270 --> 00:26:25.560 and I line it up and I do frequency distributions. 00:26:25.560 --> 00:26:29.510 Hold on, I do this all the time with my data. 00:26:29.510 --> 00:26:30.660 Is it symmetric? 00:26:30.660 --> 00:26:32.050 Is it fat tailed? 00:26:32.050 --> 00:26:33.000 Is it unimodal? 00:26:33.000 --> 00:26:35.435 So that's a very good question. 00:26:35.435 --> 00:26:36.270 Any other questions? 00:26:36.270 --> 00:26:37.770 AUDIENCE: [INAUDIBLE] have we talked 00:26:37.770 --> 00:26:39.830 about the [? standard t ?] distribution? 00:26:39.830 --> 00:26:43.511 PROFESSOR: We Introduced it in the last lecture. 00:26:43.511 --> 00:26:48.540 And the problems set this week does relate to that. 00:26:48.540 --> 00:26:51.180 KENNETH ABBOTT: All right, perfect lead-in. 00:26:51.180 --> 00:26:54.090 So the statement I made, it's 1% of the time 00:26:54.090 --> 00:26:59.340 I'd expect to lose more than 4.2 pesos on 100 peso position. 00:26:59.340 --> 00:27:03.870 That's my inferential statement. 00:27:03.870 --> 00:27:08.020 In fact, over the same time period 00:27:08.020 --> 00:27:14.760 I lost 4.2% 1.5% of the time instead of 1% of the time. 00:27:14.760 --> 00:27:19.220 What that tells me, what that suggests to me, is my data set 00:27:19.220 --> 00:27:20.860 has fat tails. 00:27:20.860 --> 00:27:24.820 What that means is the likelihood of a loss-- 00:27:24.820 --> 00:27:27.340 a simple way of thinking about it [INAUDIBLE] care 00:27:27.340 --> 00:27:30.590 whether what that means in a metaphysical sense, 00:27:30.590 --> 00:27:32.410 a way to interpret it. 00:27:32.410 --> 00:27:35.290 The likelihood of a loss is greater 00:27:35.290 --> 00:27:38.491 than would be implied by the normal distribution. 00:27:38.491 --> 00:27:38.990 All right? 00:27:38.990 --> 00:27:41.330 So when you hear people say fat tails, 00:27:41.330 --> 00:27:43.870 generally, that's what they're talking about. 00:27:43.870 --> 00:27:46.370 There are different ways you could interpret that statement, 00:27:46.370 --> 00:27:49.860 but when somebody is talking about a financial time series, 00:27:49.860 --> 00:27:51.252 it has fat tails. 00:27:51.252 --> 00:27:52.960 Roughly 3/4 of your financial time series 00:27:52.960 --> 00:27:53.829 will have fat tails. 00:27:53.829 --> 00:27:55.620 They will also have time series properties, 00:27:55.620 --> 00:27:57.110 they won't be true random walks. 00:27:57.110 --> 00:27:59.080 True random walks says that I don't 00:27:59.080 --> 00:28:01.840 know whether it's going to go up or down based on the data 00:28:01.840 --> 00:28:03.150 I have. 00:28:03.150 --> 00:28:05.659 The time series has no memory. 00:28:05.659 --> 00:28:07.950 When we start introducing time series properties, which 00:28:07.950 --> 00:28:12.950 many financial time series have, then there's seasonality, 00:28:12.950 --> 00:28:15.130 there's mean reversion, there's all kinds 00:28:15.130 --> 00:28:17.130 of other stuff, other ways that we have to think 00:28:17.130 --> 00:28:19.250 about modeling the data. 00:28:19.250 --> 00:28:20.764 Make sense? 00:28:20.764 --> 00:28:25.930 AUDIENCE: [INAUDIBLE] higher standard deviation than 00:28:25.930 --> 00:28:26.430 [INAUDIBLE]. 00:28:26.430 --> 00:28:28.410 KENNETH ABBOTT: Say it once again. 00:28:28.410 --> 00:28:30.605 AUDIENCE: Better yield, does it mean 00:28:30.605 --> 00:28:33.030 that we have a higher standard deviation than [INAUDIBLE]? 00:28:33.030 --> 00:28:33.570 KENNETH ABBOTT: No. 00:28:33.570 --> 00:28:35.611 The standard deviation is the standard deviation. 00:28:37.820 --> 00:28:42.100 No matter what I do, this is standard deviation, that's it. 00:28:42.100 --> 00:28:44.790 Don't have a higher standard deviation. 00:28:44.790 --> 00:28:46.730 But the likelihood of-- the put it 00:28:46.730 --> 00:28:52.740 this way-- the likelihood of a move of 2.33 00:28:52.740 --> 00:28:54.860 standard deviations is more than 1%. 00:28:57.890 --> 00:28:59.670 That's the way I think of it. 00:28:59.670 --> 00:29:02.585 Make sense? 00:29:02.585 --> 00:29:05.091 AUDIENCE: Is there any way for you to [INAUDIBLE] to-- 00:29:05.091 --> 00:29:05.966 KENNETH ABBOTT: What? 00:29:05.966 --> 00:29:07.340 AUDIENCE: Sorry, is there any way 00:29:07.340 --> 00:29:09.690 to put into that graph what a fatter tail looks like? 00:29:09.690 --> 00:29:11.890 KENNETH ABBOTT: Oh, well, be patient. 00:29:11.890 --> 00:29:13.365 If we have time. 00:29:13.365 --> 00:29:14.740 In fact, we do that all the time. 00:29:14.740 --> 00:29:16.700 And one of our techniques doesn't care. 00:29:16.700 --> 00:29:18.290 It goes to the empirical distribution. 00:29:18.290 --> 00:29:20.620 So it captures the fat tails completely. 00:29:20.620 --> 00:29:25.210 In fact, the homework assignment which I usually 00:29:25.210 --> 00:29:27.010 precede this lecture by has people 00:29:27.010 --> 00:29:28.740 graphing all kinds of distributions 00:29:28.740 --> 00:29:31.384 to see what these things look like. 00:29:31.384 --> 00:29:32.550 We won't have time for that. 00:29:32.550 --> 00:29:35.130 But if you have questions, send them to me. 00:29:35.130 --> 00:29:37.520 I'll send you some stuff to read about this. 00:29:37.520 --> 00:29:39.228 All right, so now you know one asset VaR, 00:29:39.228 --> 00:29:41.240 now you're qualified to go work for a big bank. 00:29:41.240 --> 00:29:42.470 All right? 00:29:42.470 --> 00:29:44.530 Get your data, calculate returns. 00:29:44.530 --> 00:29:50.100 Now I usually put in step 2b, graph your data and look at it. 00:29:50.100 --> 00:29:50.630 All right? 00:29:50.630 --> 00:29:54.350 Because everybody's data has dirt in it. 00:29:54.350 --> 00:29:55.410 Don't trust anyone else. 00:29:55.410 --> 00:29:56.850 If you're going to get fired, get 00:29:56.850 --> 00:29:58.266 fired for being incompetent, don't 00:29:58.266 --> 00:30:01.310 get fired for using someone else's bad data. 00:30:01.310 --> 00:30:02.260 Don't trust anyone. 00:30:02.260 --> 00:30:05.100 My mother gives me data, Mom, I'm graphing it. 00:30:05.100 --> 00:30:09.290 Because I think you let some poop slip into my data. 00:30:09.290 --> 00:30:15.430 Mother Teresa could come to me with a thumb drive: "Ken, S&P 00:30:15.430 --> 00:30:17.316 500." 00:30:17.316 --> 00:30:18.190 Sorry, Mother Teresa. 00:30:18.190 --> 00:30:20.500 I'm graphing it before I use it. 00:30:20.500 --> 00:30:21.070 All right? 00:30:21.070 --> 00:30:23.236 So I don't want to say that this is usually in here. 00:30:23.236 --> 00:30:24.855 We do extensive error testing. 00:30:24.855 --> 00:30:27.670 Because there could be bad data, there could be missing data. 00:30:27.670 --> 00:30:31.210 And missing data is a whole other lecture that I give. 00:30:31.210 --> 00:30:32.723 You might be shocked at [INAUDIBLE]. 00:30:36.590 --> 00:30:40.590 So for one asset VaR, get my data, create my return series. 00:30:40.590 --> 00:30:42.010 Percentage changes, log changes. 00:30:42.010 --> 00:30:43.720 Sometimes that's absolute differences. 00:30:43.720 --> 00:30:46.053 Take the variance, take the square root of the variance, 00:30:46.053 --> 00:30:47.100 multiply by 2.33. 00:30:47.100 --> 00:30:47.800 Done and dusted. 00:30:47.800 --> 00:30:49.990 Go home, take your shoes off, relax. 00:30:53.730 --> 00:30:54.640 OK. 00:30:54.640 --> 00:30:57.290 Percentage changes versus log changes. 00:30:57.290 --> 00:31:00.530 For all intents and purposes, it doesn't really matter 00:31:00.530 --> 00:31:04.161 and I will often use one or the other. 00:31:04.161 --> 00:31:19.030 The way I think about this-- all right, 00:31:19.030 --> 00:31:21.110 there'll be a little bit of bias at the ends. 00:31:21.110 --> 00:31:23.240 But for the overwhelming bulk of the observations 00:31:23.240 --> 00:31:25.800 whether you use percentage changes or log changes 00:31:25.800 --> 00:31:27.200 doesn't matter. 00:31:27.200 --> 00:31:30.250 Generally, even though I know the data is closer 00:31:30.250 --> 00:31:32.850 to log-normally distributed than normally distributed, 00:31:32.850 --> 00:31:36.082 I'll use percentage changes just because it's easier. 00:31:36.082 --> 00:31:37.790 Why would we use log-normal distribution? 00:31:37.790 --> 00:31:42.587 Well, when we're doing simulation, 00:31:42.587 --> 00:31:44.920 the log-normal distribution has this very nifty property 00:31:44.920 --> 00:31:48.430 of keeping your yields from going negative. 00:31:48.430 --> 00:31:50.440 But, even that-- I can call that into 00:31:50.440 --> 00:31:52.510 question because there are instances 00:31:52.510 --> 00:31:54.436 of yields going negative. 00:31:54.436 --> 00:31:55.670 It's happened. 00:31:55.670 --> 00:31:58.180 Doesn't happen a lot, but it happens. 00:31:58.180 --> 00:31:58.824 All right. 00:31:58.824 --> 00:32:00.240 So I talked about bad data, talked 00:32:00.240 --> 00:32:02.420 about one-sided versus two-sided. 00:32:02.420 --> 00:32:05.727 I'll talk about longs and shorts a little bit later when we 00:32:05.727 --> 00:32:06.810 we're talking multi-asset. 00:32:11.040 --> 00:32:14.280 I'm going to cover a fixed income piece. 00:32:14.280 --> 00:32:17.410 We use this thing called a PV01 because what I measure in fixed 00:32:17.410 --> 00:32:18.730 income markets isn't a price. 00:32:18.730 --> 00:32:20.290 I usually measure a yield. 00:32:20.290 --> 00:32:23.592 I have to get from a change of yield to a change of price. 00:32:23.592 --> 00:32:25.050 Hmm, sounds like a Jacobian, right? 00:32:25.050 --> 00:32:27.990 With kind of a poor man's Jacobian. 00:32:27.990 --> 00:32:30.560 It's a measure that captures the fact 00:32:30.560 --> 00:32:36.340 that my price-yield relationship-- 00:32:36.340 --> 00:32:40.180 price, yield-- is non-linear. 00:32:40.180 --> 00:32:44.700 For any small approximation I look at the tangent. 