WEBVTT 00:03.180 --> 00:09.320 Professor Robert Shiller: Today I want to spend--The 00:09.322 --> 00:15.572 title of today's lecture is: The Universal Principle of Risk 00:15.571 --> 00:21.141 Management, Pooling and the Hedging of Risk. 00:21.140 --> 00:25.010 What I'm really referring to is what I think is the very 00:25.008 --> 00:29.438 original, the deep concept that underlies theoretical finance--I 00:29.440 --> 00:31.480 wanted to get that first. 00:31.480 --> 00:36.660 It really is probability theory and the idea of spreading risk 00:36.657 --> 00:38.607 through risk pooling. 00:38.610 --> 00:44.550 So, this idea is an intellectual construct that 00:44.553 --> 00:51.663 appeared at a certain point in history and it has had an 00:51.659 --> 00:59.799 amazing number of applications and finance is one of these. 00:59.800 --> 01:03.830 Some of you--This incidentally will be a more technical of my 01:03.829 --> 01:07.929 lectures and it's a little bit unfortunate that it comes early 01:07.926 --> 01:10.676 in the semester. For those of you who have had a 01:10.684 --> 01:12.974 course in probability and statistics, there will be 01:12.967 --> 01:13.877 nothing new here. 01:13.879 --> 01:16.379 Well, nothing in terms of the math. 01:16.380 --> 01:18.220 The probability theory is new. 01:18.220 --> 01:21.370 Others though, I want to tell you that it 01:21.374 --> 01:26.184 doesn't--if you're shopping--I had a student come by yesterday 01:26.184 --> 01:30.924 and ask--he's a little rusty in his math skills--if he should 01:30.916 --> 01:32.726 take this course. 01:32.730 --> 01:35.160 I said, "Well if you can understand tomorrow's 01:35.163 --> 01:38.143 lecture--that's today's lecture--then you should have no 01:38.137 --> 01:38.837 problem." 01:38.840 --> 01:42.960 01:42.959 --> 01:47.669 I want to start with the concept of probability. 01:47.670 --> 01:50.910 01:50.910 --> 01:53.040 Do you know what a probability is? 01:53.040 --> 01:58.290 We attach a probability to an event. 01:58.290 --> 02:02.960 What is the probability that the stock market will go up this 02:02.955 --> 02:07.195 year? I would say--my personal 02:07.198 --> 02:10.278 probability is .45. 02:10.280 --> 02:15.360 That's because I'm a bear but--Do you know what that 02:15.362 --> 02:18.302 means? That 45 times out of 100 the 02:18.304 --> 02:22.824 stock market will go up and the other 55 times out of 100 it 02:22.822 --> 02:25.352 will stay the same or go down. 02:25.350 --> 02:28.530 That's a probability. 02:28.530 --> 02:32.750 Now, you're familiar with that concept, right? 02:32.750 --> 02:36.490 If someone says the probability is .55 or .45, 02:36.490 --> 02:39.150 well you know what that means. 02:39.150 --> 02:45.210 I want to emphasize that it hasn't always been that way and 02:45.208 --> 02:51.788 that probability is really a concept that arose in the 1600s. 02:51.789 --> 02:55.759 Before that, nobody ever said that. 02:55.759 --> 02:59.319 Ian Hacking, who wrote a history of 02:59.317 --> 03:03.607 probability theory, searched through world 03:03.607 --> 03:09.777 literature for any reference to a probability and could find 03:09.781 --> 03:13.131 none anywhere before 1600. 03:13.129 --> 03:18.689 There was an intellectual leap that occurred in the seventeenth 03:18.687 --> 03:23.797 century and it became very fashionable to talk in terms of 03:23.796 --> 03:26.996 probabilities. It spread throughout the 03:26.999 --> 03:29.779 world--the idea of quoting probabilities. 03:29.780 --> 03:34.470 But it was--It's funny that such a simple idea hadn't been 03:34.468 --> 03:37.838 used before. Hacking points out that the 03:37.842 --> 03:41.792 word probability--or probable--was already in the 03:41.793 --> 03:43.443 English language. 03:43.440 --> 03:48.790 In fact, Shakespeare used it, but what do you think it meant? 03:48.789 --> 03:53.679 He gives an example of a young woman, who was describing a man 03:53.683 --> 03:55.933 that she liked, and she said, 03:55.929 --> 04:00.019 I like him very much, I find him very probable. 04:00.020 --> 04:01.990 What do you think she means? 04:01.990 --> 04:03.410 Can someone answer that? 04:03.409 --> 04:07.349 Does anyone know Elizabethan English well enough to tell me? 04:07.350 --> 04:11.140 What is a probable young man? 04:11.140 --> 04:13.310 I'm asking for an answer. 04:13.310 --> 04:17.120 04:17.120 --> 04:21.320 It sounds like people have no idea. 04:21.320 --> 04:24.860 Can anyone venture a guess? 04:24.860 --> 04:26.320 No one wants to venture a guess? 04:26.320 --> 04:28.990 Student: fertile? 04:28.990 --> 04:32.560 Professor Robert Shiller: That he can father 04:32.563 --> 04:36.923 children? I don't think that's what she 04:36.924 --> 04:43.654 meant but maybe. No, what apparently she meant 04:43.653 --> 04:47.753 is trustworthy. That's a very important quality 04:47.748 --> 04:48.888 in a person I suppose. 04:48.889 --> 04:53.289 So, if something is probable you mean that you can trust it 04:53.294 --> 04:56.564 and so probability means trustworthiness. 04:56.560 --> 05:01.520 You can see how they moved from that definition of probability 05:01.522 --> 05:03.802 to the current definition. 05:03.800 --> 05:07.170 05:07.170 --> 05:09.950 But Ian Hacking, being a good historian, 05:09.949 --> 05:14.299 thought that someone must have had some concept of probability 05:14.298 --> 05:17.298 going before, even if they didn't quote it as 05:17.297 --> 05:20.757 a number the way--it must have been in their head or in their 05:20.758 --> 05:23.388 idea. He searched through world 05:23.389 --> 05:28.509 literature to try to find some use of the term that preceded 05:28.511 --> 05:33.891 the 1600s and he concluded that there were probably a number of 05:33.892 --> 05:39.262 people who had the idea, but they didn't publish it, 05:39.261 --> 05:45.601 and it never became part of the established literature partly 05:45.598 --> 05:48.808 because, he said, throughout human 05:48.806 --> 05:53.466 history, there has been a love of gambling and probability 05:53.474 --> 05:57.654 theory is extremely useful if you are a gambler. 05:57.649 --> 06:02.709 Hacking believes that there were many gambling theorists who 06:02.713 --> 06:07.693 invented probability theory at various times in history but 06:07.690 --> 06:11.810 never wrote it down and kept it as a secret. 06:11.810 --> 06:22.260 He gives an example--I like to--he gives an example from a 06:22.258 --> 06:30.138 book that--or it's a collection--I think, 06:30.139 --> 06:35.119 a collection of epic poems written in Sanskrit that goes 06:35.121 --> 06:40.651 back--it was actually written over a course of 1,000 years and 06:40.645 --> 06:44.535 it was completed in the fourth century. 06:44.540 --> 06:50.520 Well, there's a story--there's a long story in the Mahabarahta 06:50.517 --> 06:56.497 about an emperor called Nala and he had a wife named Damayanti 06:56.495 --> 07:01.195 and he was a very pure and very good person. 07:01.199 --> 07:07.849 There was an evil demon called Kali who hated Nala and wanted 07:07.849 --> 07:13.719 to bring his downfall, so he had to find a weakness of 07:13.722 --> 07:16.492 Nala. He found finally some, 07:16.493 --> 07:21.013 even though Nala was so pure and so perfect--he found one 07:21.008 --> 07:23.748 weakness and that was gambling. 07:23.750 --> 07:27.230 Nala couldn't resist the opportunity to gamble; 07:27.230 --> 07:33.120 so the evil demon seduced him into gambling aggressively. 07:33.120 --> 07:36.330 You know sometimes when you're losing and you redouble and you 07:36.334 --> 07:38.604 keep hoping to win back what you've lost? 07:38.600 --> 07:43.540 In a fit of gambling, Nala finally gambled his entire 07:43.538 --> 07:48.568 kingdom and lost--it's a terrible story--and Nala then 07:48.571 --> 07:52.561 had to leave the kingdom and his wife. 07:52.560 --> 07:54.260 They wandered for years. 07:54.259 --> 07:57.009 He separated from her because of dire necessity. 07:57.009 --> 08:00.689 They were wandering in the forests and he was in despair, 08:00.688 --> 08:02.328 having lost everything. 08:02.329 --> 08:08.959 But then he meets someone by the name of--we have Nala and he 08:08.962 --> 08:13.942 meets this man, Rituparna, and this is where a 08:13.936 --> 08:18.686 probability theory apparently comes in. 08:18.689 --> 08:23.699 Rituparna tells Nala that he knows the science of gambling 08:23.703 --> 08:28.663 and he will teach it to Nala, but that it has to be done by 08:28.663 --> 08:33.673 whispering it in his ear because it's a deep and extreme secret. 