1 00:00:01,017 --> 00:00:05,537 Alright. So, we're going to talk about random variables and probability review. 2 00:00:05,537 --> 00:00:10,618 And so, again, this is a course about modeling financial data. And the primary 3 00:00:10,618 --> 00:00:15,023 thing that we're going to be looking at are asset returns. And, and when you think 4 00:00:15,023 --> 00:00:19,837 about an asset return, You know, we went over return calculations before, you know? 5 00:00:19,837 --> 00:00:23,788 Say, we're investing in Microsoft stock, we buy it today, we hold it for a month, 6 00:00:23,788 --> 00:00:28,316 one month from now, we sell it, we can calculate the percentage change in price 7 00:00:28,316 --> 00:00:32,374 that's the rate of return. But from the point of view of today, the rate of return 8 00:00:32,374 --> 00:00:36,406 on this one-month investment is not known, because we need to know what the price 9 00:00:36,406 --> 00:00:41,016 next month is in order to be able to calculate that return. So, we can think of 10 00:00:41,016 --> 00:00:46,372 the rate of a return as a random variable because the future is not known and the 11 00:00:46,372 --> 00:00:52,282 return depends upon the future price, which is not known. And so you know, the 12 00:00:52,282 --> 00:00:57,598 outcome is uncertain. And so we can characterize returns as a random variable, 13 00:00:57,598 --> 00:01:02,832 which is a variable that can take on a, a possible set of values called the sample 14 00:01:02,832 --> 00:01:07,797 space. So, the future price could go up, it could go down. And so, there is a 15 00:01:07,797 --> 00:01:13,079 potential list of values of the future price and then, we attach a probability to 16 00:01:13,275 --> 00:01:18,563 each of those prices and that gives us the probability distribution over those 17 00:01:18,563 --> 00:01:23,905 potential values. So, what we are going to review today is various mathematical ways 18 00:01:23,905 --> 00:01:29,136 of describing random variables and distributions and with several examples 19 00:01:29,136 --> 00:01:33,685 towards thinking about asset returns and the properties of probability 20 00:01:33,685 --> 00:01:40,255 distributions for asset returns that look reasonable. So some examples typically, an 21 00:01:40,255 --> 00:01:44,852 uppercase letter denotes a random variable. So, uppercase X is, in this 22 00:01:44,852 --> 00:01:50,096 case, the price of Microsoft stock next month. It's a random variable cuz we don't 23 00:01:50,096 --> 00:01:54,436 know what the future price is going to be. What's the sample space of the future 24 00:01:54,436 --> 00:01:58,362 prices? Well, the sample space, this is the possible values that we can think, 25 00:01:58,362 --> 00:02:03,441 that we think the prices can take on. Now, prices can't become negative so we know 26 00:02:03,441 --> 00:02:07,858 that the sample space is going to be t hose say positive values. And you know, if 27 00:02:07,858 --> 00:02:12,841 the stock is trading, if the stock doesn't go out of business, then the price should 28 00:02:12,841 --> 00:02:18,053 be positive. Now, prices can't go up to infinity, so there's some realistic upper 29 00:02:18,053 --> 00:02:23,779 bound on prices and that upper bound might be a $1,000 or something like that. So, we 30 00:02:23,779 --> 00:02:29,399 can think future prices can lie anywhere, any real number, say, between zero and 31 00:02:29,399 --> 00:02:35,351 some big number M like a 1,000. So, that would be a, a characterization of a random 32 00:02:35,351 --> 00:02:40,545 variable, the future price and its sample space. Another random variable could be 33 00:02:40,545 --> 00:02:45,369 the rate of return on the investment and the rate of return, which is the 34 00:02:45,369 --> 00:02:50,866 percentage change in price. Now what is this appropriate example space for a rate 35 00:02:50,866 --> 00:02:56,077 of return. Now, what's the smallest value that a rate of return can take on? The 36 00:02:56,077 --> 00:03:02,414 worst you can do is lose all your money, right? So, if you, you buy Microsoft today 37 00:03:02,414 --> 00:03:07,644 for 30, and if Microsoft goes bankrupt, and its price goes to zero in the future, 38 00:03:07,644 --> 00:03:12,078 then the percentage change in price is minus a 100%. So, the returns are bounded 39 00:03:12,078 --> 00:03:17,088 from below by -one, so it can't be any more than that. And the upper bound, 40 00:03:17,088 --> 00:03:22,454 again, is some big positive number. You know, say, Microsoft, you know, I