00:32:44.700 --> 00:32:49.020 And I use my PV01 which has a similar notion to duration, 00:32:49.020 --> 00:32:51.390 but PV01 is a little more practical. 00:32:51.390 --> 00:32:53.019 The slope of that tells me how much 00:32:53.019 --> 00:32:55.060 my price will change for a given change of yield. 00:32:55.060 --> 00:32:55.690 See, there it is. 00:32:55.690 --> 00:32:57.280 You knew you were going to use the calculus, right? 00:32:57.280 --> 00:32:58.730 You're always using the calculus. 00:32:58.730 --> 00:33:01.280 You can't escape it. 00:33:01.280 --> 00:33:03.380 But the price-yield line is non-linear. 00:33:03.380 --> 00:33:05.710 But for all intents and purposes, what I'm doing is 00:33:05.710 --> 00:33:07.940 I'm shifting the price-yield relationship-- 00:33:07.940 --> 00:33:10.120 I'm shifting my yield change into price change 00:33:10.120 --> 00:33:14.560 by multiplying my yield change by my PV01 which 00:33:14.560 --> 00:33:21.131 is my price sensitivity to 1/100th percent move in yields. 00:33:21.131 --> 00:33:22.380 Think about that for a second. 00:33:22.380 --> 00:33:23.880 We don't have time to-- I would love 00:33:23.880 --> 00:33:26.270 to spend an hour on this, and about trading strategies, 00:33:26.270 --> 00:33:28.524 and about bull steepeners and bear steepeners 00:33:28.524 --> 00:33:30.690 and barbell trades, but we don't have time for that. 00:33:30.690 --> 00:33:35.090 Suffice to say if I'm measuring yields the thing 00:33:35.090 --> 00:33:41.960 is going to trade as a 789 or a 622 or a 401 yield. 00:33:41.960 --> 00:33:43.880 How do I get that in the change in price? 00:33:43.880 --> 00:33:46.046 Because I can't tell my boss, hey, I had a good day. 00:33:46.046 --> 00:33:48.360 I bought it at 402 and sold it at 401. 00:33:48.360 --> 00:33:50.170 No, how much money did you make? 00:33:50.170 --> 00:33:51.920 Yield to coffee break yield to lunch time, 00:33:51.920 --> 00:33:53.880 yield to go home at the end of day. 00:33:53.880 --> 00:33:56.710 How do I get from change in yield to change in price? 00:33:56.710 --> 00:33:58.330 Usually PV01. 00:33:58.330 --> 00:34:00.446 I could use duration. 00:34:00.446 --> 00:34:02.820 Bond traders who think in terms of yield to coffee break, 00:34:02.820 --> 00:34:04.515 yield to lunch time, yield to go home at the end of the day 00:34:04.515 --> 00:34:05.525 typically think in terms of PV01. 00:34:05.525 --> 00:34:07.130 Do you agree with that statement? 00:34:07.130 --> 00:34:08.420 AUDIENCE: [INAUDIBLE] 00:34:08.420 --> 00:34:10.503 KENNETH ABBOTT: How often on the fixed income desk 00:34:10.503 --> 00:34:11.957 did you use duration measures? 00:34:11.957 --> 00:34:15.880 AUDIENCE: Well, actually, [INAUDIBLE]. 00:34:15.880 --> 00:34:18.070 KENNETH ABBOTT: Because of the investor horizon? 00:34:18.070 --> 00:34:20.761 OK, the insurance companies. 00:34:20.761 --> 00:34:23.219 Very important point I want to reach here as a quick aside. 00:34:23.219 --> 00:34:25.389 You're going to hear this notion of PV01, 00:34:25.389 --> 00:34:28.239 which is called PVBP or DV01. 00:34:28.239 --> 00:34:31.460 That's the price sensitivity to a one basis point move. 00:34:31.460 --> 00:34:35.840 One basis point is 1/100th of a percent in yield. 00:34:35.840 --> 00:34:41.360 Duration is the half life, essentially, of my cash flow. 00:34:41.360 --> 00:34:44.870 What's the weighted expected time to owe my cash flows? 00:34:44.870 --> 00:34:51.070 If my duration is 7.9 years, my PV01 is probably about $790 00:34:51.070 --> 00:34:53.500 per million. 00:34:53.500 --> 00:34:56.069 In terms of significant digits, they're roughly the same 00:34:56.069 --> 00:34:58.610 but they have different meanings and the units are different. 00:34:58.610 --> 00:35:03.530 Duration is measured in yield, PV01 is measured in dollars. 00:35:03.530 --> 00:35:06.280 In bond space I typically think in PV01. 00:35:06.280 --> 00:35:08.580 If I'm selling to long term investors 00:35:08.580 --> 00:35:11.530 they have particular demands because they've got cash flow 00:35:11.530 --> 00:35:13.310 payments they have to hedge. 00:35:13.310 --> 00:35:15.740 So they may think of it in terms of duration. 00:35:15.740 --> 00:35:20.560 For our purposes, we're talking DV01 or PV01 or PVBP, 00:35:20.560 --> 00:35:23.050 those three terms more or less equal. 00:35:23.050 --> 00:35:23.710 Make sense? 00:35:23.710 --> 00:35:24.553 Yes? 00:35:24.553 --> 00:35:27.018 AUDIENCE: [INAUDIBLE] in terms of [INAUDIBLE] versus 00:35:27.018 --> 00:35:28.259 [INAUDIBLE]? 00:35:28.259 --> 00:35:29.300 KENNETH ABBOTT: We could. 00:35:29.300 --> 00:35:32.850 In some instances, in some areas and options 00:35:32.850 --> 00:35:34.800 we might look at an overall 1% move. 00:35:34.800 --> 00:35:38.000 But we have to look at what trades in the market. 00:35:38.000 --> 00:35:40.970 What trades in the market is the yield. 00:35:40.970 --> 00:35:42.710 When we quote the yield, I'm going 00:35:42.710 --> 00:35:45.380 to quote it going from 702 to 701. 00:35:45.380 --> 00:35:47.840 I'm not going to have the calculator handy to say, 00:35:47.840 --> 00:35:49.410 a 702 move to a 701. 00:35:49.410 --> 00:35:52.990 What's 702 minus 701 divided by 702? 00:35:52.990 --> 00:35:54.000 Make sense? 00:35:54.000 --> 00:35:56.159 It's the path of least resistance. 00:35:56.159 --> 00:35:58.450 What's the difference between a bond and a bond trader? 00:35:58.450 --> 00:36:00.116 A bond matures. 00:36:00.116 --> 00:36:01.892 A little fixed income humor for you. 00:36:01.892 --> 00:36:02.850 Apparently very little. 00:36:06.659 --> 00:36:08.450 I don't want to spend too much time on this 00:36:08.450 --> 00:36:11.570 because we just don't have the time. 00:36:11.570 --> 00:36:12.930 I provide an example here. 00:36:12.930 --> 00:36:15.141 If you guys want examples, contact me. 00:36:15.141 --> 00:36:17.390 I'll send you the spreadsheets I use for other classes 00:36:17.390 --> 00:36:19.056 if you just want to play around with it. 00:36:21.320 --> 00:36:26.360 When I talk about PV01, when I talk about yields, 00:36:26.360 --> 00:36:28.509 I usually have some kind of risk-free rate. 00:36:28.509 --> 00:36:30.800 Although this whole notion of the risk-free rate, which 00:36:30.800 --> 00:36:33.680 is-- so much of modern finance is predicated 00:36:33.680 --> 00:36:35.800 on this assumption that there is a risk-free rate, 00:36:35.800 --> 00:36:37.460 which used to be considered the US treasury. 00:36:37.460 --> 00:36:38.918 It used to be considered risk-free. 00:36:38.918 --> 00:36:43.390 Well, there's a credit spread out there for US Treasury. 00:36:43.390 --> 00:36:45.750 I don't mean to throw a monkey wrench into the works. 00:36:45.750 --> 00:36:47.739 But there's no such thing. 00:36:47.739 --> 00:36:50.030 I'm not going to question 75 years of academic finance. 00:36:50.030 --> 00:36:52.210 But it's troublesome. 00:36:52.210 --> 00:36:55.440 Just like when I was taking economics 30 years ago, 00:36:55.440 --> 00:36:57.850 inflation just mucked with everything. 00:36:57.850 --> 00:36:59.555 All of the models fell apart. 00:36:59.555 --> 00:37:01.260 There were appendices to every chapter 00:37:01.260 --> 00:37:04.610 on how you have to change this model to address inflation. 00:37:04.610 --> 00:37:07.500 And then inflation went away and everything was better. 00:37:07.500 --> 00:37:09.140 But this may not go away. 00:37:09.140 --> 00:37:10.650 I've got two components here. 00:37:10.650 --> 00:37:17.530 If the yield is 6%, I might have a 450 treasury rate and 150 00:37:17.530 --> 00:37:19.820 basis point credit spread. 00:37:19.820 --> 00:37:22.430 The credit spread reflects the probability of default. 00:37:22.430 --> 00:37:24.770 And I don't want to get into measures of risk neutrality 00:37:24.770 --> 00:37:25.830 here. 00:37:25.830 --> 00:37:29.840 But if I'm an issuer and I have a chance of default, 00:37:29.840 --> 00:37:33.010 I have to pay my investors more. 00:37:33.010 --> 00:37:40.767 Usually when we measure sensitivity 00:37:40.767 --> 00:37:42.600 we talk about that credit spread sensitivity 00:37:42.600 --> 00:37:45.340 and the risk-free sensitivity. 00:37:45.340 --> 00:37:47.492 We say, well, how could they possibly be different? 00:37:47.492 --> 00:37:49.200 And I don't want to get into detail here, 00:37:49.200 --> 00:37:52.380 but the notion is, when credit spreads start getting high, 00:37:52.380 --> 00:37:56.964 it implies a higher probability of default. 00:37:56.964 --> 00:37:59.380 You have to think about credit spread sensitivity a little 00:37:59.380 --> 00:37:59.750 differently. 00:37:59.750 --> 00:38:01.541 Because when you get to 1,000 basis points, 00:38:01.541 --> 00:38:03.450 1,500 basis points credit spread, 00:38:03.450 --> 00:38:04.920 it's a high probability of default. 00:38:04.920 --> 00:38:07.230 And your credit models will think differently. 00:38:07.230 --> 00:38:09.270 Your credit models will say, ah, that means 00:38:09.270 --> 00:38:11.260 I'm not going to get my next three payments. 00:38:11.260 --> 00:38:13.690 There's an expected, there's a probability of default, 00:38:13.690 --> 00:38:16.400 there's a loss given default, and there's recovery. 00:38:16.400 --> 00:38:18.660 A bunch of other stochastic measures come into play. 00:38:18.660 --> 00:38:20.180 I don't want to spend any more time on it because it's just 00:38:20.180 --> 00:38:21.820 going to confuse you now. 00:38:21.820 --> 00:38:25.720 Suffice to say we have these yields and yields 00:38:25.720 --> 00:38:29.390 are composed of risk-free rates and credit spreads. 00:38:29.390 --> 00:38:31.597 And I apologize for rushing through that, 00:38:31.597 --> 00:38:32.930 but we don't have time to do it. 00:38:36.290 --> 00:38:38.450 Typically you have more than one asset. 