08:33.670 --> 08:35.550 Nala is skeptical. 08:35.549 --> 08:38.239 How does Rituparna know how to gamble? 08:38.240 --> 08:43.270 So Rituparna tries to prove to him his abilities and he says, 08:43.274 --> 08:47.224 see that tree there, I can estimate how many leaves 08:47.220 --> 08:51.210 there are on that tree by counting leaves on one branch. 08:51.210 --> 08:56.960 Rituparna looked at one branch and estimated the number of 08:56.960 --> 09:01.500 leaves on the tree, but Nala was skeptical. 09:01.500 --> 09:06.300 He stayed up all night and counted every leaf on the tree 09:06.297 --> 09:10.407 and it came very close to what Rituparna said; 09:10.409 --> 09:18.319 so he--the next morning--believed Rituparna. 09:18.320 --> 09:20.600 Now this is interesting, Hacking says, 09:20.597 --> 09:23.977 because it shows that sampling theory was part of Nala's 09:23.982 --> 09:26.482 theory. You don't have to count all the 09:26.482 --> 09:29.302 leaves on the tree, you can take a sample and you 09:29.303 --> 09:31.423 count that and then you multiply. 09:31.419 --> 09:36.179 Anyway, the story ends and Nala goes back and is now armed with 09:36.175 --> 09:38.625 probability theory, we assume. 09:38.629 --> 09:42.389 He goes back and gambles again, but he has nothing left to 09:42.387 --> 09:47.857 wager except his wife; so he puts her and gambles her. 09:47.860 --> 09:50.390 But remember, now he knows what he's doing 09:50.389 --> 09:54.209 and so he really wasn't gambling his wife--he was really a very 09:54.214 --> 09:55.884 pure and honorable man. 09:55.879 --> 10:01.739 So he won back the entire kingdom and that's the ending. 10:01.740 --> 10:05.130 Anyway, that shows that I think probability theory does have a 10:05.127 --> 10:07.737 long history, but--it not being an 10:07.740 --> 10:12.100 intellectual discipline--it didn't really inform a 10:12.101 --> 10:14.951 generation of finance theory. 10:14.950 --> 10:18.360 When you don't have a theory, then you don't have a way to be 10:18.364 --> 10:21.844 rigorous. So, it was in the 1600s that 10:21.840 --> 10:28.050 probability theory started to get written down as a theory and 10:28.050 --> 10:33.140 many things then happened in that century that, 10:33.139 --> 10:36.539 I think, are precursors both to finance and insurance. 10:36.539 --> 10:41.269 One was in the 1600s when people started constructing life 10:41.272 --> 10:43.362 tables. What is a life table? 10:43.360 --> 10:47.780 It's a table showing the probability of dying at each 10:47.779 --> 10:50.159 age, for each age and sex. 10:50.159 --> 10:53.249 That's what you need to know if you're going to do life 10:53.252 --> 10:55.442 insurance. So, they started to do 10:55.435 --> 10:59.445 collecting of data on mortality and they developed something 10:59.446 --> 11:03.056 called actuarial science, which is estimating the 11:03.055 --> 11:05.275 probability of people living. 11:05.280 --> 11:09.270 11:09.269 --> 11:15.429 That then became the basis for insurance. 11:15.429 --> 11:20.549 Actually, insurance goes back to ancient Rome in some form. 11:20.549 --> 11:23.899 In ancient Rome they had something called burial 11:23.903 --> 11:27.363 insurance. You could buy a policy that 11:27.361 --> 11:32.761 protected you against your family not having the money to 11:32.757 --> 11:35.067 bury you if you died. 11:35.070 --> 11:38.510 In ancient culture people worried a great deal about being 11:38.510 --> 11:40.930 properly buried, so that's an interesting 11:40.925 --> 11:43.175 concept. They were selling that in 11:43.182 --> 11:45.682 ancient Rome; but you might think, 11:45.679 --> 11:47.729 but why just for burial? 11:47.730 --> 11:50.710 Why don't you make it into full-blown life insurance? 11:50.710 --> 11:52.580 You kind of wonder why they didn't. 11:52.580 --> 11:55.870 I think maybe it's because they didn't have the concepts down. 11:55.870 --> 11:59.790 In Renaissance Italy they started writing insurance 11:59.789 --> 12:03.629 policies--I read one of the insurance policies, 12:03.629 --> 12:07.409 it's in the Journal of Risk and Insurance--and they translate a 12:07.410 --> 12:11.070 Renaissance insurance policy and it's very hard to understand 12:11.068 --> 12:13.018 what this policy was saying. 12:13.019 --> 12:17.459 I guess they didn't have our language, they didn't--they were 12:17.461 --> 12:21.681 intuitively halfway there but they couldn't express it, 12:21.679 --> 12:25.079 so I think the industry didn't get really started. 12:25.080 --> 12:29.090 I think it was the invention of probability theory that really 12:29.093 --> 12:32.843 started it and that's why I think theory is very important 12:32.843 --> 12:33.833 in finance. 12:33.830 --> 12:38.010 12:38.009 --> 12:42.929 Some people date fire insurance with the fire of London in 1666. 12:42.929 --> 12:45.949 The whole city burned down, practically, 12:45.947 --> 12:50.507 in a terrible fire and fire insurance started to proliferate 12:50.511 --> 12:52.911 right after that in London. 12:52.909 --> 12:55.339 But you know, you kind of wonder if that's a 12:55.339 --> 12:58.839 good example for fire insurance because if the whole city burns 12:58.841 --> 13:00.841 down, then insurance companies would 13:00.839 --> 13:02.179 go bankrupt anyway, right? 13:02.179 --> 13:05.549 London insurance companies would because the whole concept 13:05.554 --> 13:08.814 of insurance is pooling of independent probabilities. 13:08.810 --> 13:12.230 13:12.230 --> 13:14.700 Nonetheless, that was the beginning. 13:14.700 --> 13:20.040 We're also going to recognize, however, that insurance got a 13:20.042 --> 13:25.482 slow start because--I believe it is because--people could not 13:25.476 --> 13:29.276 understand the concept of probability. 13:29.279 --> 13:35.019 They didn't have the concept firmly in mind. 13:35.020 --> 13:36.670 There are lots of aspects to it. 13:36.669 --> 13:41.129 In order to understand probability, you have to take 13:41.126 --> 13:46.366 things as coming from a random event and people don't clearly 13:46.369 --> 13:51.349 have that in their mind from an intuitive standpoint. 13:51.350 --> 13:55.610 They have maybe a sense that I can influence events by willing 13:55.606 --> 13:59.096 or wishing and if I think that--if I have kind of a 13:59.095 --> 14:03.415 mystical side to me, then probabilities don't have a 14:03.418 --> 14:07.408 clear meaning. It has been shown that even 14:07.413 --> 14:10.733 today people seem to think that. 14:10.730 --> 14:16.840 They don't really take, at an intuitive level, 14:16.835 --> 14:20.765 probabilities as objective. 14:20.769 --> 14:23.019 For example, if you ask people how much they 14:23.017 --> 14:25.157 would be willing to bet on a coin toss, 14:25.159 --> 14:29.579 they will typically bet more if they can toss the coin or they 14:29.577 --> 14:33.267 will bet more if the coin hasn't been tossed yet. 14:33.269 --> 14:36.089 It could have been already tossed and concealed. 14:36.090 --> 14:37.920 Why would that be? 14:37.919 --> 14:43.439 It might be that there's just some intuitive sense that I 14:43.436 --> 14:49.346 can--I don't know--I have some magical forces in me and I can 14:49.346 --> 14:52.966 change things. The idea of probability theory 14:52.966 --> 14:56.346 is that no, you can't change things, there are all these 14:56.350 --> 15:00.350 objective laws of probability out there that guide everything. 15:00.350 --> 15:10.380 Most languages around the world have a different word for luck 15:10.378 --> 15:15.638 and risk--or luck and fortune. 15:15.639 --> 15:19.839 Luck seems to mean something about you: like I'm a lucky 15:19.842 --> 15:21.862 person. I don't know what that 15:21.860 --> 15:25.190 means--like God or the gods favor me and so I'm lucky or 15:25.188 --> 15:26.638 this is my lucky day. 15:26.639 --> 15:31.219 Probability theory is really a movement away from that. 15:31.220 --> 15:38.660 We then have a mathematically rigorous discipline. 15:38.659 --> 15:45.489 Now, I'm going to go through some of the terms of probability 15:45.487 --> 15:50.377 and--this will be review for many of you, 15:50.379 --> 15:56.009 but it will be something that we're going to use in the--So 15:56.012 --> 16:02.032 I'll use the symbol P or I can sometimes write it out as 16:02.033 --> 16:06.213 prob to represent a probability. 