don't 41 00:03:22,454 --> 00:03:27,017 know, in, invent some product that revolutionizes the world and its price is 42 00:03:27,017 --> 00:03:30,512 going to shoot up. And but, you know, it's not going to go off to infinity. So, 43 00:03:30,512 --> 00:03:35,834 there's some reasonable upper bound associated with that. Another random 44 00:03:35,834 --> 00:03:41,740 variable we can think of, and that's used a lot in probability modelling in finance, 45 00:03:41,740 --> 00:03:46,559 is a just a, a discrete random variable that just takes on two values. So, we're 46 00:03:46,559 --> 00:03:51,465 going to set x to be equal to one if the stock price goes up and we'll say, x is 47 00:03:51,465 --> 00:03:55,479 equal to zero if the stock price goes down. This is sort of like a coin flipping 48 00:03:55,479 --> 00:04:00,261 example. You flip the coin and it lands heads, that's like the stock prices going 49 00:04:00,261 --> 00:04:04,445 up. You flip the coin, it lands tails, it's like the stock prices going down, and 50 00:04:04,445 --> 00:04:09,031 then we're just coding the random variable to be 1,0 based upon those events. And 51 00:04:09,031 --> 00:04:12,620 here, the sample space is very simple with just, just two va lues zero and one. So, 52 00:04:12,620 --> 00:04:17,081 those are examples of, of random variables that we'll be looking at. So, once we have 53 00:04:17,081 --> 00:04:21,085 these random variables, we need to characterize their probability 54 00:04:21,085 --> 00:04:26,488 distribution. Now we in, in probability models, we usually distinguish between 55 00:04:26,488 --> 00:04:31,014 what are call discrete random variables. And the discrete random variables is a 56 00:04:31,014 --> 00:04:35,700 random variables that can only take on finite set of values. So, in the last 57 00:04:35,700 --> 00:04:40,737 example, the up-down indicator, we took on two values, zero and one, that's the 58 00:04:40,737 --> 00:04:45,954 discrete random variables and its sample space is two discrete points, okay? The 59 00:04:45,954 --> 00:04:51,609 probability distribution of a discrete random variable is a function, say, P(x). 60 00:04:51,609 --> 00:04:58,481 Such that P(x) is the probability that the random variable is equal to little x. So 61 00:04:58,481 --> 00:05:04,525 typically, in notation capital letter denotes the random variable and a lower 62 00:05:04,525 --> 00:05:10,668 case letter denotes a value in the sample space that the random variable can take 63 00:05:10,668 --> 00:05:16,819 on. Now, this probability function must satisfy certain conditions in order to be 64 00:05:16,819 --> 00:05:21,685 a valid probability function. So, probabilities are greater and equal to 65 00:05:21,685 --> 00:05:27,795 zero for all values in the sample space. Probabilities are equal to zero for values 66 00:05:27,795 --> 00:05:33,961 outside of the samples space. You know, we assume that all the values in the sample 67 00:05:33,961 --> 00:05:38,306 space, you know, again are, are discrete, distinct points. The sum of the 68 00:05:38,306 --> 00:05:42,635 probabilities of the values in the sample space is equal to a 100%. And 69 00:05:42,822 --> 00:05:48,730 probabilities have to be less than one, as well. So, as an example of a simple 70 00:05:48,730 --> 00:05:56,804 discrete random variable and a probability distribution, here's a case where we can 71 00:05:56,804 --> 00:06:09,692 have a random variable x here and X is going to denote the annual rate of return 72 00:06:09,692 --> 00:06:19,340 on Microsoft stock. Alright. So, we have a random variable X that's going to 73 00:06:19,340 --> 00:06:24,310 represent the annual rate of return on Microsoft stock. And this is an example 74 00:06:24,310 --> 00:06:28,957 that's kind of like the case where you might have a stock analyst who's working 75 00:06:28,957 --> 00:06:33,814 for an investment bank, and they're doing some fundamental an analysis and they need 76 00:06:33,978 --> 00:06:38,255 you know, make some forecast of what the rate of the return might be over the next 77 00:06:38,255 --> 00:06:42,837 year. And the analysis might have a very simplified way of viewing the world that 78 00:06:43,004 --> 00:06:47,797 you know, over the next year the price of Microsoft stock is primarily contingent 79 00:06:47,797 --> 00:06:51,927 upon what's going to happen to the state of the economy. And so, the analyst says, 80 00:06:51,927 --> 00:06:56,859 well, I think there are really one, two, three, four, five potential states of