00:38:38.450 --> 00:38:43.030 So in this framework where I take 2.33 standard deviations 00:38:43.030 --> 00:38:46.370 times my dollar investment, or my renminbi investment 00:38:46.370 --> 00:38:47.705 or my sterling investment. 00:38:50.520 --> 00:38:52.060 That example was with one asset. 00:38:52.060 --> 00:38:55.010 If I want to expand this, I can expand 00:38:55.010 --> 00:39:01.276 this using this notion of covariance and correlation. 00:39:01.276 --> 00:39:03.650 You guys covered correlation and covariance at some point 00:39:03.650 --> 00:39:04.940 in your careers? 00:39:04.940 --> 00:39:05.780 Yes, no? 00:39:05.780 --> 00:39:08.460 All right? 00:39:08.460 --> 00:39:12.820 Both of them measure the way one asset moves vis-à-vis another 00:39:12.820 --> 00:39:14.310 asset. 00:39:14.310 --> 00:39:19.040 Correlation is scaled between negative 1 and positive 1. 00:39:19.040 --> 00:39:23.280 So I think of correlation as an index of linearity. 00:39:23.280 --> 00:39:24.650 Covariance is not scaled. 00:39:24.650 --> 00:39:26.290 I'll give you an example of the difference between covariance 00:39:26.290 --> 00:39:27.240 and correlation. 00:39:27.240 --> 00:39:31.430 What if I have 50 years of data on crop yields 00:39:31.430 --> 00:39:35.900 and that same 50 years of data on tons of fertilizer used? 00:39:35.900 --> 00:39:38.550 I would expect a positive correlation 00:39:38.550 --> 00:39:41.709 between tons of fertilizer used and crop yields. 00:39:41.709 --> 00:39:43.750 So the correlation would exist between negative 1 00:39:43.750 --> 00:39:45.360 and positive 1. 00:39:45.360 --> 00:39:47.072 The covariance could be any number, 00:39:47.072 --> 00:39:48.780 and that covariance will change depending 00:39:48.780 --> 00:39:50.670 on whether I measure my fertilizer 00:39:50.670 --> 00:39:56.950 in tons, or in pounds, or in ounces, or in kilos. 00:39:56.950 --> 00:39:59.500 The correlation will always be exactly the same. 00:39:59.500 --> 00:40:03.630 The linear relationship is captured by the correlation. 00:40:03.630 --> 00:40:08.440 But the units-- in covariance, the units count. 00:40:08.440 --> 00:40:14.100 If I have covariance-- here it is. 00:40:23.050 --> 00:40:25.450 Covariance matrices are symmetric. 00:40:25.450 --> 00:40:28.940 They have the variance along the diagonal. 00:40:28.940 --> 00:40:31.500 And the covariance is on the off-diagonal. 00:40:31.500 --> 00:40:33.980 Which is to say that the variance is the covariance 00:40:33.980 --> 00:40:36.510 of an item with itself. 00:40:36.510 --> 00:40:39.750 The correlation matrix, also symmetric, 00:40:39.750 --> 00:40:44.800 is the same thing scaled, with correlations, 00:40:44.800 --> 00:40:47.900 where the diagonal is 1.0. 00:40:47.900 --> 00:40:51.430 If I have covariance-- because correlation 00:40:51.430 --> 00:41:01.560 is covariance-- covariance divided 00:41:01.560 --> 00:41:04.100 by the product of the standard deviations 00:41:04.100 --> 00:41:07.650 gets me-- sorry-- correlation hat. 00:41:12.280 --> 00:41:15.410 This is like the apostrophe in French. 00:41:15.410 --> 00:41:16.980 You forget it all the time. 00:41:16.980 --> 00:41:20.060 But the one time you really need it, you won't do it 00:41:20.060 --> 00:41:22.820 and you'll be in trouble. 00:41:22.820 --> 00:41:25.650 If you have the covariances, you can get to the correlations. 00:41:25.650 --> 00:41:27.090 If you have the correlations, you 00:41:27.090 --> 00:41:29.700 can't get to the covariances unless you know the variances. 00:41:29.700 --> 00:41:31.750 That's a classic mid-term question. 00:41:31.750 --> 00:41:37.300 I give that almost-- not every year, maybe every other year. 00:41:37.300 --> 00:41:39.350 Don't have time to spend much more time on it. 00:41:39.350 --> 00:41:41.500 Suffice to say this measure of covariance 00:41:41.500 --> 00:41:43.870 says when x is a certain distance from its mean, 00:41:43.870 --> 00:41:47.131 how far is y from its mean and in what direction? 00:41:47.131 --> 00:41:47.630 Yes? 00:41:50.520 --> 00:41:52.320 Now this is just a little empirical stuff 00:41:52.320 --> 00:41:54.400 because I'm not as clever as you guys. 00:41:54.400 --> 00:41:56.530 And I don't trust anyone. 00:41:56.530 --> 00:42:00.550 I read it in the textbook, I don't trust anyone. 00:42:00.550 --> 00:42:03.707 a, b, here's a plus b. 00:42:03.707 --> 00:42:06.040 Variance of a plus b is variance of a plus variance of b 00:42:06.040 --> 00:42:07.340 plus 2 times covariance a b. 00:42:07.340 --> 00:42:10.180 It's not just a good idea, it's the law. 00:42:10.180 --> 00:42:12.350 I saw it in a thousand statistics textbooks, 00:42:12.350 --> 00:42:13.762 I tested it anyway. 00:42:13.762 --> 00:42:15.220 Because if I want to get fired, I'm 00:42:15.220 --> 00:42:17.170 going to get fired for making my own mistake, 00:42:17.170 --> 00:42:18.590 not making someone else's mistake. 00:42:18.590 --> 00:42:20.280 I do this all the time. 00:42:20.280 --> 00:42:22.610 And I just prove it empirically here. 00:42:22.610 --> 00:42:26.752 The proof of which will be left to the reader as an exercise. 00:42:26.752 --> 00:42:28.313 I hated when books said that. 00:42:28.313 --> 00:42:29.771 PROFESSOR: I actually kind of think 00:42:29.771 --> 00:42:31.385 that's a proven point, that you really 00:42:31.385 --> 00:42:34.719 should never trust output from computer programs or packages-- 00:42:34.719 --> 00:42:36.760 KENNETH ABBOTT: Or your mother, or Mother Teresa. 00:42:36.760 --> 00:42:38.218 PROFESSOR: It's good to check them. 00:42:38.218 --> 00:42:39.344 Check all the calculations. 00:42:39.344 --> 00:42:41.717 KENNETH ABBOTT: Mother Teresa will slip you some bad data 00:42:41.717 --> 00:42:42.335 if she can. 00:42:42.335 --> 00:42:43.550 I'm telling you, she will. 00:42:43.550 --> 00:42:45.420 She's tricky that way. 00:42:45.420 --> 00:42:47.370 Don't trust anyone. 00:42:47.370 --> 00:42:51.010 I've caught mistakes in software, all right? 00:42:51.010 --> 00:42:56.930 I had a programmer-- it's one of my favorite stories-- 00:42:56.930 --> 00:42:59.260 we're doing one of our first Monte Carlo simulations, 00:42:59.260 --> 00:43:00.990 and we're factoring a matrix. 00:43:00.990 --> 00:43:05.640 If we have time, we'll get-- so I factor a covariance matrix 00:43:05.640 --> 00:43:13.900 into E transpose lambda E. It's our friend the quadratic form. 00:43:13.900 --> 00:43:16.330 We're going to see this again. 00:43:16.330 --> 00:43:20.872 And this is a diagonal matrix of eigenvalues. 00:43:20.872 --> 00:43:22.330 And I take the square root of that. 00:43:22.330 --> 00:43:28.060 So I can say this is E transpose lambda to the 1/2 lambda 00:43:28.060 --> 00:43:30.860 to the 1/2 E. 00:43:30.860 --> 00:43:33.850 And so my programmer had gotten this, and I said, 00:43:33.850 --> 00:43:34.910 do me a favor. 00:43:34.910 --> 00:43:38.040 I said, take this, and transpose and multiply by itself. 00:43:38.040 --> 00:43:39.730 So take the square root and multiply it 00:43:39.730 --> 00:43:43.650 by the other square root, and show me that you get this. 00:43:43.650 --> 00:43:44.410 Just show me. 00:43:44.410 --> 00:43:45.130 He said I got it. 00:43:45.130 --> 00:43:45.880 I said you got it? 00:43:45.880 --> 00:43:48.450 He said out to 16 decimals. 00:43:48.450 --> 00:43:50.510 I said stop. 00:43:50.510 --> 00:43:56.075 On my block, the square root of 2 times the square root of 2 00:43:56.075 --> 00:43:58.820 equals 2.0. 00:43:58.820 --> 00:43:59.860 All right? 00:43:59.860 --> 00:44:04.470 2.0000000, what do you mean out to 16 decimal places? 00:44:04.470 --> 00:44:06.980 What planet are you on? 00:44:06.980 --> 00:44:09.640 And I scratched the surface, and I dug, 00:44:09.640 --> 00:44:11.570 and I asked a bunch of questions. 00:44:11.570 --> 00:44:13.385 And it turned out in this code he 00:44:13.385 --> 00:44:16.390 was passing a float to a fixed. 00:44:16.390 --> 00:44:17.760 All right? 00:44:17.760 --> 00:44:20.480 Don't trust anyone's software. 00:44:20.480 --> 00:44:21.330 Check it yourself. 00:44:24.690 --> 00:44:27.610 Someday when I'm dead and you guys are in my position, 00:44:27.610 --> 00:44:29.880 you'll be thanking me for that. 00:44:29.880 --> 00:44:31.657 Put a stone on my grave or something. 00:44:34.500 --> 00:44:36.154 All right so covariance. 00:44:36.154 --> 00:44:37.820 Covariance tells me some measure of when 00:44:37.820 --> 00:44:41.250 x moves, how far does y move? [? Or ?] for any other asset? 00:44:41.250 --> 00:44:42.750 Could I have a piece of your cookie? 00:44:42.750 --> 00:44:43.670 I hardly had lunch. 00:44:43.670 --> 00:44:45.461 You want me to have a piece of this, right? 00:44:48.143 --> 00:44:49.559 It's just looking very good there. 00:44:49.559 --> 00:44:50.059 Thank you. 00:44:54.250 --> 00:44:55.140 It's foraging. 00:44:55.140 --> 00:44:57.940 I'm convinced 10 million years ago, 00:44:57.940 --> 00:44:59.620 my ape ancestors were the first one 00:44:59.620 --> 00:45:03.350 at the dead antelope on the planes. 00:45:03.350 --> 00:45:05.314 All right. 00:45:05.314 --> 00:45:08.960 So we're talking about correlation, covariance. 00:45:08.960 --> 00:45:10.190 Covariance is not unit free. 00:45:12.840 --> 00:45:16.420 I can use either, but I have to make sure I get my units right. 00:45:16.420 --> 00:45:18.130 Units screw me up every time. 00:45:18.130 --> 00:45:21.610 They still screw me up. 00:45:21.610 --> 00:45:23.570 That was a good cookie. 00:45:23.570 --> 00:45:26.030 All right. 00:45:26.030 --> 00:45:28.220 So more facts. 00:45:28.220 --> 00:45:31.030 Variance of xa times yb; x squared 00:45:31.030 --> 00:45:35.467 variance a, y squared variance b plus 2xy covariance ab. 00:45:35.467 --> 00:45:36.550 You guys seen this before? 00:45:36.550 --> 00:45:37.530 I assume you have. 00:45:40.240 --> 00:45:43.390 Now I can get pretty silly with this if I want. 00:45:43.390 --> 00:45:46.970 x, a, y, b you get the picture, right? 