16:06.210 --> 16:11.980 It is always a number that lies between zero and one, 16:11.977 --> 16:14.747 or between 0% and 100%. 16:14.750 --> 16:20.580 "Percent" means divided by 100 in Latin, so 100% is one. 16:20.580 --> 16:28.510 If the probability is zero that means the event can't happen. 16:28.509 --> 16:32.519 If the probability is one, it means that it's certain to 16:32.523 --> 16:34.673 happen. If the probability is--Can 16:34.671 --> 16:36.681 everyone see this from over there? 16:36.679 --> 16:41.099 I can probably move this or can't I? 16:41.100 --> 16:44.100 Yes, I can. Now, can you now--you're the 16:44.102 --> 16:47.442 most disadvantaged person and you can see it, 16:47.437 --> 16:57.127 right? So that's the basic idea. 16:57.129 --> 17:00.429 One of the first principles of probability is the idea of 17:00.434 --> 17:01.324 independence. 17:01.320 --> 17:08.340 17:08.339 --> 17:17.579 The idea is that probability measures the likelihood of some 17:17.583 --> 17:20.753 outcome. Let's say the outcome of an 17:20.751 --> 17:22.711 experiment, like tossing a coin. 17:22.710 --> 17:26.150 You might say the probability that you toss a coin and it 17:26.145 --> 17:29.515 comes up heads is a half, because it's equally likely to 17:29.519 --> 17:30.929 be heads and tails. 17:30.930 --> 17:36.280 Independent experiments are experiments that occur without 17:36.282 --> 17:38.632 relation to each other. 17:38.630 --> 17:42.480 If you toss a coin twice and the first experiment doesn't 17:42.477 --> 17:45.977 influence the second, we say they're independent and 17:45.981 --> 17:48.731 there's no relation between the two. 17:48.730 --> 17:53.790 One of the first principles of probability theory is called the 17:53.793 --> 17:55.593 multiplication rule. 17:55.590 --> 18:00.570 18:00.569 --> 18:03.739 That says that if you have independent probabilities, 18:03.735 --> 18:07.205 then the probability of two events is equal to the product 18:07.206 --> 18:08.846 of their probabilities. 18:08.849 --> 18:22.019 So, the Prob(A and B) = Prob(A)*Prob(B). 18:22.020 --> 18:28.310 18:28.309 --> 18:32.869 That wouldn't hold if they're not independent. 18:32.869 --> 18:38.079 The theory of insurance is that ideally an insurance company 18:38.081 --> 18:41.351 wants to insure independent events. 18:41.349 --> 18:44.459 Ideally, life insurance is insuring people--or fire 18:44.464 --> 18:47.454 insurance is insuring people--against independent 18:47.454 --> 18:50.384 events; so it's not the fire of London. 18:50.380 --> 18:55.520 It's the problem that sometimes people knock over an oil lamp in 18:55.516 --> 18:59.426 their home and they burn their own house down. 18:59.430 --> 19:02.390 It's not going to burn any other houses down since it's 19:02.391 --> 19:04.971 just completely independent of anything else. 19:04.970 --> 19:08.720 So, the probability that the whole city burns down is 19:08.722 --> 19:10.962 infinitesimally small, right? 19:10.960 --> 19:13.570 This will generalize to probability of A and 19:13.574 --> 19:16.774 B and C equals the probability of A times 19:16.765 --> 19:19.425 the probability of B times the probability of 19:19.432 --> 19:20.742 C and so on. 19:20.740 --> 19:24.520 If the probability is 1 in 1,000 that a house burns down 19:24.524 --> 19:29.224 and there are 1,000 houses, then the probability that they 19:29.215 --> 19:33.255 all burn down is 1/1000 to the 1000th power, 19:33.260 --> 19:34.800 which is virtually zero. 19:34.799 --> 19:37.639 So insurance companies then–Basically, 19:37.635 --> 19:40.665 if they write a lot of policies, then they have 19:40.669 --> 19:42.119 virtually no risk. 19:42.119 --> 19:48.639 That is the fundamental idea that may seem simple and obvious 19:48.644 --> 19:55.174 to you, but it certainly wasn't back when the idea first came 19:55.168 --> 19:57.098 up. Incidentally, 19:57.102 --> 20:01.882 we have a problem set, which I want you to start today 20:01.881 --> 20:06.031 and it will be due not in a week this time, 20:06.029 --> 20:10.319 because we have Martin Luther King Day coming up, 20:10.319 --> 20:14.519 but it will be due the Monday following that. 20:14.520 --> 20:24.020 20:24.019 --> 20:27.619 If you follow through from the independent theory, 20:27.616 --> 20:31.136 there's one of the basic relations in probability 20:31.139 --> 20:34.809 theory--it's called the binomial distribution. 20:34.809 --> 20:42.879 I'm not going to spend a whole lot of time on this but it gives 20:42.880 --> 20:50.950 the probability of x successes in n trials or, 20:50.950 --> 20:54.580 in the case of insurance x, if you're insuring 20:54.579 --> 20:57.719 against an accident, then the probability that 20:57.720 --> 21:01.490 you'll get x accidents and n trials. 21:01.490 --> 21:12.410 The binomial distribution gives the probability as a function of 21:12.414 --> 21:22.654 x and it's given by the formula where P is the 21:22.645 --> 21:33.565 probability of the accident: P^(X) (1-P)^(N-X) [n!/(n-x)!]. 21:33.569 --> 21:37.989 That is the formula that insurance companies use when 21:37.990 --> 21:41.220 they have independent probabilities, 21:41.220 --> 21:47.230 to estimate the likelihood of having a certain number of 21:47.228 --> 21:50.478 accidents. They're concerned with having 21:50.484 --> 21:53.444 too many accidents, which might exhaust their 21:53.435 --> 21:55.915 reserves. An insurance company has 21:55.924 --> 21:59.714 reserves and it has enough reserves to cover them for a 21:59.714 --> 22:01.894 certain number of accidents. 22:01.890 --> 22:06.080 It uses the binomial distribution to calculate the 22:06.075 --> 22:11.025 probability of getting any specific number of accidents. 22:11.029 --> 22:16.619 So, that is the binomial distribution. 22:16.620 --> 22:21.110 22:21.109 --> 22:25.409 I'm not going to expand on this because I can't get into--This 22:25.409 --> 22:29.709 is not a course in probability theory but I'm hopeful that you 22:29.708 --> 22:32.878 can see the formula and you can apply it. 22:32.880 --> 22:34.420 Any questions? Is this clear enough? 22:34.420 --> 22:35.480 Can you read my handwriting? 22:35.480 --> 22:40.460 22:40.460 --> 22:43.980 Another important concept in probability theory that we will 22:43.980 --> 22:45.770 use a lot is expected value, 22:45.770 --> 22:51.170 22:51.170 --> 22:56.450 the mean, or average--those are all roughly interchangeable 22:56.448 --> 23:02.368 concepts. We have expected value, 23:02.372 --> 23:06.012 mean or average. 23:06.010 --> 23:14.370 23:14.369 --> 23:18.509 We can define it in a couple of different ways depending on 23:18.512 --> 23:22.872 whether we're talking about sample mean or population mean. 23:22.869 --> 23:28.199 The basic definition--the expected value of some random 23:28.198 --> 23:34.508 variable x--E(x)--I guess I should have said that a random 23:34.512 --> 23:39.152 variable is a quantity that takes on value. 23:39.150 --> 23:43.470 If you have an experiment and the outcome of the experiment is 23:43.471 --> 23:47.791 a number, then a random variable is the number that comes from 23:47.793 --> 23:49.983 the experiment. For example, 23:49.980 --> 23:52.780 the experiment could be tossing a coin; 23:52.779 --> 23:56.559 I will call the outcome heads the number one, 23:56.556 --> 24:00.556 and I'll call the outcome tails the number zero, 24:00.555 --> 24:03.735 so I've just defined a random variable. 24:03.740 --> 24:07.250 You have discrete random variables, like the one I just 24:07.245 --> 24:10.295 defined, or there are also--which take on only a 24:10.296 --> 24:14.386 finite number of values--and we have continuous random variables 24:14.385 --> 24:18.405 that can take on any number of values along a continuum. 24:18.410 --> 24:22.370 Another experiment would be to mix two chemicals together and 24:22.372 --> 24:25.742 put a thermometer in and measure the temperature. 24:25.740 --> 24:28.850 That's another invention of the 1600s, by the way--the 24:28.848 --> 24:31.148 thermometer. And they learned that 24:31.152 --> 24:34.472 concept--perfectly natural to us--temperature. 24:34.470 --> 24:37.340 But it was a new idea in the 1600s. 24:37.339 --> 24:39.939 So anyway, that's continuous, right? 24:39.940 --> 24:42.890 When you mix two chemicals together, it could be any 24:42.892 --> 24:46.192 number, there's an infinite number of possible numbers and 24:46.191 --> 24:47.871 that would be continuous. 