the 81 00:06:56,859 --> 00:07:02,431 economy, you know, depression, you know, it's a very bad state. And if the economy 82 00:07:02,431 --> 00:07:07,163 is in a depression, then Microsoft is going to lose 30%. And the analyst puts a 83 00:07:07,163 --> 00:07:11,966 probability of five% on that event occurring. And then, the other state of 84 00:07:11,966 --> 00:07:17,399 the world could be, say, a recession. And if a recession happens, then Microsoft has 85 00:07:17,399 --> 00:07:22,682 an annual return of zero%, and the analyst puts a probability of twenty% on that. And 86 00:07:22,682 --> 00:07:27,068 then similarly, normal mild boom, major booms, these are other states of the 87 00:07:27,068 --> 00:07:32,030 world, and then we see as the economy gets better, the rate of return goes up, and, 88 00:07:32,190 --> 00:07:36,881 and then we have these probabilities. So notice that the, the normal state of the 89 00:07:36,881 --> 00:07:42,049 world is the state that gets the highest probability associated with it. Now this 90 00:07:42,049 --> 00:07:47,791 is a valid probability distribution, all the probabilities are between zero and one 91 00:07:47,791 --> 00:07:53,897 and the sum of all the probabilities add up to one. Now, in this case, you know, 92 00:07:53,897 --> 00:07:58,059 where do these probabilities come from? You now, in, in this case, the 93 00:07:58,059 --> 00:08:04,018 probabilities are the subjective beliefs of the analyst. They may have nothing to 94 00:08:04,018 --> 00:08:08,090 do with the real world, you know, in quotations. But there are just, you know, 95 00:08:08,090 --> 00:08:12,610 views or opinions associated with, with the analyst. In, in probability theory, 96 00:08:12,610 --> 00:08:17,523 there are generally two types of ways that we view probability. One way, the 97 00:08:17,523 --> 00:08:22,788 subjective approach is probabilities are opinions or degrees of belief. And this is 98 00:08:22,788 --> 00:08:28,439 associated with what is often referred to as Basian statistics. The other approach 99 00:08:28,439 --> 00:08:33,876 views probabilities of, actually, actually, real physical objective things . 100 00:08:33,876 --> 00:08:39,706 So, the, the, the objective view of probability. Think of the coin flipping 101 00:08:39,706 --> 00:08:45,060 example. So, if you have a fair coin that is, it's weighted such that the 102 00:08:45,060 --> 00:08:50,701 probability that the coin lands heads or tails is 50%. So, where doe s that 103 00:08:50,701 --> 00:08:56,354 probability of 50% come from? Well, it's an objective feature of the coin, and the 104 00:08:56,354 --> 00:09:01,459 experiment of flipping the coin and the idea is that, you know, when you flip a 105 00:09:01,459 --> 00:09:06,530 fair coin, you know, and you, you do this in an experimental setting, say, a million 106 00:09:06,530 --> 00:09:11,502 times, the probability that the coin lands heads is a fraction of the times in your 107 00:09:11,502 --> 00:09:16,237 experiment that you actually observe the coin landing heads. And the probability 108 00:09:16,237 --> 00:09:21,033 that lands tails is the fraction of times you actually observe the coin landing 109 00:09:21,033 --> 00:09:25,480 tails. So there, you think of probability as being a property of the coin, the 110 00:09:25,480 --> 00:09:29,641 experimental setting, and something that you can repeat over and over again and, 111 00:09:29,641 --> 00:09:34,054 and reproduce. Whereas, the opinion approach is, there's no experiment going 112 00:09:34,054 --> 00:09:39,049 on here. You can't repeat, you know, the idea of the views going over and over 113 00:09:39,049 --> 00:09:44,283 again. It's just a degree of belief. Both views of probability theory are, are, are 114 00:09:44,283 --> 00:09:48,213 equally valid, and but, you know, within the statistics literature, you know, 115 00:09:48,213 --> 00:09:53,575 there's often very strong opinions about what is the right way or the superior way 116 00:09:53,575 --> 00:09:58,998 to view probability. In this class, we won't take a, make a judgement on that, 117 00:09:58,998 --> 00:10:06,733 but we'll, you know, essentially just take probability as given and do computations. 118 00:10:06,733 --> 00:10:15,663 So one of the things I want to, to show you is an Excel spreadsheet that I have 119 00:10:15,663 --> 00:10:26,063 and now, Excel is not necessarily a good tool for doing probability calculations. 