00:45:46.970 --> 00:45:51.160 But what you should be thinking, this 00:45:51.160 --> 00:45:58.860 is a covariance matrix, sigma squared, sigma squared, 00:45:58.860 --> 00:46:00.840 sigma squared. 00:46:00.840 --> 00:46:04.050 It's the sum of the variances plus 2 times the sum 00:46:04.050 --> 00:46:05.500 of the covariances. 00:46:05.500 --> 00:46:09.100 So if I have one unit of every asset, I've got n assets, 00:46:09.100 --> 00:46:12.300 all have to do to get the portfolio variance is sum up 00:46:12.300 --> 00:46:13.890 the whole covariance matrix. 00:46:13.890 --> 00:46:16.700 Now, you never get only one unit, but just saying. 00:46:16.700 --> 00:46:20.840 But you notice that this is kind of a regular pattern 00:46:20.840 --> 00:46:22.770 that we see here. 00:46:22.770 --> 00:46:24.210 And so what I can do is I can use 00:46:24.210 --> 00:46:27.230 a combination of my correlation matrix 00:46:27.230 --> 00:46:31.180 and a little bit of linear algebra legerdemain, 00:46:31.180 --> 00:46:33.330 to do some very convenient calculations. 00:46:33.330 --> 00:46:35.580 And here I just give an example of a covariance matrix 00:46:35.580 --> 00:46:37.060 and a correlation matrix. 00:46:37.060 --> 00:46:39.607 Note the correlation matrices between negative 1 00:46:39.607 --> 00:46:41.430 and positive 1. 00:46:41.430 --> 00:46:41.930 All right. 00:46:44.932 --> 00:46:46.140 Let me cut to the chase here. 00:46:51.724 --> 00:46:53.140 I'll draw it here because I really 00:46:53.140 --> 00:46:57.180 want to get into some of the other stuff. 00:46:57.180 --> 00:47:04.160 What this means, if I have a covariance structure sigma. 00:47:04.160 --> 00:47:06.830 And I have a vector of positions, 00:47:06.830 --> 00:47:10.430 x dollars in dollar/yen, y dollars in gold, 00:47:10.430 --> 00:47:13.800 z dollars in oil. 00:47:13.800 --> 00:47:17.140 And let's say I've got a position vector, 00:47:17.140 --> 00:47:22.560 x_1, x_2, x_3, x_n. 00:47:28.730 --> 00:47:31.580 If I have all my positions recorded 00:47:31.580 --> 00:47:34.450 as a vector-- this is asset one, asset two, and this 00:47:34.450 --> 00:47:41.000 is in dollars-- and I have the covariance structure, 00:47:41.000 --> 00:47:44.490 the variance of this portfolio that 00:47:44.490 --> 00:47:47.460 has these assets and this covariance structure-- this 00:47:47.460 --> 00:48:01.780 is where the magic happens-- is x transpose sigma x equals 00:48:01.780 --> 00:48:04.260 sigma squared hat portfolio. 00:48:07.500 --> 00:48:10.900 Now you really could go work for a bank. 00:48:10.900 --> 00:48:14.580 This is how portfolio variance, using 00:48:14.580 --> 00:48:17.640 the variance/covariance method, is done. 00:48:17.640 --> 00:48:22.740 In fact, when we were doing it this way 20 years ago, 00:48:22.740 --> 00:48:25.380 spreadsheets only have 256 columns. 00:48:25.380 --> 00:48:27.600 So we tried to simplify everything into 256-- 00:48:27.600 --> 00:48:29.270 or sometimes you had to sum it up 00:48:29.270 --> 00:48:31.560 using two different spreadsheets. 00:48:31.560 --> 00:48:33.620 We didn't have multitab spreadsheets. 00:48:33.620 --> 00:48:36.740 That was a dream, multitab spreadsheets. 00:48:36.740 --> 00:48:40.530 This was Lotus 1-2-3 we're talking about here, OK? 00:48:40.530 --> 00:48:42.600 You guys don't even know what Lotus 1-2-3 is. 00:48:42.600 --> 00:48:45.121 It's like an abacus but on the screen. 00:48:45.121 --> 00:48:45.620 Yes? 00:48:45.620 --> 00:48:47.107 AUDIENCE: What's x again in this? 00:48:47.107 --> 00:48:48.440 KENNETH ABBOTT: Position vector. 00:48:48.440 --> 00:48:52.050 Let's say I tell you that you've got dollar/yen, gold, and oil. 00:48:52.050 --> 00:48:58.490 You've got $100 of dollar/yen, $50 of oil, and $25 of gold. 00:48:58.490 --> 00:49:01.370 It would be 100, 50, 25. 00:49:01.370 --> 00:49:05.189 Now, I should say $100 of dollar/yen, 00:49:05.189 --> 00:49:06.980 your position vector would actually show up 00:49:06.980 --> 00:49:10.240 as negative 100, 50, 25. 00:49:10.240 --> 00:49:11.150 Why is that? 00:49:11.150 --> 00:49:14.330 Because if I'm measuring my dollar/yen-- 00:49:14.330 --> 00:49:18.830 and this is just a little aside-- typically, 00:49:18.830 --> 00:49:22.740 I measure dollar/yen in yen per dollar. 00:49:22.740 --> 00:49:25.800 So dollar/yen might be 95. 00:49:25.800 --> 00:49:30.640 If I own yen and I'm a dollar investor and I own yen, 00:49:30.640 --> 00:49:34.340 and yen go from 95 per dollar to 100 per dollar, 00:49:34.340 --> 00:49:37.270 do I make or lose money? 00:49:37.270 --> 00:49:38.540 I lose money. 00:49:38.540 --> 00:49:42.090 Negative 100. 00:49:42.090 --> 00:49:43.840 Just store that. 00:49:43.840 --> 00:49:47.699 You won't be tested on that, but we think about that 00:49:47.699 --> 00:49:48.240 all the time. 00:49:48.240 --> 00:49:49.410 Same thing with yields. 00:49:49.410 --> 00:49:51.694 Typically, when I record my PV01-- 00:49:51.694 --> 00:49:53.860 and I'll record some version, something like my PV01 00:49:53.860 --> 00:49:55.630 in that vector, my interest rate sensitivity, 00:49:55.630 --> 00:49:57.171 I'm going to record it as a negative. 00:49:57.171 --> 00:50:01.190 Because when yields go up and I own the bond, I lose money. 00:50:01.190 --> 00:50:04.920 Signs, very important. 00:50:04.920 --> 00:50:07.030 And, again, we've covered-- usually 00:50:07.030 --> 00:50:10.220 I do this in a two hour lecture. 00:50:10.220 --> 00:50:13.180 And we've covered it in less than an hour, so pretty good. 00:50:13.180 --> 00:50:14.437 All right. 00:50:14.437 --> 00:50:16.270 I spent a lot more time on the fixed income. 00:50:16.270 --> 00:50:17.230 [STUDENT COUGHING] 00:50:17.230 --> 00:50:18.966 Are you taking something for that? 00:50:18.966 --> 00:50:21.691 That does not sound healthy. 00:50:21.691 --> 00:50:22.940 I don't mean to embarrass you. 00:50:22.940 --> 00:50:25.106 But I just want to make sure that you're taking care 00:50:25.106 --> 00:50:27.468 of yourself because grad students don't-- I was a grad 00:50:27.468 --> 00:50:29.468 student, I didn't take care of myself very well. 00:50:29.468 --> 00:50:29.967 I worry. 00:50:33.600 --> 00:50:36.350 All right. 00:50:36.350 --> 00:50:38.150 Big picture, variance/covariance. 00:50:38.150 --> 00:50:40.400 Collect data, calculate returns, test 00:50:40.400 --> 00:50:45.680 the data, matrix construction, get my position vector, 00:50:45.680 --> 00:50:48.230 multiply my matrices. 00:50:48.230 --> 00:50:48.830 All right? 00:50:48.830 --> 00:50:51.630 Quick and dirty, that's how we do it. 00:50:51.630 --> 00:50:54.030 That's the simplified approach to measuring 00:50:54.030 --> 00:50:57.469 this order statistic called value at risk using 00:50:57.469 --> 00:50:58.552 this particular technique. 00:51:01.930 --> 00:51:03.890 Questions, comments? 00:51:03.890 --> 00:51:06.004 Anyone? 00:51:06.004 --> 00:51:08.536 Anything you think I need to elucidate on that? 00:51:11.320 --> 00:51:18.070 And this is, in fact, how we did this up until the late '90s. 00:51:18.070 --> 00:51:19.860 Firms used variance/covariance. 00:51:19.860 --> 00:51:24.210 I heard a statistic in Europe in 1996 00:51:24.210 --> 00:51:27.367 that 80% of the European banks were using this technique 00:51:27.367 --> 00:51:28.450 to do their value at risk. 00:51:28.450 --> 00:51:31.780 It was no more complicated than this. 00:51:31.780 --> 00:51:33.165 I use a little flow diagram. 00:51:35.740 --> 00:51:38.210 Get your data returns, graph your data 00:51:38.210 --> 00:51:40.670 to make sure you don't screw it up. 00:51:40.670 --> 00:51:44.280 Get your covariance matrix, multiply your matrices out. 00:51:44.280 --> 00:51:47.230 x transpose sigma x. 00:51:47.230 --> 00:51:50.340 Using the position vectors and then you can do your analysis. 00:51:50.340 --> 00:51:52.010 Normally I would spend some more time 00:51:52.010 --> 00:51:54.468 on that bottom row and different things you can do with it, 00:51:54.468 --> 00:51:58.566 but that will have to suffice for now. 00:51:58.566 --> 00:52:01.044 A couple of points I want to make before we move on 00:52:01.044 --> 00:52:01.960 about the assumptions. 00:52:04.147 --> 00:52:06.230 Actually, I'll fly through this here so we can get 00:52:06.230 --> 00:52:09.310 into Monte Carlo simulation. 00:52:09.310 --> 00:52:10.790 Where am I going to get my data? 00:52:10.790 --> 00:52:11.748 Where do I get my data? 00:52:11.748 --> 00:52:13.660 I often get a lot of my data from Bloomberg, 00:52:13.660 --> 00:52:16.036 I get it from public sources, I get it from the internet. 00:52:16.036 --> 00:52:17.660 Especially when you get it from-- look, 00:52:17.660 --> 00:52:19.730 if it says so on the internet, it must be true. 00:52:19.730 --> 00:52:20.861 Right? 00:52:20.861 --> 00:52:24.530 Didn't Abe Lincoln say, don't believe everything 00:52:24.530 --> 00:52:26.230 you read on the internet? 00:52:26.230 --> 00:52:29.760 That was a quote, I saw that some place. 00:52:29.760 --> 00:52:32.367 You get data from people, you check it. 00:52:32.367 --> 00:52:34.200 There's some sources that are very reliable. 00:52:34.200 --> 00:52:37.190 If you're looking for yield data or foreign exchange data, 00:52:37.190 --> 00:52:38.590 the Federal Reserve has it. 00:52:38.590 --> 00:52:40.910 And they have it back 20 years, daily data. 00:52:40.910 --> 00:52:45.860 It's the H.15 and the H.10. 00:52:45.860 --> 00:52:48.490 It's there, it's free, it's easy to download, just 00:52:48.490 --> 00:52:49.175 be aware of it. 00:52:49.175 --> 00:52:49.675 Exchange-- 00:52:49.675 --> 00:52:52.630 PROFESSOR: [INAUDIBLE] study posted 00:52:52.630 --> 00:52:58.720 on the website that goes through computations 00:52:58.720 --> 00:53:01.840 for regression analysis and asset pricing models 00:53:01.840 --> 00:53:04.490 and the data that's used there is from the Federal 00:53:04.490 --> 00:53:05.560 Reserve for yields. 