24:47.869 --> 24:53.229 For discrete random variables, we can define the expected 24:58.397 --> 25:03.467 mu--as the summation i = 1 to infinity of. 25:03.470 --> 25:12.970 [P(x=x_i) times (x_i)]. 25:12.970 --> 25:16.690 I have it down that there might be an infinite number of 25:16.690 --> 25:20.140 possible values for the random variable x. 25:20.140 --> 25:22.830 In the case of the coin toss, there are only two, 25:22.832 --> 25:26.312 but I'm saying in general there could be an infinite number. 25:26.309 --> 25:29.869 But they're accountable and we can list all possible values 25:29.869 --> 25:33.739 when they're discrete and form a probability weighted average of 25:33.736 --> 25:36.266 the outcomes. That's called the expected 25:36.269 --> 25:39.159 value. People also call that the mean 25:39.161 --> 25:44.621 or the average. But, note that this is based on 25:44.615 --> 25:47.415 theory. These are probabilities. 25:47.420 --> 25:53.320 In order to compute using this formula you have to know the 25:53.315 --> 25:55.445 true probabilities. 25:55.450 --> 26:00.060 There's another formula that applies for a continuous random 26:00.060 --> 26:04.900 variables and it's the same idea except that--I'll also call it 26:09.410 --> 26:17.940 We have the integral from minus infinity to plus infinity of 26:17.942 --> 26:22.582 F(x)*x*dx, and that's really--you see it's 26:22.579 --> 26:25.689 the same thing because an integral is analogous to a 26:25.687 --> 26:26.477 summation. 26:26.480 --> 26:32.020 26:32.019 --> 26:35.319 Those are the two population definitions. 26:35.319 --> 26:40.149 F(x) is the continuous probability distribution for 26:40.148 --> 26:41.078 x. 26:41.080 --> 26:45.590 26:45.589 --> 26:49.209 That's different when you have continuous values--you don't 26:49.211 --> 26:52.021 have P (x = x_i) because it's 26:52.021 --> 26:54.831 always zero. The probability that the 27:02.534 --> 27:05.634 an infinite number of possibilities. 27:05.630 --> 27:10.660 We have instead what's called a probability density when we have 27:10.663 --> 27:13.063 continuous random variables. 27:13.059 --> 27:16.469 You're not going to need to know a lot about this for this 27:16.470 --> 27:20.060 course, but this is--I wanted to get the basic ideas down. 27:20.059 --> 27:25.599 These are called population measures because they refer to 27:25.604 --> 27:31.544 the whole population of possible outcomes and they measure the 27:31.538 --> 27:35.138 probabilities. It's the truth, 27:35.143 --> 27:39.223 but there are also sample means. 27:39.220 --> 27:43.380 When you get--this is Rituparna, counting the leaves 27:43.375 --> 27:46.795 on a tree--you can estimate, from a sample, 27:46.798 --> 27:49.648 the population expected values. 27:49.650 --> 27:56.720 The population mean is often written "x-bar." 27:56.720 --> 28:00.790 If you have a sample with n observations, 28:00.787 --> 28:04.677 it's the summation i = 1 to n of 28:04.682 --> 28:08.752 x_i/n--that's the average. 28:08.750 --> 28:10.290 You know that formula, right? 28:10.289 --> 28:14.239 You count n leaves--you count the number of leaves. 28:14.240 --> 28:21.020 You have n branches on the tree and you count the 28:21.017 --> 28:25.327 number of leaves and sum them up. 28:25.329 --> 28:29.769 One would be--I'm having a little trouble putting this into 28:29.770 --> 28:33.140 the Rituparna story, but you see the idea. 28:33.140 --> 28:34.560 You know the average, I assume. 28:34.559 --> 28:40.049 That's the most elementary concept and you could use it to 28:40.046 --> 28:45.626 estimate either a discreet or continuous expected value. 28:45.630 --> 28:49.870 In finance, there's often reference to another kind of 28:49.869 --> 28:53.869 average, which I want to refer you to and which, 28:53.869 --> 28:58.049 in the Jeremy Siegel book, a lot is made of this. 28:58.049 --> 29:05.279 The other kind of average is called the geometric average. 29:05.279 --> 29:16.139 We'll call that--I'll only show the sample version of it G(x) = 29:16.142 --> 29:25.432 the product i = 1 to n of (x_i 29:25.427 --> 29:29.007 )^(1/n). Does everyone--Can you see that? 29:29.010 --> 29:35.170 29:35.170 --> 29:38.680 Instead of summing them and dividing by M, 29:38.676 --> 29:43.376 I multiply them all together and take the n^(th) root of 29:43.380 --> 29:46.940 them. This is called the geometric 29:46.942 --> 29:51.832 average and it's used only for positive numbers. 29:51.829 --> 29:54.779 So, if you have any negative numbers you'd have a problem, 29:54.776 --> 29:56.466 right? If you had one negative number 29:56.473 --> 29:58.873 in it, then the product would be a negative number and, 29:58.869 --> 30:01.969 if you took a root of that, then you might get an imaginary 30:01.969 --> 30:06.129 number. We don't want to use it in that 30:06.128 --> 30:09.098 case. There's an appendix to one of 30:09.101 --> 30:13.861 the chapters in Jeremy Siegel's book where he says that one of 30:13.859 --> 30:18.769 the most important applications of this theory is to measure how 30:18.772 --> 30:21.192 successful an investor is. 30:21.190 --> 30:26.120 30:26.119 --> 30:30.029 Suppose someone is managing money. 30:30.030 --> 30:33.840 Have they done well? 30:33.839 --> 30:36.249 If so, you would say, "Well, they've been investing 30:36.252 --> 30:38.232 money over a number of different years. 30:38.230 --> 30:40.930 Let's take the average over all the different years." 30:40.930 --> 30:44.330 Suppose someone has been investing money for n 30:44.327 --> 30:48.307 years and x_i is the return on the investment 30:48.313 --> 30:50.743 in a given year. What is their average 30:50.744 --> 30:53.004 performance? The natural thing to do would 30:52.998 --> 30:54.608 be to average them up, right? 30:54.609 --> 31:01.529 But Jeremy says that maybe that's not a very good thing to 31:01.532 --> 31:05.112 do. What he says you should do 31:05.107 --> 31:11.837 instead is to take the geometric average of gross returns. 31:11.839 --> 31:17.329 The return on an investment is how much you made from the 31:17.332 --> 31:22.042 investment as a percent of the money invested. 31:22.039 --> 31:27.349 The gross return is the return plus one. 31:27.349 --> 31:31.479 The worst you can ever do investing is lose all of your 31:31.483 --> 31:33.323 investment--lose 100%. 31:33.319 --> 31:38.349 If we add one to the return, then you've got a number that's 31:38.350 --> 31:43.040 never negative and we can then use geometric returns. 31:43.039 --> 31:47.379 Jeremy Siegel says that in finance we should be using 31:47.375 --> 31:50.705 geometric and not arithmetic averages. 31:50.710 --> 31:52.580 Why is that? Well I'll tell you in very 31:52.583 --> 31:53.643 simple terms, I think. 31:53.640 --> 31:57.570 Suppose someone is investing your money and he announces, 31:57.572 --> 31:59.752 I have had very good returns. 31:59.750 --> 32:03.720 I have invested and I've produced 20% a year for nine out 32:03.719 --> 32:05.419 of the last ten years. 32:05.420 --> 32:09.670 You think that's great, but what about the last year. 32:09.670 --> 32:13.630 The guy says, "Oh I lost 100% in that year." 32:13.630 --> 32:16.480 You might say, "Alright, that's good." 32:16.480 --> 32:21.130 I would add up 20% a year for nine years and than put in a 32:21.130 --> 32:24.640 zero–no, 120 because it's gross return 32:24.638 --> 32:28.798 for nine years--and put in a zero for one year. 32:28.799 --> 32:31.319 Maybe that doesn't look bad, right? 32:31.319 --> 32:33.889 But think about it, if you were investing your 32:33.892 --> 32:37.152 money with someone like that, what did you end up with? 32:37.150 --> 32:39.300 You ended up with nothing. 32:39.299 --> 32:41.639 If they have one year when they lose everything, 32:41.641 --> 32:44.531 it doesn't matter how much they made in the other years. 32:44.529 --> 32:49.419 Jeremy says in the text that the geometric return is always 32:49.421 --> 32:54.651 lower than the arithmetic return unless all the numbers are the 32:54.650 --> 33:00.910 same. It's a less optimistic version. 33:00.910 --> 33:04.390 So, we should use that, but people in finance resist 33:04.393 --> 33:08.013 using that because it's a lower number and when you're 33:08.012 --> 33:11.912 advertising your return you want to make it look as big as 33:11.905 --> 33:14.915 possible. We also need some measure 33:14.917 --> 33:19.047 of--We've been talking here about measures of central 33:19.050 --> 33:22.230 tendency only and in finance we need, 33:22.230 --> 33:28.180 as well, measures of dispersion, which is how much 33:28.182 --> 33:30.492 something varies. 