120 00:10:26,063 --> 00:10:32,065 And so, the point here is just to illustrate how to do certain computations 121 00:10:32,065 --> 00:10:37,833 in Excel and, and, and show some, and show graphics and, and, and things like that. R 122 00:10:37,833 --> 00:10:43,153 is a much better environment for doing a probability calculations and, and so on. 123 00:10:43,153 --> 00:10:49,037 But in, for my case, you know, I have my example I just put out an Excel 124 00:10:49,037 --> 00:10:54,423 spreadsheet, I have my returns, I have my probabilities, and then I can do a nice 125 00:10:54,423 --> 00:10:59,059 simple bar chart to represent the probability distribution. So, in the 126 00:10:59,059 --> 00:11:03,076 graphical representation of the distribution, I put the values in the 127 00:11:03,076 --> 00:11:08,046 sample space on this axis. And then, the heights of the bars just represent the 128 00:11:08,046 --> 00:11:13,023 probabilities. And so, from this graphic al representation, we can clearly see the 129 00:11:13,023 --> 00:11:17,893 most likely value is around one%. The shape of the distribution is symmetric and 130 00:11:17,893 --> 00:11:22,294 that this is the middle of the distribution. The shape to the left of the 131 00:11:22,294 --> 00:11:27,712 middle, and the shape to the right of the middle is the same and, and so on. We can 132 00:11:27,712 --> 00:11:34,814 do the same computation in, in R. So, here is my this is new, this is again, just 133 00:11:34,814 --> 00:11:41,405 based on the examples in my lecture notes. So, if I wanted to plot the probability 134 00:11:41,405 --> 00:11:49,229 distribution in R I would just create a vector of values that represent the values 135 00:11:49,229 --> 00:11:58,229 in the sample space, create a vector of probabilities, and then do a bar plot. And 136 00:11:58,698 --> 00:12:09,074 that gives me the, the values here. Okay. A more mathematical model for a discreet 137 00:12:09,074 --> 00:12:15,016 random variable is based on what's called the Bernoulli distribution And the 138 00:12:15,016 --> 00:12:20,142 Bernoulli distribution is a probability model that essentially describes the 139 00:12:20,142 --> 00:12:26,006 coin-flipping experiment. So, we have two mutually exclusive events generically 140 00:12:26,006 --> 00:12:31,522 called success and failure, right? So, in modeling stock prices, a success event is 141 00:12:31,522 --> 00:12:36,554 the stock price goes up, a fail event is the stock price goes down. That's assuming 142 00:12:36,554 --> 00:12:41,464 you have a long position in, in, in, in the asset. If you have a short position, 143 00:12:41,464 --> 00:12:46,754 then stock price going up is the failure, and the stock price going down is the 144 00:12:46,754 --> 00:12:51,615 success. What, so, in investments, a long position means you buy something today. 145 00:12:51,615 --> 00:12:56,564 You hold it, you sell it in the future. A short position means you sell it today, 146 00:12:56,564 --> 00:13:01,153 you, and then you buy it back in the future. Okay, and so, when your long 147 00:13:01,153 --> 00:13:05,871 something, you're hoping the price will go up. When you're short something, you hope 148 00:13:05,871 --> 00:13:10,569 the price is going to go down. Typically shorting works, I just, just as an aside, 149 00:13:10,569 --> 00:13:15,891 because this is going to come up lat er on. When you short a stock, typically, you 150 00:13:15,891 --> 00:13:20,921 open a brokerage account in, you know, like at Fidelity or E-trade or something 151 00:13:20,921 --> 00:13:25,156 like that. Say, you want to short Microsoft. Well, you are going to sell 152 00:13:25,156 --> 00:13:29,330 something you don't own. So, how do you do that? Well, if you have a brokerage 153 00:13:29,330 --> 00:13:33,498 account, you can borrow the stock from somebody who owns it and you borrow it, 154 00:13:33,498 --> 00:13:37,564 and you sell it, you get the proceeds that you hold on to it. But because you 155 00:13:37,564 --> 00:13:42,127 borrowed it, you have to give it back at some point. So, when you close out the 156 00:13:42,127 --> 00:13:46,608 short position, you go back in the market, you buy it back, and then you return the 157 00:13:46,608 --> 00:13:51,293 stock to who you borrowed it from. So, you want to think of that transaction is 158 00:13:51,293 --> 00:13:57,670 taking place when you do a short sale. Alright. So we're going to calling this in 159 00:13:57,670 --> 00:14:03,379 this, go back to the Bernoulli example. Let's say X = one, if a success occurs and 160 00:14:03,379 --> 00:14:08,075 X = zero, if a failure