00:53:05.560 --> 00:53:06.610 KENNETH ABBOTT: It's H.15 It's for yields, 00:53:06.610 --> 00:53:07.690 it's probably from the H.15. 00:53:07.690 --> 00:53:07.930 [INTERPOSING VOICES] 00:53:07.930 --> 00:53:09.670 PROFESSOR: Those files, you can see how to actually get 00:53:09.670 --> 00:53:10.750 that data for yourselves. 00:53:10.750 --> 00:53:12.970 KENNETH ABBOTT: Now, another great source of data 00:53:12.970 --> 00:53:15.080 is Bloomberg. 00:53:15.080 --> 00:53:16.890 Now the good thing about Bloomberg data 00:53:16.890 --> 00:53:19.660 is everybody uses it, so it's clean. 00:53:19.660 --> 00:53:20.840 Relatively clean. 00:53:20.840 --> 00:53:22.990 I still find errors in it from time to time. 00:53:22.990 --> 00:53:25.490 But what happens is when you find an error in your Bloomberg 00:53:25.490 --> 00:53:27.200 data, you get on the phone to Bloomberg right away 00:53:27.200 --> 00:53:28.270 and say I found an error in your data. 00:53:28.270 --> 00:53:29.270 They say, oh, what date? 00:53:29.270 --> 00:53:32.320 June 14, you know, 2012. 00:53:32.320 --> 00:53:34.370 And they'll say, OK, we'll fix it. 00:53:34.370 --> 00:53:34.900 All right? 00:53:34.900 --> 00:53:37.340 So everybody does that, and the data set is pretty clean. 00:53:37.340 --> 00:53:40.430 I found consistently that Bloomberg data is 00:53:40.430 --> 00:53:43.497 the cleanest in my experience. 00:53:43.497 --> 00:53:45.080 How much data do we use in doing this? 00:53:45.080 --> 00:53:47.460 I could use one year of data, I can use two weeks of data. 00:53:47.460 --> 00:53:49.626 Now, times series, we usually want 100 observations. 00:53:49.626 --> 00:53:52.020 That's always been my rule of thumb. 00:53:52.020 --> 00:53:53.320 I can use one year of data. 00:53:53.320 --> 00:53:54.880 There are regulators that require you 00:53:54.880 --> 00:53:57.110 to use at least a year of data. 00:53:57.110 --> 00:53:58.560 You could use two years of data. 00:53:58.560 --> 00:54:00.790 In fact, some firms use one year of data. 00:54:00.790 --> 00:54:04.410 There's one firm that uses five years of data. 00:54:04.410 --> 00:54:07.000 And there, we could say, well, am I going to weight it. 00:54:07.000 --> 00:54:09.060 Am I going to weight my more recent data heavily? 00:54:09.060 --> 00:54:11.480 I could do that with exponential smoothing, which we 00:54:11.480 --> 00:54:12.730 won't have time to talk about. 00:54:12.730 --> 00:54:17.550 It's a technique I can use to lend more credence to the more 00:54:17.550 --> 00:54:18.680 recent data. 00:54:18.680 --> 00:54:22.450 Now, I'm a relatively simple guy. 00:54:22.450 --> 00:54:24.000 I tend to use equally weighted data 00:54:24.000 --> 00:54:26.050 because I believe in Occam's razor, which 00:54:26.050 --> 00:54:29.920 is, the simplest explanation is usually the best. 00:54:29.920 --> 00:54:31.319 I think we get too clever by half 00:54:31.319 --> 00:54:32.485 when we try to parameterize. 00:54:36.140 --> 00:54:39.110 How much more does last week's data 00:54:39.110 --> 00:54:41.550 have an impact than from two weeks ago, three weeks ago. 00:54:41.550 --> 00:54:44.680 I'm not saying that it doesn't, what I am saying is, 00:54:44.680 --> 00:54:48.980 I'm not smart enough to know exactly how much it does. 00:54:48.980 --> 00:54:51.120 And assuming that everything's equally 00:54:51.120 --> 00:54:55.300 weighted throughout time is just as strong an assumption. 00:54:55.300 --> 00:54:59.050 But it's a very simple assumption, and I love simple. 00:54:59.050 --> 00:55:00.002 Yes? 00:55:00.002 --> 00:55:04.140 AUDIENCE: [INAUDIBLE] calculate covariance matrix? 00:55:04.140 --> 00:55:05.132 KENNETH ABBOTT: Yes. 00:55:05.132 --> 00:55:07.317 All right, quickly. 00:55:07.317 --> 00:55:09.150 Actually I think I have some slides on that. 00:55:11.770 --> 00:55:14.850 Let me just finish this and I'll get to that. 00:55:14.850 --> 00:55:15.640 Gaps in data. 00:55:15.640 --> 00:55:17.011 Missing data is a problem. 00:55:17.011 --> 00:55:18.260 How do I fill in missing data? 00:55:18.260 --> 00:55:22.420 I can do a linear interpolation, I can use the prior day's data. 00:55:22.420 --> 00:55:24.780 I can do a Brownian bridge, which is I just 00:55:24.780 --> 00:55:26.800 do a Monte Carlo between them. 00:55:26.800 --> 00:55:28.880 I can do a regression based, I can use regression 00:55:28.880 --> 00:55:31.901 to project changes from one onto changes in another. 00:55:31.901 --> 00:55:33.400 That's usually a whole other lecture 00:55:33.400 --> 00:55:34.860 I gave on how to do missing data. 00:55:34.860 --> 00:55:37.670 Now you've got that lecture for free. 00:55:37.670 --> 00:55:39.320 That's all you need to know. 00:55:39.320 --> 00:55:42.320 It's not only a lecture, it's a very hard homework assignment. 00:55:42.320 --> 00:55:47.292 But how frequently do I update my data? 00:55:47.292 --> 00:55:49.500 Some people update their covariance structures daily. 00:55:49.500 --> 00:55:50.830 I think that's an overkill. 00:55:50.830 --> 00:55:54.760 We update our data set weekly. 00:55:54.760 --> 00:55:55.520 That's what we do. 00:55:55.520 --> 00:56:00.650 And I think that's overkill, but tell that to my regulators. 00:56:00.650 --> 00:56:02.850 And we use daily data, weekly data, monthly data. 00:56:02.850 --> 00:56:04.400 We typically use daily data. 00:56:04.400 --> 00:56:05.957 Some firms may do it differently. 00:56:12.160 --> 00:56:13.770 All right. 00:56:13.770 --> 00:56:17.060 Here's your exponential smoothing. 00:56:17.060 --> 00:56:22.210 Remember, I usually measure covariance, sum of x_i 00:56:22.210 --> 00:56:27.640 minus x bar times y minus y bar divided by n minus 1. 00:56:27.640 --> 00:56:31.670 What if I stuck an omega in there? 00:56:31.670 --> 00:56:34.390 And I use this calculation instead, 00:56:34.390 --> 00:56:38.310 where the denominator is the sum of all the omegas-- you should 00:56:38.310 --> 00:56:39.470 be thinking finite series. 00:56:42.310 --> 00:56:44.841 You have to realize, I was a decent math student, 00:56:44.841 --> 00:56:46.090 I wasn't a great math student. 00:56:46.090 --> 00:56:49.360 And what I found when I was studying this, I was like, wow, 00:56:49.360 --> 00:56:51.340 all that stuff that I learned, it actually-- 00:56:51.340 --> 00:56:52.890 finite series, who knew? 00:56:52.890 --> 00:56:55.130 Who knew that I'd actually use it? 00:56:55.130 --> 00:56:58.470 So I take this, and let's say I'm working backwards in time. 00:56:58.470 --> 00:57:02.070 So today's observations is t_0. 00:57:02.070 --> 00:57:05.880 Yesterday's observation is t_1, t_2, t_3. 00:57:05.880 --> 00:57:10.630 So today's observation would get-- and let's 00:57:10.630 --> 00:57:12.820 assume for the time being that this omega 00:57:12.820 --> 00:57:15.320 is on the order 0.95. 00:57:15.320 --> 00:57:17.090 It could be anything. 00:57:17.090 --> 00:57:21.850 So today would be 0.95 to the 0 divided 00:57:21.850 --> 00:57:23.400 by the sum of all the omegas. 00:57:23.400 --> 00:57:26.900 Tomorrow it will be 0.95 divided by the sum of the omegas. 00:57:26.900 --> 00:57:29.270 The next would be 0.95 squared divided 00:57:29.270 --> 00:57:31.950 by the sum of the omegas. 00:57:31.950 --> 00:57:33.780 0.95 cubed and get smaller and smaller. 00:57:36.960 --> 00:57:42.760 For example, if you use 0.94, 99% of your weight 00:57:42.760 --> 00:57:46.220 will be in the last 76 days. 00:57:46.220 --> 00:57:48.390 76 observations, I shouldn't say 76 days. 00:57:48.390 --> 00:57:49.800 76 observations. 00:57:49.800 --> 00:57:55.270 So there's this notion that the impact declines exponentially. 00:57:55.270 --> 00:57:57.580 Does that make sense? 00:57:57.580 --> 00:58:01.530 People use this pretty commonly, but what scares me about it-- 00:58:01.530 --> 00:58:03.275 somebody stuck these fancy transitions 00:58:03.275 --> 00:58:04.656 in between these slides. 00:58:04.656 --> 00:58:07.940 Anyway, is that here's my standard deviation 00:58:07.940 --> 00:58:09.960 with a rolling six-month window. 00:58:09.960 --> 00:58:13.910 And here's my standard deviation using different weights. 00:58:13.910 --> 00:58:15.660 The point I want to make here, and it's 00:58:15.660 --> 00:58:20.840 an important point, my assumption about my weighting 00:58:20.840 --> 00:58:24.520 coefficient has a material impact 00:58:24.520 --> 00:58:29.110 on the size of my measured volatility. 00:58:29.110 --> 00:58:31.400 Now when I see this, and this is just me. 00:58:31.400 --> 00:58:33.510 There's no finance or statistics theory 00:58:33.510 --> 00:58:38.470 behind this, any time the choice-- any time an assumption 00:58:38.470 --> 00:58:41.900 has this material an impact, bells and whistles go off 00:58:41.900 --> 00:58:43.040 and sirens. 00:58:43.040 --> 00:58:44.840 All right, and red lights flash. 00:58:44.840 --> 00:58:47.190 Be very, very careful. 00:58:47.190 --> 00:58:49.660 Now, lies, damn lies, and statistics. 00:58:49.660 --> 00:58:51.610 You tell me the outcome you want, 00:58:51.610 --> 00:58:54.280 and I'll tell you what statistics to use. 00:58:54.280 --> 00:58:55.720 That's where this could be abused. 00:58:55.720 --> 00:58:57.480 Oh, you want to show high volatility? 00:58:57.480 --> 00:58:59.510 Well let's use this. 00:58:59.510 --> 00:59:01.710 You want to show low volatility, let's use this? 00:59:01.710 --> 00:59:07.780 See, I choose to just take the simplest approach. 00:59:07.780 --> 00:59:09.890 And that's me. 00:59:09.890 --> 00:59:12.460 That's not a terribly scientific opinion, 00:59:12.460 --> 00:59:13.613 but that's what I think. 00:59:17.570 --> 00:59:21.470 Daily versus weekly, percentage changes log changes. 00:59:21.470 --> 00:59:23.130 Units. 00:59:23.130 --> 00:59:27.290 Just like dollar/yen, interest rates. 