33:30.490 --> 33:35.250 Central tendency is a measure of the center of a probability 33:35.252 --> 33:40.022 distribution of the--Central tendency is a measure--Variance 33:40.015 --> 33:44.935 is a measure of how much things change from one observation to 33:44.939 --> 33:49.799 another. We have variance and it's often 33:49.795 --> 33:56.765 represented by σ², that's the Greek letter sigma, 33:56.765 --> 33:59.525 lower case, squared. 33:59.529 --> 34:08.819 Or, especially when talking about estimates of the variance, 34:19.679 --> 34:22.909 The standard deviation is the square root of the variance. 34:22.909 --> 34:31.829 For population variance, the variance of some random 34:31.828 --> 34:42.668 variable x is defined as the summation i = 1 to infinity 34:42.671 --> 34:50.721 of the Prob (x = x_i) times 34:59.809 --> 35:05.829 So mu is the mean--we just defined it of x--that's 35:05.827 --> 35:11.197 the expectation of x or also E(x), 35:11.199 --> 35:15.609 so it's the probability weighted average of the squared 35:15.606 --> 35:17.806 deviations from the mean. 35:17.809 --> 35:21.759 If it moves a lot--either way from the mean--then this number 35:21.758 --> 35:23.468 squared is a big number. 35:23.469 --> 35:28.269 The more x moves, the bigger the variance is. 35:28.269 --> 35:35.369 There's also another variance measure, which we use in the 35:35.370 --> 35:41.470 sample--or also Var is used sometimes--and this is 35:41.474 --> 35:44.344 ∑². 35:44.340 --> 35:48.410 There's also another variance measure, which is for the 35:48.411 --> 35:52.601 sample. When we have n 35:52.595 --> 36:01.935 observations it's just the summation i = 1 to n of 36:06.950 --> 36:12.870 36:12.870 --> 36:17.070 That is the sample variance. 36:17.070 --> 36:20.040 Some people will divide by n–1. 36:20.039 --> 36:25.509 I suppose I would accept either answer. 36:25.510 --> 36:27.870 I'm just keeping it simple here. 36:27.869 --> 36:34.399 They divide by n-1 to make it an unbiased estimator of 36:34.400 --> 36:38.900 the population variance; but I'm just going to show it 36:38.896 --> 36:40.156 in a simple way here. 36:40.159 --> 36:43.529 So you see what it is--it's a measure of how much x 36:43.531 --> 36:46.601 deviates from the mean; but it's squared. 36:46.599 --> 36:50.069 It weights big deviations a lot because the square of a big 36:50.072 --> 36:51.452 number is really big. 36:51.450 --> 36:57.680 So, that's the variance. 36:57.679 --> 37:02.789 So, that completes central tendency and dispersion. 37:02.789 --> 37:06.799 We're going to be talking about these in finance in regards to 37:06.803 --> 37:10.753 returns because--generally the idea here is that we want high 37:10.750 --> 37:13.710 returns. We want a high expected value 37:13.708 --> 37:16.698 of returns, but we don't like variance. 37:16.699 --> 37:21.599 Expected value is good and variance is bad because that's 37:21.597 --> 37:23.027 risk; that's uncertainty. 37:23.030 --> 37:27.810 That's what this whole theory is about: how to get a lot of 37:27.812 --> 37:31.772 expected return without getting a lot of risk. 37:31.769 --> 37:35.919 Another concept that's very basic here is covariance. 37:35.920 --> 37:41.160 Covariance is a measure of how much two variables move 37:41.162 --> 37:47.752 together. Covariance is--we'll call 37:47.754 --> 37:59.644 it--now we have two random variables, so I'll just talk 37:59.644 --> 38:06.034 about it in a sample term. 38:06.030 --> 38:15.520 It's the summation i = 1 to n of [(x – 38:15.523 --> 38:22.963 x-bar) times (y – y-bar)]/n. 38:22.960 --> 38:28.770 So x is the deviation for the i-subscript, 38:28.769 --> 38:34.049 meaning we have a separate x_i and 38:34.050 --> 38:38.170 y_i for each observation. 38:38.170 --> 38:41.000 So we're talking about an experiment when you 38:41.002 --> 38:44.672 generate--Each experiment generates both an x and a 38:44.671 --> 38:48.471 y observation and we know when x is high, 38:48.469 --> 38:51.699 y also tends to be high, or whether it's the other way 38:51.696 --> 38:54.396 around. If they tend to move together, 38:54.400 --> 38:58.620 when x is high and y is high together at the 38:58.617 --> 39:01.557 same time, then the covariance will tend 39:01.556 --> 39:03.256 to be a positive number. 39:03.260 --> 39:07.100 If when x is low, y also tends to be low, 39:07.099 --> 39:10.649 then this will be negative number and so will this, 39:10.653 --> 39:13.003 so their product is positive. 39:13.000 --> 39:18.260 A positive covariance means that the two move together. 39:18.260 --> 39:21.720 A negative covariance means that they tend to move opposite 39:21.715 --> 39:24.215 each other. If x is high relative to 39:24.216 --> 39:27.166 x-bar--this is positive--then y tends to 39:27.169 --> 39:30.009 be low relative to its mean y-bar and this is 39:30.010 --> 39:31.930 negative. So the product would be 39:31.933 --> 39:34.083 negative. If you get a lot of negative 39:34.083 --> 39:36.803 products, that makes the covariance negative. 39:36.800 --> 39:41.410 Then I want to move to correlation. 39:41.410 --> 39:47.090 39:47.090 --> 39:51.800 So this is a measure--it's a scaled covariance. 39:51.800 --> 39:56.980 We tend to use the Greek letter rho. 39:56.980 --> 40:01.420 If you were to use Excel, it would be correl or 40:01.424 --> 40:03.944 sometimes I say corr. 40:03.940 --> 40:05.000 That's the correlation. 40:05.000 --> 40:12.760 This number always lies between -1 and +1. 40:12.760 --> 40:19.470 40:19.469 --> 40:26.139 It is defined as rho= [cov(x_iy 40:26.143 --> 40:32.983 _i)/S_x S_y] 40:32.983 --> 40:39.493 That's the correlation coefficient. 40:39.489 --> 40:43.539 That has kind of almost entered the English language in the 40:43.540 --> 40:47.800 sense that you'll see it quoted occasionally in newspapers. 40:47.800 --> 40:51.510 I don't know how much you're used to it--Where would you see 40:51.509 --> 40:53.999 that? They would say there is a low 40:53.999 --> 40:58.029 correlation between SAT scores and grade point averages in 40:58.029 --> 41:01.209 college, or maybe it's a high correlation. 41:01.210 --> 41:03.010 Does anyone know what it is? 41:03.010 --> 41:06.390 But you could estimate the corr--it's probably positive. 41:06.389 --> 41:12.339 I bet it's way below one, but it has some correlation, 41:12.337 --> 41:14.467 so maybe it's .3. 41:14.469 --> 41:19.369 That would mean that people who have high SAT scores tend to get 41:19.366 --> 41:22.196 higher grades. If it were negative--it's very 41:22.202 --> 41:25.072 unlikely that it's negative--it couldn't be negative. 41:25.070 --> 41:29.080 It couldn't be that people who have high SAT scores tend to do 41:29.076 --> 41:30.386 poorly in college. 41:30.389 --> 41:34.949 If you quantify how much they relate, then you could look at 41:34.949 --> 41:36.339 the correlation. 41:36.340 --> 41:42.580 41:42.580 --> 41:46.840 I want to move to regression. 41:46.840 --> 41:52.720 This is another concept that is very basic to statistics, 41:52.722 --> 41:58.502 but it has particular use in finance, so I'll give you a 41:58.499 --> 42:00.809 financial example. 42:00.809 --> 42:07.759 The concept of regression goes back to the mathematician Gauss, 42:07.755 --> 42:14.695 who talked about fitting a line through a scatter of points. 42:14.699 --> 42:22.309 Let's draw a line through a scatter of points here. 42:22.309 --> 42:31.269 I want to put down on this axis the return on the stock market 42:31.269 --> 42:39.489 and on this axis I want to put the return on one company, 42:39.494 --> 42:43.024 let's say Microsoft. 42:43.020 --> 42:46.690 42:46.690 --> 42:49.990 I'm going to have each observation as a year. 42:49.989 --> 42:53.309 I shouldn't put down a name of a company because I can't 42:53.313 --> 42:55.673 reproduce this diagram for Microsoft. 42:55.670 --> 43:02.490 Let's not say Microsoft, let's say Shiller, 43:02.494 --> 43:04.704 Inc. There's no such company, 43:04.701 --> 43:06.601 so I can be completely hypothetical. 43:06.599 --> 43:11.839 Let's put zero here because these are not gross returns 43:11.844 --> 43:16.414 these are returns, so they're often negative. 43:16.409 --> 43:20.869 Suppose that in a given year--and say this is minus five 43:20.865 --> 43:25.325 and this is plus five, this is minus five and this is 43:25.330 --> 43:30.120 plus five--Suppose that in the first year in our sample, 43:30.120 --> 43:34.030 the company Shiller, Inc. 43:34.030 --> 43:36.140 and the market both did 5%. 