00:59:27.290 --> 00:59:30.080 Am I long or am I short? 00:59:30.080 --> 00:59:32.490 If I'm long gold, I show it as a positive number. 00:59:32.490 --> 00:59:34.600 And if I'm short gold, in my position vector, 00:59:34.600 --> 00:59:36.550 I show it as a negative number. 00:59:36.550 --> 00:59:40.970 If I'm long yen, and yen is measured in yen per dollar, 00:59:40.970 --> 00:59:42.470 then I show it as a negative number. 00:59:42.470 --> 00:59:46.370 If I'm long yen, but my covariance matrix measures yen 00:59:46.370 --> 00:59:51.740 as dollars per yen-- 0.000094, whatever-- 00:59:51.740 --> 00:59:54.620 then I show it as a positive number. 00:59:54.620 --> 00:59:58.766 It's just like physics only worse 00:59:58.766 --> 01:00:00.140 because it'll cost you real-- no, 01:00:00.140 --> 01:00:01.312 I guess physics would be worse because if you 01:00:01.312 --> 01:00:03.500 get the units wrong, you blow up, right? 01:00:03.500 --> 01:00:06.282 This will just cost you money. 01:00:06.282 --> 01:00:07.240 I've made this mistake. 01:00:07.240 --> 01:00:10.000 I've made the units mistake. 01:00:10.000 --> 01:00:12.283 All right, we talked about fixed income. 01:00:17.830 --> 01:00:21.770 So that's what I want to cover from the bare bones 01:00:21.770 --> 01:00:22.870 setup for VaR. 01:00:22.870 --> 01:00:24.850 Now I'm going to skip the historical simulation 01:00:24.850 --> 01:00:26.110 and go right to the Monte Carlo because I 01:00:26.110 --> 01:00:28.693 want to show you another way we can use covariance structures. 01:00:35.555 --> 01:00:37.052 [POWERPOINT SOUND EFFECT] 01:00:38.380 --> 01:00:42.040 That's going to happen two or three more times. 01:00:42.040 --> 01:00:44.710 Somebody did this, somebody made my presentation cute 01:00:44.710 --> 01:00:45.460 some years ago. 01:00:45.460 --> 01:00:47.309 And I just-- I apologize. 01:00:50.620 --> 01:00:53.185 All right, see, there's a lot to meat in this presentation 01:00:53.185 --> 01:00:56.670 that we don't have time to get to. 01:00:56.670 --> 01:01:01.120 Another approach to doing value at risk 01:01:01.120 --> 01:01:05.620 is rather than use this parametric approach, 01:01:05.620 --> 01:01:09.270 is to simulate the outcomes. 01:01:09.270 --> 01:01:13.950 Simulate the outcomes 100 times, 1,000 times, 10,000 times, 01:01:13.950 --> 01:01:16.920 a million times, and say, these are all the possible outcomes 01:01:16.920 --> 01:01:19.640 based on my simulation assumptions. 01:01:19.640 --> 01:01:23.309 And let's say I simulate 10,000 times, 01:01:23.309 --> 01:01:25.350 and I have 10,000 possible outcomes for tomorrow. 01:01:25.350 --> 01:01:29.840 And I wanted to measure my value at risk at the 1% significance 01:01:29.840 --> 01:01:31.270 level. 01:01:31.270 --> 01:01:34.790 All I would do is take my 10,000 outcomes 01:01:34.790 --> 01:01:39.170 and I would sort them and take my hundredth worst. 01:01:39.170 --> 01:01:41.366 Put it in your pocket, go home. 01:01:41.366 --> 01:01:43.310 That's it. 01:01:43.310 --> 01:01:45.970 This is a different way of getting 01:01:45.970 --> 01:01:47.010 to that order statistic. 01:01:50.330 --> 01:01:52.026 Lends a lot more flexibility. 01:01:52.026 --> 01:01:54.400 So I can go and I can tweak the way I do that simulation, 01:01:54.400 --> 01:01:56.145 I can relax my assumptions of normality. 01:01:58.304 --> 01:01:59.970 I don't have to use normal distribution, 01:01:59.970 --> 01:02:01.740 I could use a t distribution, I could do lots, 01:02:01.740 --> 01:02:03.906 I could tweak my distribution, I could customize it. 01:02:03.906 --> 01:02:06.150 I could put mean reversion in there, 01:02:06.150 --> 01:02:08.410 I could do all kinds of stuff. 01:02:08.410 --> 01:02:12.300 So another way we do value at risk is we 01:02:12.300 --> 01:02:15.110 simulate possible outcomes. 01:02:15.110 --> 01:02:19.670 We rank the outcomes, and we just count them. 01:02:19.670 --> 01:02:21.763 If I've got the 10,000 observations 01:02:21.763 --> 01:02:23.240 and I want my 5% order statistic, 01:02:23.240 --> 01:02:25.230 well I just take my 500th. 01:02:27.830 --> 01:02:28.820 Make sense? 01:02:28.820 --> 01:02:30.890 It's that simple. 01:02:30.890 --> 01:02:32.306 Well, I don't want to make it seem 01:02:32.306 --> 01:02:33.200 like it's that simple because it actually 01:02:33.200 --> 01:02:35.140 gets a little messy in here. 01:02:35.140 --> 01:02:37.680 But when we do Monte Carlo simulation, 01:02:37.680 --> 01:02:40.449 we're simulating what we think is going to happen 01:02:40.449 --> 01:02:41.740 all subject to our assumptions. 01:02:44.640 --> 01:02:46.880 And we run through this Monte Carlo simulation. 01:02:46.880 --> 01:02:49.830 Simulation of method using sequences of random numbers. 01:02:49.830 --> 01:02:51.550 Coined during the Manhattan Project, 01:02:51.550 --> 01:02:54.430 similar to games of chance. 01:02:54.430 --> 01:02:57.610 You need to describe your system in terms of probability density 01:02:57.610 --> 01:02:58.480 functions. 01:02:58.480 --> 01:03:00.900 What type of distribution? 01:03:00.900 --> 01:03:02.040 Is this normal? 01:03:02.040 --> 01:03:02.540 Is it t? 01:03:02.540 --> 01:03:03.470 Is it chi squared? 01:03:03.470 --> 01:03:04.500 Is it F? 01:03:04.500 --> 01:03:05.360 All right? 01:03:05.360 --> 01:03:07.800 That's the way we do it. 01:03:07.800 --> 01:03:09.720 So quickly, how do I do that? 01:03:14.570 --> 01:03:16.200 I have to have random numbers. 01:03:16.200 --> 01:03:19.760 Now they're truly random numbers. 01:03:19.760 --> 01:03:22.457 Somewhere at MIT you could buy-- I used to say tape, 01:03:22.457 --> 01:03:23.540 but people don't use tape. 01:03:23.540 --> 01:03:28.840 They'll give you a website where you can get the atomic decay. 01:03:28.840 --> 01:03:30.450 That's random. 01:03:30.450 --> 01:03:31.706 All right? 01:03:31.706 --> 01:03:36.480 Anything else is pseudo-random. 01:03:36.480 --> 01:03:39.190 What you see when you go into MATLAB, 01:03:39.190 --> 01:03:41.570 you have a random number generator, it's an algorithm. 01:03:41.570 --> 01:03:45.110 It probably takes some number and takes the square root 01:03:45.110 --> 01:03:47.910 of that number and then goes 54 decimal places to the right 01:03:47.910 --> 01:03:51.690 and takes the 55 decimal places to the right, 01:03:51.690 --> 01:03:55.170 multiplies those two numbers together 01:03:55.170 --> 01:03:58.110 and then takes the fifth root, and then goes 16 decimal places 01:03:58.110 --> 01:04:01.610 to the right to get that-- it's some algorithm. 01:04:01.610 --> 01:04:05.590 True story, before I came to appreciate that these were all 01:04:05.590 --> 01:04:09.225 highly algorithmically driven, I was in my 20's, I 01:04:09.225 --> 01:04:11.390 was taking a computer class, I saw two computers, 01:04:11.390 --> 01:04:13.830 they were both running random number of generators 01:04:13.830 --> 01:04:17.190 and they were generating the same random numbers. 01:04:17.190 --> 01:04:19.870 And I thought I was at the event horizon. 01:04:19.870 --> 01:04:22.760 I thought that light was bending and the world 01:04:22.760 --> 01:04:24.610 was coming to an end, all right? 01:04:24.610 --> 01:04:27.767 Because this this stuff can't happen, all right? 01:04:27.767 --> 01:04:29.350 It was happening right in front of me. 01:04:29.350 --> 01:04:31.860 It was a pseudo-random number generator. 01:04:31.860 --> 01:04:34.860 I didn't know, I was 24. 01:04:34.860 --> 01:04:35.360 Anyway. 01:04:38.320 --> 01:04:42.810 quasi-random numbers, it's sort of a way of imposing some order 01:04:42.810 --> 01:04:43.790 on your random numbers. 01:04:43.790 --> 01:04:47.020 You random numbers, one particular set of draws 01:04:47.020 --> 01:04:49.895 may not have enough draws in a particular area 01:04:49.895 --> 01:04:51.270 to give you the numbers you want. 01:04:51.270 --> 01:04:53.767 I can impose some conditions upon that. 01:04:53.767 --> 01:04:56.100 I don't want to get into a discussion of random numbers. 01:04:58.850 --> 01:05:02.250 How do I get from random uniform-- 01:05:02.250 --> 01:05:05.610 most random number generators give you random uniform number 01:05:05.610 --> 01:05:07.280 between 0 and 1. 01:05:07.280 --> 01:05:10.080 What you'll typically do is you'll take that random uniform 01:05:10.080 --> 01:05:13.190 number, you'll map it over to the cumulative density 01:05:13.190 --> 01:05:16.340 function, and map it down. 01:05:16.340 --> 01:05:19.720 So this gets you from random uniform space 01:05:19.720 --> 01:05:23.150 into standard deviation space. 01:05:23.150 --> 01:05:24.860 We used to worry about how we did this, 01:05:24.860 --> 01:05:27.534 now your software does it for you. 01:05:27.534 --> 01:05:29.450 I've gotten comfortable enough, truth be told. 01:05:29.450 --> 01:05:36.040 I usually trust my random number generators in Excel, in MATLAB. 01:05:36.040 --> 01:05:38.980 So I kind of violate my own rules, I don't check. 01:05:38.980 --> 01:05:43.940 But I think most of your standard random number 01:05:43.940 --> 01:05:46.074 of generators are decent enough now. 01:05:46.074 --> 01:05:47.490 And you can go straight to normal, 01:05:47.490 --> 01:05:50.150 you don't have to do random uniform and back 01:05:50.150 --> 01:05:51.280 into random normal. 01:05:51.280 --> 01:05:53.910 You can get it distributed in any way you want. 01:05:59.550 --> 01:06:05.524 What I do when I do a Monte Carlo simulation-- and this 01:06:05.524 --> 01:06:08.065 is going to be rushed because we've only got like 20 minutes. 01:06:10.800 --> 01:06:14.165 If I take a covariance matrix-- you're 01:06:14.165 --> 01:06:15.540 going to have to trust me on this 01:06:15.540 --> 01:06:18.460 because again, I'm covering like eight hours of lecture 01:06:18.460 --> 01:06:19.907 in an hour and a half. 01:06:19.907 --> 01:06:21.740 You guys go to MIT so I have no doubt you're 01:06:21.740 --> 01:06:22.823 going to be all over this. 01:06:48.134 --> 01:06:52.080 Let's take this out of here for a second. 01:06:52.080 --> 01:06:56.250 I can factor my covariance structure. 01:06:59.070 --> 01:07:02.716 I can factor my covariance structure like this. 01:07:02.716 --> 01:07:05.300 And this is the transpose of this. 01:07:05.300 --> 01:07:08.920 I didn't realize that the first time we did this commercially 01:07:08.920 --> 01:07:12.690 I saw this instead of this and I thought we had 01:07:12.690 --> 01:07:13.940 sent bad data to the customer. 01:07:13.940 --> 01:07:15.280 I got physically sick. 01:07:15.280 --> 01:07:21.190 And then I remembered AB transpose 01:07:21.190 --> 01:07:25.840 equals B transpose A-- these things keep happening. 01:07:25.840 --> 01:07:28.720 My high school math keeps coming back to me. 01:07:28.720 --> 01:07:31.114 But I had forgotten this and I got physically sick 01:07:31.114 --> 01:07:33.030 because I thought we'd sent bad data because I 01:07:33.030 --> 01:07:36.400 was looking at this when it's just the transpose of this. 01:07:36.400 --> 01:07:39.900 Anyway, I can factor this into this where this 01:07:39.900 --> 01:07:44.550 is a matrix of eigenvectors. 01:07:44.550 --> 01:07:48.195 This is a diagonal matrix of the eigenvalues. 01:07:48.195 --> 01:07:50.640 All right? 01:07:50.640 --> 01:07:56.100 This is the vaunted Gaussian copula. 01:07:56.100 --> 01:07:57.930 This is it. 01:07:57.930 --> 01:07:59.560 Most people view it as a black box. 01:07:59.560 --> 01:08:01.890 If you've had any more than introductory statistics, 01:08:01.890 --> 01:08:03.360 this should be a glass box to you. 01:08:03.360 --> 01:08:04.765 That's why I wanted to go through this even though I'd 01:08:04.765 --> 01:08:07.872 love to spend another hour and a half and do about 50 examples. 01:08:07.872 --> 01:08:09.330 Because this is how I learned this, 01:08:09.330 --> 01:08:12.880 I didn't learn it from looking at this equation and saying, 01:08:12.880 --> 01:08:13.530 oh, I get it. 01:08:13.530 --> 01:08:15.870 I learned it from actually doing it about 1,000 times 01:08:15.870 --> 01:08:20.020 in a spreadsheet, and sunk in like water into a stone. 01:08:20.020 --> 01:08:23.649 So I factor this matrix, and then 01:08:23.649 --> 01:08:33.250 I take this, which is the square root matrix, which 01:08:33.250 --> 01:08:36.020 is my transpose of my eigenvector matrix 01:08:36.020 --> 01:08:39.490 and diagonal matrix contain the square root of my eigenvalues. 01:08:39.490 --> 01:08:42.750 Now, could this ever be negative and take me 01:08:42.750 --> 01:08:44.910 into imaginary root land? 01:08:44.910 --> 01:08:50.140 Well, if my variances are positive or zero, 01:08:50.140 --> 01:08:52.176 then that will be a problem. 01:08:52.176 --> 01:08:53.800 So here we get into this-- remember you 01:08:53.800 --> 01:08:55.899 guys studied positive semidefinite, 01:08:55.899 --> 01:08:56.649 positive definite. 01:08:56.649 --> 01:08:59.232 Once again, it's another one of these high school math things. 01:08:59.232 --> 01:09:00.130 Like, here it is. 01:09:00.130 --> 01:09:02.035 I had to know this. 01:09:02.035 --> 01:09:04.160 Suddenly I care whether it's positive semidefinite. 01:09:04.160 --> 01:09:08.270 Covariance structures have to be positive semidefinite. 01:09:08.270 --> 01:09:10.450 If you don't have a complete data set, 01:09:10.450 --> 01:09:13.439 let's say you've got 100 observations, 100 observations, 01:09:13.439 --> 01:09:17.740 100 observations, 25 observations, 100 observations, 01:09:17.740 --> 01:09:19.486 you may have a negative eigenvalue. 01:09:19.486 --> 01:09:21.569 If you just measure the covariance with the amount 01:09:21.569 --> 01:09:23.160 of data that you have. 01:09:23.160 --> 01:09:26.430 My intuition-- and I doubt this is the [INAUDIBLE]-- 01:09:26.430 --> 01:09:28.880 is that you're measuring with error and you have fewer 01:09:28.880 --> 01:09:31.470 observations you measure with more error. 01:09:31.470 --> 01:09:35.160 So it's possible if some of your covariance measures 01:09:35.160 --> 01:09:38.000 have 25 observations and some of them 01:09:38.000 --> 01:09:41.590 have 100 observations that there's more error in some 01:09:41.590 --> 01:09:42.229 than in others. 01:09:42.229 --> 01:09:44.529 And so there's the theoretical possibility 01:09:44.529 --> 01:09:46.550 for negative variance. 01:09:46.550 --> 01:09:51.060 True story, we didn't know this in the '90s. 01:09:51.060 --> 01:09:54.075 I took this problem to the chairman of the statistics 01:09:54.075 --> 01:09:58.290 department at NYU said, I'm getting negative eigenvalues. 01:09:58.290 --> 01:10:00.140 And he didn't know. 01:10:00.140 --> 01:10:03.370 He had no idea, he's a smart guy. 01:10:03.370 --> 01:10:06.260 You have to fill in your missing data. 01:10:06.260 --> 01:10:08.500 You have to fill in your missing data. 01:10:08.500 --> 01:10:10.500 If you've got 1,000 observations, 1,000 01:10:10.500 --> 01:10:12.654 observations, 1,000 observations, 200 observations, 01:10:12.654 --> 01:10:14.320 and you want to make sure you won't have 01:10:14.320 --> 01:10:16.300 a negative eigenvalue, you've got 01:10:16.300 --> 01:10:17.810 to fill in those observations. 01:10:17.810 --> 01:10:20.510 Which is why missing data is a whole other thing 01:10:20.510 --> 01:10:22.999 we talk about. 01:10:22.999 --> 01:10:24.790 Again, I could spend a lot of time on that. 01:10:24.790 --> 01:10:26.123 And I learned that the hard way. 01:10:28.690 --> 01:10:32.640 But anyway, so I take this square root matrix, 01:10:32.640 --> 01:10:40.240 if I pre-multiply that square root matrix by row after row 01:10:40.240 --> 01:10:46.700 of normals, I will get out an array 01:10:46.700 --> 01:10:49.580 that has the same covariance structure as that 01:10:49.580 --> 01:10:50.606 with which I started. 01:10:54.140 --> 01:10:56.130 Another story here, I've been using 01:10:56.130 --> 01:10:58.710 the same eigenvalue-- I believe in full attribution, 01:10:58.710 --> 01:10:59.660 I'm not a clever guy. 01:10:59.660 --> 01:11:02.660 I have not an original thought in my head. 01:11:02.660 --> 01:11:04.580 And whenever I use someone else's stuff, 01:11:04.580 --> 01:11:06.830 I give them credit for it. 01:11:06.830 --> 01:11:08.606 And the guy who wrote the code that 01:11:08.606 --> 01:11:10.230 did the eigenvalue decomposition-- this 01:11:10.230 --> 01:11:13.740 is something that was translated from Fortran IV. 01:11:13.740 --> 01:11:15.820 It wasn't even [INAUDIBLE], there's 01:11:15.820 --> 01:11:18.570 a dichotomy in the world. 01:11:18.570 --> 01:11:21.100 There are people that have written Fortran, 01:11:21.100 --> 01:11:22.100 and people that haven't. 01:11:22.100 --> 01:11:24.730 I'm guessing that there are two people in this room that have 01:11:24.730 --> 01:11:27.020 ever written a line of Fortran. 01:11:27.020 --> 01:11:29.580 Anyone here? 01:11:29.580 --> 01:11:31.530 Just saying. 01:11:31.530 --> 01:11:34.882 Yeah, with cards or without cards? 01:11:34.882 --> 01:11:36.740 PROFESSOR: [INAUDIBLE]. 01:11:36.740 --> 01:11:38.410 KENNETH ABBOTT: I didn't use cards. 01:11:38.410 --> 01:11:40.410 See, you're an old-timer because you used cards. 01:11:44.226 --> 01:11:47.701 The punch line is, I've been using this guy's code. 01:11:47.701 --> 01:11:48.950 And I could show you the code. 01:11:48.950 --> 01:11:50.650 It's like the Lone Ranger, I didn't even 01:11:50.650 --> 01:11:52.370 get a chance to thank him. 01:11:52.370 --> 01:11:55.360 Because he didn't put his name on the code. 01:11:55.360 --> 01:11:57.430 On the internet now, if you do something clever 01:11:57.430 --> 01:11:58.610 on the quant newsgroups, you're going 01:11:58.610 --> 01:11:59.980 to post your name all over it. 01:11:59.980 --> 01:12:02.840 I've been wanting to thank this guy for like 20 years 01:12:02.840 --> 01:12:04.040 and I haven't been able to. 01:12:04.040 --> 01:12:06.100 Anyway, eigenvalue code that's been translated. 01:12:06.100 --> 01:12:07.660 Let me show you what this means. 01:12:10.841 --> 01:12:11.840 Here's some source data. 01:12:14.770 --> 01:12:17.446 Here's some percentage changes. 01:12:17.446 --> 01:12:20.120 Just like we talked about. 01:12:20.120 --> 01:12:23.880 Here is the empirical correlation 01:12:23.880 --> 01:12:26.970 of those percentage changes. 01:12:26.970 --> 01:12:30.570 So the correlation of my government 10 year to my AAA 10 01:12:30.570 --> 01:12:32.590 year is 0.83. 01:12:32.590 --> 01:12:33.852 To my AA, 0.84. 01:12:33.852 --> 01:12:36.310 All right, you see this. 01:12:36.310 --> 01:12:38.690 And I have this covariance matrix 01:12:38.690 --> 01:12:42.320 which is the-- the correlation matrix is a scaled version 01:12:42.320 --> 01:12:44.470 of the covariance matrix. 01:12:44.470 --> 01:12:46.720 And I do a little bit of statistical legerdemain. 01:12:51.582 --> 01:12:52.790 Eigenvalues and eigenvectors. 01:12:55.320 --> 01:12:56.985 Take the square root of that. 01:12:56.985 --> 01:12:59.660 And again, I'd love to spend a lot more time on this, 01:12:59.660 --> 01:13:03.090 but we just don't-- suffice to say, 01:13:03.090 --> 01:13:05.820 I call this a transformation matrix, that's my term. 01:13:05.820 --> 01:13:12.235 This matrix here is this. 01:13:12.235 --> 01:13:13.760 If we had another hour and a half 01:13:13.760 --> 01:13:15.710 I'd take the step by step to get you there. 01:13:15.710 --> 01:13:18.400 The proof of which is left to the reader as an exercise. 01:13:18.400 --> 01:13:20.930 I'll leave this spreadsheet for you, I'll send it to you. 01:13:20.930 --> 01:13:22.600 I have this matrix. 01:13:22.600 --> 01:13:24.672 This matrix is like a prism. 01:13:24.672 --> 01:13:26.380 I'm going to pass white light through it, 01:13:26.380 --> 01:13:29.390 I'm going to get a beautiful rainbow. 