43:36.139 --> 43:44.069 That puts a point right there at five and five. 43:44.070 --> 43:48.900 In another year, however, the stock market lost 43:48.900 --> 43:51.210 5% and Shiller, Inc. 43:51.210 --> 43:55.310 lost 7%. We would have a point, 43:55.313 --> 43:59.063 say, down here at five and seven. 43:59.059 --> 44:02.979 This could be 1979, this could be 1980, 44:02.978 --> 44:09.678 and we keep adding points so we have a whole scatter of points. 44:09.679 --> 44:14.409 It's probably upward sloping, right? 44:14.409 --> 44:19.469 Probably when the overall stock market does well so does 44:19.469 --> 44:22.589 Shiller, Inc. What Gauss did was said, 44:22.585 --> 44:26.125 let's fit a line through the point--the scatter of 44:26.127 --> 44:29.667 points--and that's called the regression line. 44:29.670 --> 44:35.880 He chose the line so that--this is Gauss--he chose the line to 44:35.877 --> 44:41.877 minimize the sum of squared distances of the points from the 44:41.882 --> 44:44.242 lines. So these distances are the 44:44.240 --> 44:45.890 lengths of these line segments. 44:45.889 --> 44:49.159 To get the best fitting line, you find the line that 44:49.160 --> 44:51.790 minimizes the sum of squared distances. 44:51.789 --> 44:56.129 That's called the regression line and the intercept is called 44:56.132 --> 44:58.812 alpha--there's alpha. 44:58.809 --> 45:01.369 And the slope is called beta. 45:01.369 --> 45:06.599 That may be a familiar enough concept to you, 45:06.601 --> 45:13.261 but in the context of finance this is a major concept. 45:13.260 --> 45:18.680 The way I've written it, the beta of Shiller, 45:18.682 --> 45:21.382 Inc. is the slope of this line. 45:21.380 --> 45:29.200 The alpha is just the intercept of this curve. 45:29.200 --> 45:33.170 45:33.170 --> 45:35.410 We can also do this with excess returns. 45:35.409 --> 45:38.789 I will get to this later, where I have the return minus 45:38.790 --> 45:42.610 the interest rate on this axis and the market return minus the 45:42.609 --> 45:44.549 interest rate on this axis. 45:44.550 --> 45:49.050 In that case, alpha is a measure of 45:49.049 --> 45:51.349 how much Shiller, Inc. 45:51.354 --> 45:54.474 outperforms. We'll come back to this, 45:54.468 --> 45:57.918 but beta of the stock is a measure of how much it moves 45:57.921 --> 46:00.921 with the market and the alpha of a stock is how 46:00.922 --> 46:02.962 much it outperforms the market. 46:02.960 --> 46:05.680 We'll have to come back to that--these are basic concepts. 46:05.680 --> 46:12.860 46:12.860 --> 46:18.170 I want to--another concept--I guess I've just been implicit in 46:18.166 --> 46:22.686 what I have--There's a distribution called the normal 46:22.690 --> 46:27.910 distribution and that is--I'm sure you've heard of this, 46:27.910 --> 46:33.060 right? If you have a distribution that 46:33.059 --> 46:39.479 looks like this--it's bell-shaped--this is x 46:39.478 --> 46:46.538 and--I have to make it look symmetric which I may not be 46:46.539 --> 46:52.059 able to do that well--this is f(x), 46:52.060 --> 46:53.950 the normal distribution. 46:53.949 --> 46:59.459 f(x) = [1/(√ (2π)σ)] 46:59.460 --> 47:06.860 times e to minus [(x-µ)^(2) / 2σ]. 47:06.860 --> 47:12.060 It's a famous formula, which is due to Gauss again. 47:12.059 --> 47:15.689 We often assume in finance that random variables, 47:15.691 --> 47:19.021 such as returns, are normally distributed. 47:19.019 --> 47:24.249 This is called the normal distribution or the Gaussian 47:24.245 --> 47:28.875 distribution--it's a continuous distribution. 47:28.880 --> 47:30.120 I think you've heard of this, right? 47:30.119 --> 47:33.159 This is high school raw material. 47:33.159 --> 47:38.129 But I want to emphasize that there are also other bell-shaped 47:38.133 --> 47:40.373 curves. This is the most famous 47:40.371 --> 47:43.421 bell-shaped curve, but there are other ones with 47:43.418 --> 47:45.038 different mathematics. 47:45.039 --> 47:49.019 A particular interest in finance is fat-tailed 47:49.022 --> 47:50.352 alternatives. 47:50.350 --> 47:55.840 47:55.840 --> 47:59.250 It could be that a random distribution--I don't have 47:59.249 --> 48:01.589 colored chalk here I don't think, 48:01.590 --> 48:05.280 so I will use a dash line to represent the fat-tailed 48:05.276 --> 48:08.286 distribution. Suppose the distribution looks 48:08.286 --> 48:08.996 like this. 48:09.000 --> 48:13.020 48:13.019 --> 48:17.649 Then I have to try to do that on the other side, 48:17.651 --> 48:20.411 as symmetrically as I can. 48:20.409 --> 48:24.229 These are the tails of the distribution; 48:24.230 --> 48:30.970 this is the right tail and this is the left tail. 48:30.969 --> 48:37.339 You can see that the dash distribution I drew has more out 48:37.338 --> 48:41.918 in the tails, so we call it fat-tailed. 48:41.920 --> 48:45.710 This refers to random variables that have fat-tailed 48:45.710 --> 48:49.950 distributions--random variables that occasionally give you 48:49.946 --> 48:51.726 really big outcomes. 48:51.730 --> 48:55.410 You have a chance of being way out here with a fat-tailed 48:55.413 --> 48:58.043 distribution. It's a very important 48:58.035 --> 49:02.545 observation in finance that returns on a lot of speculative 49:02.554 --> 49:05.674 assets have fat-tailed distributions. 49:05.670 --> 49:09.750 That means that you can go through twenty years of a career 49:09.751 --> 49:13.971 on Wall Street and all you've observed is observations in the 49:13.973 --> 49:16.733 central region. So you feel that you know 49:16.729 --> 49:19.999 pretty well how things behave; but then, all of a sudden, 49:19.996 --> 49:21.846 there's something way out here. 49:21.849 --> 49:25.519 This would be good luck if you were long and now suddenly you 49:25.517 --> 49:29.247 got a huge return that you would not have thought was possible 49:29.246 --> 49:31.566 since you've never seen it before. 49:31.570 --> 49:35.510 But you can also have an incredibly bad return. 49:35.510 --> 49:38.620 This complicates finance because it means that you never 49:38.623 --> 49:41.423 know. You never have enough 49:41.417 --> 49:46.437 experience to get through all these things. 49:46.440 --> 49:51.350 It's a big complication in finance. 49:51.349 --> 49:53.859 My friend Nassim Talib has just written a book about it 49:53.858 --> 49:56.368 called--maybe I'll talk about that--called The Black 49:56.366 --> 49:59.996 Swan. It's about how so many plans in 49:59.999 --> 50:05.859 finance are messed up by rare events that suddenly appear out 50:05.862 --> 50:08.502 of nowhere. He called it The Black 50:08.500 --> 50:11.600 Swan because if you look at swans, they're always white. 50:11.600 --> 50:13.860 You've never seen a black swan. 50:13.860 --> 50:16.540 So, you end up going through life assuming that there are no 50:16.535 --> 50:18.125 black swans. But, in fact, 50:18.132 --> 50:21.172 there are and you might finally see one. 50:21.170 --> 50:24.130 You don't want to predicate making complicated gambles under 50:24.126 --> 50:26.076 the assumption that they don't exist. 50:26.079 --> 50:31.079 Talib, who's a Wall Street professional, 50:31.083 --> 50:38.143 talks about these black swans as being the real story of 50:38.140 --> 50:40.160 finance. Now. 50:40.159 --> 50:44.939 I want to move away from statistics and talk about 50:44.944 --> 50:49.054 present values, which is another concept in 50:49.045 --> 50:52.165 finance that is fundamental. 50:52.170 --> 50:58.890 And so, let me--And then this will conclude today's lecture. 50:58.890 --> 51:02.040 What is a present value? 51:02.040 --> 51:06.790 51:06.789 --> 51:09.819 This isn't really statistics anymore, but it's a concept that 51:09.815 --> 51:11.625 I want to include in this lecture. 51:11.630 --> 51:17.910 People in business often have claims on future money, 51:17.907 --> 51:21.417 not money today. For example, 51:21.424 --> 51:28.264 I may have someone who promises to pay me $1 in one year or in 51:28.261 --> 51:31.401 two years or three years. 51:31.400 --> 51:37.050 The present value is what that's worth today. 51:37.050 --> 51:41.270 I may have an "IOU" from someone or I may own a bond from 51:41.265 --> 51:45.555 someone that promises to pay me something in a year or two 51:45.557 --> 51:48.687 years. According to a time-honored 51:48.691 --> 51:53.191 tradition in finance, it says that it's a promise to 51:53.190 --> 51:56.630 pay $1, but it's not worth $1 today. 