01:13:29.390 --> 01:13:32.120 Let me show you what I mean. 01:13:32.120 --> 01:13:34.770 So remember that matrix, this matrix I'm calling t. 01:13:39.710 --> 01:13:41.410 Remember my matrix is 10 by 10. 01:13:41.410 --> 01:13:47.095 One, two, three, four, five, six, seven, eight, nine, ten. 01:13:47.095 --> 01:13:50.410 10 columns of data. 01:13:50.410 --> 01:13:53.614 10 by 10 correlation matrix. 01:13:53.614 --> 01:13:54.548 Let's check. 01:14:01.490 --> 01:14:08.980 Now I've got row vectors of sorry-- uncorrelated 01:14:08.980 --> 01:14:09.760 random normals. 01:14:15.840 --> 01:14:20.710 So what I'm doing then is I'm pre-multiplying 01:14:20.710 --> 01:14:26.860 that transformation matrix row by row by each row 01:14:26.860 --> 01:14:28.960 of uncorrelated random normals. 01:14:28.960 --> 01:14:33.940 And what I get is correlated random normals. 01:14:33.940 --> 01:14:37.110 So what I'm telling you here is this array 01:14:37.110 --> 01:14:44.080 happens to be 10 wide and 1,000 long. 01:14:44.080 --> 01:14:45.810 And I'm telling you that I started 01:14:45.810 --> 01:14:53.200 with my historical data-- let me see how much data have there. 01:14:53.200 --> 01:14:57.230 A couple hundred observations of historical data. 01:14:57.230 --> 01:15:03.940 And what I've done is once I have that covariance structure, 01:15:03.940 --> 01:15:17.210 I can create a data set here which 01:15:17.210 --> 01:15:25.300 has the same statistical properties as this. 01:15:32.530 --> 01:15:35.280 Not quite the same. 01:15:35.280 --> 01:15:39.545 It can have the same means and the same variances. 01:15:39.545 --> 01:15:41.430 This is what Monte Carlo simulation is about. 01:15:41.430 --> 01:15:43.804 I wish we had another hour because I'd like to spend time 01:15:43.804 --> 01:15:45.360 and-- this is one of these things, 01:15:45.360 --> 01:15:49.000 and again, when I first saw this, I was like, oh my god. 01:15:49.000 --> 01:15:51.176 I felt like I got the keys to the kingdom. 01:15:51.176 --> 01:15:53.550 And I did, this is manually, did it all on a spreadsheet. 01:15:53.550 --> 01:15:54.966 Didn't believe anyone else's code, 01:15:54.966 --> 01:15:57.080 did it all on a spreadsheet. 01:15:57.080 --> 01:16:03.149 But what that means-- quickly, let me just go back over here 01:16:03.149 --> 01:16:03.690 for a second. 01:16:07.660 --> 01:16:09.630 I happen to have about 800 observations here. 01:16:12.230 --> 01:16:14.450 Historical observations. 01:16:14.450 --> 01:16:22.370 What I did was I happened to generate 1,000 samples here. 01:16:22.370 --> 01:16:25.530 But I could generate 10,000 or 100,000, 01:16:25.530 --> 01:16:27.740 or a million or 10 million or a billion 01:16:27.740 --> 01:16:29.570 just by doing more random normals. 01:16:29.570 --> 01:16:31.640 I could generate-- in effect, what 01:16:31.640 --> 01:16:36.695 I'm generating here is synthetic time series that 01:16:36.695 --> 01:16:39.680 have properties similar to my underlying data. 01:16:42.250 --> 01:16:44.430 That's what Monte Carlo simulation is about. 01:16:44.430 --> 01:16:48.100 The means and the variances and the covariances of this data 01:16:48.100 --> 01:16:50.470 set are just like that. 01:16:50.470 --> 01:16:53.700 Now, again, true story, when somebody first showed me this 01:16:53.700 --> 01:16:54.680 I did not believe them. 01:16:54.680 --> 01:16:58.346 So I developed a bunch of little tests. 01:16:58.346 --> 01:17:02.210 And I said, let me just look at the correlation of my Monte 01:17:02.210 --> 01:17:07.430 Carlo data versus my original correlation matrix. 01:17:07.430 --> 01:17:13.270 So 0.83, 0.84, 0.85, 0.85, 0.67, 0.81. 01:17:13.270 --> 01:17:16.510 You look at the corresponding ones of the random numbers I 01:17:16.510 --> 01:17:21.890 just generated, 0.81, 0.82, 0.84, 0.84, 0.64, 0.52. 01:17:21.890 --> 01:17:24.641 0.54 versus 0.52. 01:17:24.641 --> 01:17:26.580 0.18 Versus 0.12. 01:17:26.580 --> 01:17:28.600 0.51 versus 0.47. 01:17:28.600 --> 01:17:32.270 Somebody want to tell me why they're not spot on? 01:17:32.270 --> 01:17:34.150 Sampling error. 01:17:34.150 --> 01:17:37.270 The more data I use the closer it will get to that. 01:17:37.270 --> 01:17:41.290 If I do 1 million, I'd better get right on top of that. 01:17:41.290 --> 01:17:43.450 Does that make sense? 01:17:43.450 --> 01:17:46.490 So what I'm telling you here is that I can 01:17:46.490 --> 01:17:48.080 generate synthetic time series. 01:17:48.080 --> 01:17:49.559 Now, why would I generate so many? 01:17:49.559 --> 01:17:51.100 Well because, remember, I care what's 01:17:51.100 --> 01:17:53.260 going on out in that tail. 01:17:53.260 --> 01:17:56.220 If I only have 100 observations and I'm looking empirically 01:17:56.220 --> 01:18:01.040 at my tail, I've only got one observation out in the 1% tail. 01:18:01.040 --> 01:18:04.040 And that doesn't tell me a whole lot about what's going on. 01:18:04.040 --> 01:18:06.100 If I can simulate that distribution exactly, 01:18:06.100 --> 01:18:08.635 I can say, you know what, I want a billion observations 01:18:08.635 --> 01:18:10.740 in that tail. 01:18:10.740 --> 01:18:12.000 Now we can look at that tail. 01:18:18.506 --> 01:18:19.880 If I have 1 billion observations, 01:18:19.880 --> 01:18:22.750 let's say I'm looking at some kind of normal distribution. 01:18:22.750 --> 01:18:27.200 I'm circling it out here, I'm seeing-- 01:18:27.200 --> 01:18:29.930 I can really dig in and see what the properties of this thing 01:18:29.930 --> 01:18:30.560 are. 01:18:30.560 --> 01:18:32.980 In fact, this can really only take two distributions, 01:18:32.980 --> 01:18:34.063 and really, it's only one. 01:18:34.063 --> 01:18:36.870 But that's another story. 01:18:36.870 --> 01:18:39.210 So what I do in Monte Carlo simulations, 01:18:39.210 --> 01:18:45.800 I'm simulating these outcomes so we can get a lot more meat 01:18:45.800 --> 01:18:48.419 in this tail to understand what's happening out there. 01:18:48.419 --> 01:18:49.460 Does it drop off quickly? 01:18:49.460 --> 01:18:50.970 Does it not drop off quickly? 01:18:50.970 --> 01:18:53.920 That's kind of what it's about. 01:18:53.920 --> 01:18:56.850 So we're about out of time. 01:18:56.850 --> 01:18:59.850 We just covered like four weeks of material, all right? 01:18:59.850 --> 01:19:00.990 But you guys are from MIT. 01:19:00.990 --> 01:19:02.680 I have complete confidence in you. 01:19:02.680 --> 01:19:03.950 I say that to the people who work for me. 01:19:03.950 --> 01:19:05.700 I have complete confidence in your ability 01:19:05.700 --> 01:19:08.270 to get that done by tomorrow morning. 01:19:11.560 --> 01:19:13.020 Questions or comments? 01:19:13.020 --> 01:19:16.050 I know you're sipping from the fire hose here. 01:19:16.050 --> 01:19:17.160 I fully appreciate that. 01:19:20.350 --> 01:19:21.610 So those are examples. 01:19:21.610 --> 01:19:24.210 When I do this with historical simulation 01:19:24.210 --> 01:19:29.870 I won't generate these Monte Carlo trials, 01:19:29.870 --> 01:19:32.070 I'll just use historical data. 01:19:32.070 --> 01:19:35.740 And my fat tails are built into it. 01:19:35.740 --> 01:19:38.250 But what I've shown you today is what 01:19:38.250 --> 01:19:42.030 we developed a one-asset VaR model, 01:19:42.030 --> 01:19:47.530 then we developed a multi-asset variance/covariance model. 01:19:47.530 --> 01:19:50.520 And then I showed you quickly, and in far less time 01:19:50.520 --> 01:19:52.050 than I would like to have shown you, 01:19:52.050 --> 01:19:57.060 is how I can use another statistical technique, which 01:19:57.060 --> 01:20:01.950 is called the Gaussian copula, to generate synthetic data 01:20:01.950 --> 01:20:05.470 sets that will have the same properties as my source 01:20:05.470 --> 01:20:07.470 historical data. 01:20:07.470 --> 01:20:08.060 All right? 01:20:10.982 --> 01:20:12.444 There you have it. 01:20:12.444 --> 01:20:12.944 [APPLAUSE] 01:20:12.944 --> 01:20:16.304 Oh you don't have to-- please, please, please. 01:20:16.304 --> 01:20:18.470 And I'll tell you, for me, one of the coolest things 01:20:18.470 --> 01:20:20.240 was actually being able to apply so much 01:20:20.240 --> 01:20:22.410 of the math I learned in high school and in college 01:20:22.410 --> 01:20:23.826 and never thought I'd apply again. 01:20:23.826 --> 01:20:25.860 One of my best moments was actually 01:20:25.860 --> 01:20:27.355 finding a use for trigonometry. 01:20:30.110 --> 01:20:34.030 If you're not an engineer, where are you going to use it? 01:20:34.030 --> 01:20:35.030 Where do you use it? 01:20:35.030 --> 01:20:36.090 Seasonals. 01:20:36.090 --> 01:20:38.210 You do seasonal estimation. 01:20:38.210 --> 01:20:41.489 And what you do is you do fast Fourier transform. 01:20:41.489 --> 01:20:43.280 Because I can describe any seasonal pattern 01:20:43.280 --> 01:20:45.990 with a linear combination of sine and cosine functions. 01:20:45.990 --> 01:20:47.475 And it actually works. 01:20:47.475 --> 01:20:49.600 I have my students do it as an exercise every year. 01:20:49.600 --> 01:20:52.080 I say, go get New York city temperature data. 01:20:52.080 --> 01:20:54.650 And show me some linear combination 01:20:54.650 --> 01:20:57.230 of sine and cosine functions that 01:20:57.230 --> 01:21:00.740 will show me the seasonal pattern of temperature data. 01:21:00.740 --> 01:21:04.952 And when I first realized I could use trigonometry, yes! 01:21:04.952 --> 01:21:07.460 It wasn't a waste of time. 01:21:07.460 --> 01:21:08.960 I still-- polar coordinates, I still 01:21:08.960 --> 01:21:11.165 haven't found a use for that one. 01:21:11.165 --> 01:21:11.790 But it's there. 01:21:11.790 --> 01:21:12.890 I know it's there. 01:21:12.890 --> 01:21:13.790 All right? 01:21:13.790 --> 01:21:15.340 Go home.