51:56.630 --> 51:59.930 It must be worth less than $1. 51:59.929 --> 52:05.999 What you could do hundreds of years ago--and can still do it 52:06.004 --> 52:11.674 today--was go to a bank and present this bond or IOU and 52:11.666 --> 52:15.916 say, "What will you give me for it?" 52:15.920 --> 52:24.540 The bank will discount it. 52:24.539 --> 52:27.649 Sometimes we say "present discounted value." 52:27.650 --> 52:31.240 52:31.239 --> 52:34.279 The banker will say, "Well you have $1 a year from 52:34.277 --> 52:38.117 now, but that's a year from now, so I won't give you $1 now. 52:38.119 --> 52:42.039 I'll give you the present discounted value for it." 52:42.039 --> 52:45.259 Now, I'm going to abstract from risk. 52:45.260 --> 52:48.030 Let's assume that we know that this thing is going to be paid, 52:48.033 --> 52:49.583 so it's a matter of simple time. 52:49.579 --> 52:53.189 Of course, the banker isn't going to give you $1 for 52:53.187 --> 52:57.497 something that is paying $1 in a year because the banker knows 52:57.502 --> 53:01.112 that $1 could be invested at the interest rate. 53:01.110 --> 53:12.310 Let's say the interest rate is r and that would be a 53:12.308 --> 53:17.708 number like .05, which is 5%, 53:17.714 --> 53:25.634 which is five divided by one hundred. 53:25.630 --> 53:37.440 Then the present value of $1--The present PDV or PV of $1 53:37.441 --> 53:41.661 = $1/(1+r). 53:41.659 --> 53:48.689 That's because the banker is thinking, if I have this amount 53:48.691 --> 53:56.081 right now and I invest it for one year, then what do I have. 53:56.079 --> 53:59.969 I have (1 + r)*(1/1+r). 53:59.969 --> 54:02.539 It's $1, so that works out exactly right. 54:02.539 --> 54:06.889 You have to discount something that's one period in the future 54:06.889 --> 54:09.099 by dividing it by 1+r. 54:09.099 --> 54:14.809 This is the present value of $1 in one time period, 54:14.807 --> 54:18.457 which I'm taking to be a year. 54:18.460 --> 54:20.450 It doesn't have to be a year. 54:20.449 --> 54:24.969 The interest rate has units of time, so I have to specify a 54:24.972 --> 54:29.342 time period over which I'm measuring an interest rate. 54:29.340 --> 54:31.010 Typically it's a year. 54:31.010 --> 54:39.140 If it's a one-year interest rate, the time period is one 54:39.140 --> 54:48.010 year, and the present value of $1 in one time period is given 54:48.010 --> 54:55.990 by this: the present value of $1 in n periods is 54:55.993 --> 55:04.423 1/(1+r)^(n) and that's all there is to this. 55:04.420 --> 55:10.210 55:10.210 --> 55:16.960 I want to talk about valuing streams of payments. 55:16.960 --> 55:22.320 Suppose someone has a contract that promises to pay an amount 55:22.316 --> 55:25.616 each period over a number of years. 55:25.619 --> 55:35.529 We have formulas for these present values and these 55:35.534 --> 55:40.694 formulas are well known. 55:40.690 --> 55:43.860 I'm just going to go through them rather quickly here. 55:43.860 --> 55:51.910 The simplest thing is something called a consol or perpetuity. 55:51.910 --> 55:57.040 55:57.039 --> 56:02.809 A perpetuity is an asset or a contract that pays a fixed 56:02.806 --> 56:07.416 amount of money each time period, forever. 56:07.420 --> 56:11.710 We call them consols because, in the early 1700s, 56:11.713 --> 56:16.723 the British Government issued what they called consols or 56:16.722 --> 56:22.452 consolidated debt of the British Crown that paid a certain amount 56:22.447 --> 56:26.827 of pound sterling every six months forever. 56:26.829 --> 56:29.399 You may say, what audacity for the British 56:29.395 --> 56:32.395 Government to promise to pay anything forever. 56:32.400 --> 56:33.810 Will they be around forever? 56:33.809 --> 56:37.229 Well as far as you're concerned, it's as good as 56:37.232 --> 56:39.832 forever, right? Maybe someday the 56:39.833 --> 56:43.443 British--United Kingdom--something will happen 56:43.442 --> 56:46.572 to it, it will fall apart or change; 56:46.570 --> 56:50.090 but that is so distant in the future that we can disregard 56:50.092 --> 56:52.442 that, so we'll take that as forever. 56:52.440 --> 56:55.260 Anyway, the government might buy them back too, 56:55.262 --> 56:57.412 so who cares if it isn't forever. 56:57.409 --> 56:59.339 Let's just talk about it as forever. 56:59.340 --> 57:06.640 Let's say this thing pays one pound a period forever. 57:06.639 --> 57:09.869 What is the present value of that? 57:09.869 --> 57:16.339 Well, the first--each payment we'll call a coupon--so it pays 57:16.343 --> 57:19.583 one pound one year from now. 57:19.579 --> 57:22.119 Let's say it's one year just to simplify things. 57:22.119 --> 57:26.729 It pays another pound two years from now, it pays another pound 57:26.729 --> 57:28.439 three years from now. 57:28.440 --> 57:32.350 The present value is equal to--remember it starts one year 57:32.354 --> 57:36.474 from now under assumption--we could do it differently but I'm 57:36.474 --> 57:38.264 assuming one year now. 57:38.260 --> 57:45.320 The present value is 1/(1+r) for the first 57:45.320 --> 57:51.550 year; plus for the second year it's 58:03.449 --> 58:08.019 That's an infinite series and you know how to sum that, 58:08.023 --> 58:10.763 I think. I'll tell you what it is: 58:10.762 --> 58:13.032 it's 1/r, or it would be 58:14.699 --> 58:18.719 Generally, if it pays c dollars for every period, 58:18.716 --> 58:21.706 the present value is c/r. 58:21.710 --> 58:27.450 That's the formula for the present value of a consol. 58:27.449 --> 58:29.909 That's one of the most basic formulas in finance. 58:29.909 --> 58:34.919 The interesting thing is that it means that the value of 58:34.919 --> 58:39.109 consol moves inversely to the interest rate. 58:39.110 --> 58:42.700 The British Government issued those consols in the early 1700s 58:42.699 --> 58:46.349 and, while they were refinanced in the late nineteenth century, 58:46.347 --> 58:47.757 they're still there. 58:47.760 --> 58:51.840 If you want to go out and buy one, you can get on your laptop 58:51.842 --> 58:55.042 right after this lecture and buy one of them. 58:55.039 --> 58:58.209 Then you've got something that will pay you something forever. 58:58.210 --> 59:02.090 But you're going to know that the value of that in the market 59:02.094 --> 59:04.494 moves opposite with interest rates. 59:04.489 --> 59:07.479 So, if interest rates go down, the value goes up; 59:07.480 --> 59:10.210 if interest rates go up, the value of your investment 59:10.210 --> 59:15.300 goes down. Another formula is--what if the 59:15.297 --> 59:23.487 consol doesn't pay--I'm sorry, the next thing is a growing 59:23.492 --> 59:24.932 consol. 59:24.930 --> 59:28.080 59:28.079 --> 59:31.599 I'm calling it a growing consol even though the British consols 59:31.596 --> 59:33.816 didn't grow. Let's say that the British 59:33.815 --> 59:37.045 Government didn't say that they'll pay one pound per year, 59:37.050 --> 59:44.000 but it'll be one pound the first year, then it will grow at 59:44.000 --> 59:51.550 the rate g and it will eventually be infinitely large. 59:51.550 --> 1:00:02.370 You get one pound the first year, you get 1+g pounds 1:00:09.838 --> 1:00:15.248 the third year and so on. 1:00:15.250 --> 1:00:19.540 The present value of this--suppose it pays--let's say 1:00:19.539 --> 1:00:24.239 it pays c pounds each year, so it would be c 1:00:24.242 --> 1:00:27.802 times this. It would be c times 1:00:31.008 --> 1:00:36.268 etc., Then the present value is equal to c/(r- 1:00:36.268 --> 1:00:41.168 g)--that's the formula for the value of a growing 1:00:41.172 --> 1:00:44.372 console. g has to be less than 1:00:44.371 --> 1:00:48.291 r for this to make sense because if g--if it's 1:00:48.294 --> 1:00:51.224 growing faster than the rate of interest, 1:00:51.219 --> 1:00:55.869 then this infinite series will not converge and the value would 1:00:55.869 --> 1:00:57.829 be infinite. You might ask, 1:00:57.827 --> 1:01:00.287 "Well then how does that make sense?" 1:01:00.289 --> 1:01:04.959 What if the British Government promised to pay 10% more each 1:01:04.962 --> 1:01:08.132 year, how would the market value that? 1:01:08.130 --> 1:01:10.570 The formula doesn't have a number. 1:01:10.570 --> 1:01:13.330 I'll tell you why it doesn't have a number: 1:01:13.331 --> 1:01:16.951 the British Government will never promise to pay you 10% 1:01:16.947 --> 1:01:19.837 more each year because they can't do it. 1:01:19.840 --> 1:01:22.960 And, the market wouldn't believe them because you can't 1:01:22.958 --> 1:01:25.728 grow every year faster than the interest rate. 1:01:25.730 --> 1:01:28.190 Now that's one of the most basic lessons, 1:01:28.185 --> 1:01:29.285 you can't do it. 1:01:29.290 --> 1:01:33.940 1:01:33.940 --> 1:01:37.120 One more thing that I think would be relevant to 1:01:37.116 --> 1:01:39.816 the--there's also the annuity formula. 1:01:39.820 --> 1:01:48.400 This is a formula that applies to--what if an asset pays a 1:01:48.398 --> 1:01:54.868 fixed amount every period and then stops? 1:01:54.870 --> 1:01:56.680 That's called an annuity. 1:01:56.679 --> 1:02:09.329 An annuity pays c dollars starting in t = 1:02:09.328 --> 1:02:22.678 1,2, 3, and n is the last period, then it stops. 1:02:22.679 --> 1:02:27.509 A good example of an annuity is a mortgage on a house. 1:02:27.510 --> 1:02:30.690 When you buy a house, you borrow the money and you 1:02:30.687 --> 1:02:33.867 pay it back in fixed--it would usually be monthly, 1:02:33.865 --> 1:02:36.065 but let's say annual payments. 1:02:36.070 --> 1:02:40.840 You pay every year a fixed amount on your house to the 1:02:40.844 --> 1:02:45.984 mortgage originator and then after so many--n is 30 1:02:45.979 --> 1:02:51.249 years, typically--you would then have 1:02:51.254 --> 1:02:54.874 paid it off. It used to be that mortgages 1:02:54.874 --> 1:02:57.834 had what's called a balloon payment at the end. 1:02:57.829 --> 1:03:00.369 This means that you would have to pay extra money at the end; 1:03:00.369 --> 1:03:02.979 but they decided that people have trouble doing that. 1:03:02.980 --> 1:03:06.080 It's much better to pay a fixed payment and then you're done. 1:03:06.079 --> 1:03:08.359 Otherwise, if you ask them to pay more at the end, 1:03:08.361 --> 1:03:10.411 then a lot of people won't have the money. 1:03:10.410 --> 1:03:13.840 We now have annuity mortgages. 1:03:13.840 --> 1:03:16.800 What is the present value of an annuity? 1:03:16.800 --> 1:03:27.030 That is, the present value of an annuity is equal to the 1:03:27.026 --> 1:03:34.836 amount--what did I say--c*{1 – 1:03:34.836 --> 1:03:41.526 [1/(1+r)]^(n) }/r. 1:03:41.530 --> 1:03:46.060 That is the present value of an annuity. 1:03:46.059 --> 1:03:50.919 I wanted to say one more thing because I realize that you have 1:03:50.916 --> 1:03:55.926 to--your first problem set will cover this--is to talk about the 1:03:55.932 --> 1:04:00.472 concept that applies probability theory to Economics. 1:04:00.470 --> 1:04:04.560 That is expected utility theory. 1:04:04.560 --> 1:04:07.020 Then I'll conclude with this. 1:04:07.019 --> 1:04:13.679 In Economics, it is assumed that people have 1:04:13.677 --> 1:04:21.417 a utility function, which represents how happy they 1:04:21.419 --> 1:04:30.709 are with an outcome--we typically take that as U, 1:04:30.710 --> 1:04:36.850 If I have a monetary outcome, then I have a certain amount of 1:04:36.849 --> 1:04:39.509 money, x dollars. 1:04:39.510 --> 1:04:46.210 How happy I am with x dollars is called U(x). 1:04:46.210 --> 1:04:53.600 This, I think you've gotten from other economics courses--we have 1:04:53.603 --> 1:04:59.383 something called diminishing marginal utility. 1:04:59.380 --> 1:05:07.460 1:05:07.460 --> 1:05:13.930 The idea is that for any amount of money--if this x is 1:05:13.931 --> 1:05:20.191 the amount of money that I receive--utility as the function 1:05:20.186 --> 1:05:26.546 of the amount of money I receive is downwardly-concave. 1:05:26.550 --> 1:05:30.640 The exact shape of the curve is subject to discussion, 1:05:30.639 --> 1:05:34.959 but the point of diminishing marginal utility is that, 1:05:34.960 --> 1:05:39.520 as you get more and more money, the increment in utility for 1:05:39.522 --> 1:05:41.922 each extra dollar diminishes. 1:05:41.920 --> 1:05:45.800 Usually we say it never goes down, we don't have it going 1:05:45.797 --> 1:05:47.387 down, cross that out. 1:05:47.389 --> 1:05:50.999 That would be where having more money makes you less happy. 1:05:51.000 --> 1:05:54.210 That may actually work that way, but our theory says no, 1:05:54.208 --> 1:05:55.548 you always want more. 1:05:55.550 --> 1:05:59.480 It's always upward sloping, but it may, after awhile, 1:05:59.482 --> 1:06:03.492 you get close to satiation where you've got enough. 1:06:03.489 --> 1:06:06.829 Incidentally, I mentioned this last time--I 1:06:06.828 --> 1:06:11.198 was talking about--I was philosophizing about wealth and 1:06:11.200 --> 1:06:15.970 I asked what are you going to do with a billion dollars. 1:06:15.969 --> 1:06:19.359 We have many billionaires in this country and I think that 1:06:19.359 --> 1:06:22.689 the only thing you have to do with it is philanthropy. 1:06:22.690 --> 1:06:25.960 They have to give it away because they are essentially 1:06:25.961 --> 1:06:27.571 satiated. Because, as I said, 1:06:27.569 --> 1:06:30.509 you can only drive one car at a time and if you've got ten of 1:06:30.506 --> 1:06:32.856 them in the garage, then it doesn't really do you 1:06:32.855 --> 1:06:34.835 much good. You can't do it; 1:06:34.840 --> 1:06:36.790 you can't enjoy all ten of them. 1:06:36.789 --> 1:06:41.019 It's important--that's one reason why we want policies that 1:06:41.019 --> 1:06:44.299 encourage equality of incomes--not necessarily 1:06:44.301 --> 1:06:46.161 equality, but reasonable 1:06:46.164 --> 1:06:50.094 equality--because the people with very low wealth have a very 1:06:50.085 --> 1:06:54.195 high marginal utility of income and people with very high wealth 1:06:54.202 --> 1:06:55.642 have very little. 1:06:55.639 --> 1:06:59.789 So, if you take from the rich and give to the poor you make 1:06:59.792 --> 1:07:03.872 people happier. We're not going to do that in a 1:07:03.872 --> 1:07:06.892 Robin Hood way; but in finance we're going to 1:07:06.888 --> 1:07:09.698 do that in a systematic way through risk management. 1:07:09.699 --> 1:07:13.709 We're going to be taking away from lucky--you think of 1:07:13.705 --> 1:07:17.025 yourself as randomly on any point of this. 1:07:17.030 --> 1:07:20.270 You don't want--you know that you'd like to take money away 1:07:20.274 --> 1:07:23.074 from yourself in the high-outcome years and give it 1:07:23.072 --> 1:07:25.312 to yourself in the low-income years. 1:07:25.309 --> 1:07:30.959 What finance theory is based on--and much of economics is 1:07:30.960 --> 1:07:36.910 based on--the idea that people want to maximize the expected 1:07:36.914 --> 1:07:39.744 utility of their wealth. 1:07:39.739 --> 1:07:43.869 Since this is a concave function, it's not just the 1:07:43.874 --> 1:07:47.054 expected value. To calculate the expected 1:07:47.049 --> 1:07:50.799 utility of your wealth, you might also have to look at 1:07:50.801 --> 1:07:54.771 the expected return, or the geometric expected 1:07:54.771 --> 1:07:58.051 return, or the standard deviation. 1:07:58.050 --> 1:07:59.680 Or you might have to look at the fat tail. 1:07:59.679 --> 1:08:03.189 There are so many different aspects that we can get into and 1:08:03.188 --> 1:08:06.458 this underlying theory motivates a lot of what we do. 1:08:06.460 --> 1:08:11.910 But it's not a complete theory until we specify the utility 1:08:11.906 --> 1:08:14.476 function. Of course, we will also be 1:08:14.475 --> 1:08:17.955 talking about behavioral finance in this course and we'll, 1:08:17.960 --> 1:08:21.620 at times, be saying that the utility function concept isn't 1:08:21.624 --> 1:08:25.224 always right--the idea that people are actually maximizing 1:08:25.224 --> 1:08:28.514 expected utility might not be entirely accurate. 1:08:28.510 --> 1:08:31.610 But, in terms of the basic theory, that's the core concept. 1:08:31.609 --> 1:08:35.829 I have one question on the problem set that asks you to 1:08:35.830 --> 1:08:40.910 think about how you would handle a decision: whether to gamble, 1:08:40.909 --> 1:08:45.679 based on efficient--based on expected utility theory. 1:08:45.680 --> 1:08:50.270 That's a little bit of a tricky question but--So, 1:08:50.270 --> 1:08:55.910 do the best you can on it and think--try to think about what 1:08:55.913 --> 1:09:01.463 this kind of theory would imply for gambling behavior. 1:09:01.460 --> 1:09:04.580 I will see you on Friday. 1:09:04.579 --> 1:09:06.999 That's two days from now in this room.