occurs, okay? So, that's, that's coin lands head, success, 161 00:14:08,075 --> 00:14:13,809 coin lands tail failure. Now, the probability that we have a success, that X 162 00:14:13,809 --> 00:14:18,537 = one is were, is been equal to pi and pi is some number between zero and one. And 163 00:14:18,537 --> 00:14:23,963 then the probability of a failure that X = zero is then one - pi. Alright, cuz we 164 00:14:23,963 --> 00:14:28,609 have two events. The sum of the pi will always have to add to one, so pi + one - 165 00:14:28,609 --> 00:14:36,994 pi = one. Now, a simple mathematical model for this probability distribution are 166 00:14:36,994 --> 00:14:44,008 P(x), we can write as pi^x one - pi^1 - x where x only takes two values, zero and 167 00:14:44,008 --> 00:14:49,185 one. So, this P(x) function gives us our probabilities. Notice that when X is zero, 168 00:14:49,185 --> 00:14:57,298 P of zero is pi^0 one - pi^1 - zero. So, that's one - pi. So then, we have pi^0 is 169 00:14:57,298 --> 00:15:05,918 one. One - pi^1 is one - pi. And then, when X = one, my P of one is pi^1 one - 170 00:15:05,918 --> 00:15:10,005 pi^1 - one so that's equal to -pi. So, this very simple mathematical 171 00:15:10,005 --> 00:15:15,664 representation gives us our probability function for the Bernoulli distribution. 172 00:15:15,664 --> 00:15:24,512 Now, the other type of random variables that we work at, look at, are called 173 00:15:24,512 --> 00:15:29,489 continuous random variables. A continuous random variable is one, is a random 174 00:15:29,489 --> 00:15:34,725 variable that can take on any real value, alright? And so now, we talk about the 175 00:15:34,725 --> 00:15:39,956 probab ility density function of a continuous random variable. We're going to 176 00:15:39,956 --> 00:15:44,924 have a probability function that we're going to denote as f(x) to distinguish it 177 00:15:44,924 --> 00:15:49,536 from P(x) for the discrete random variable. And this f(x) represents what's 178 00:15:49,536 --> 00:15:55,460 often referred to as the probability curve, okay? Now, the probability curve 179 00:15:55,460 --> 00:16:01,642 satisfies such that, if A is any inte rval on the real line, the probability that the 180 00:16:01,642 --> 00:16:07,064 continuous random variable is in this interval is equal to the interval of the 181 00:16:07,064 --> 00:16:13,076 probability curve over that interval. So, in other words, the probability that X is, 182 00:16:13,076 --> 00:16:20,001 is in this interval, is the area under the probability curve over this particular 183 00:16:20,001 --> 00:16:26,004 interval. So, we have a continuous random variable, we have a probability curve, and 184 00:16:26,004 --> 00:16:31,077 probabilities are associated with areas under the curve. Now, this probability 185 00:16:31,077 --> 00:16:35,951 curve must satisfy, it always, is always positive cuz we want to compute areas 186 00:16:35,951 --> 00:16:41,553 under a curve. And the total area under the probability curve is equal to a 100%. 187 00:16:41,553 --> 00:16:48,491 So, that's the ideas that the, all the probabilities add to one. So, if we think 188 00:16:48,491 --> 00:16:54,052 of a probability curve here, let's say, X is continuous random variable, here, its 189 00:16:54,052 --> 00:16:59,261 the probability curve, and I've witnessed, suggestively like a, a bell shape curve 190 00:16:59,261 --> 00:17:04,636 like a normal distribution which we will talk about later. And we want to say, what 191 00:17:04,636 --> 00:17:09,660 is the probability that this random variables between -two and one. So, our 192 00:17:09,660 --> 00:17:15,356 intervals between -two and one and the probability of this event is equal to the 193 00:17:15,356 --> 00:17:20,862 area under the probability curve over this interval. So, we see that one of the 194 00:17:20,862 --> 00:17:25,698 reasons why probability theory with Calculus is useful because in order to 195 00:17:25,698 --> 00:17:30,581 calculate probabilities, we have to find area under a curve. In order to find area 196 00:17:30,581 --> 00:17:35,544 under the curve, we have to integrate the probability function. So, when you take a 197 00:17:35,544 --> 00:17:40,228 more mathematically-oriented probability theory course, you do a lot of C`alculus 198 00:17:40,228 --> 00:17:44,910 to do these types of calculations. In this class, we are not going to do the 199 00:17:44,910 --> 00:17:49,496 Calculus, calculations, right? We need to know the concept of doing this and then 200 00:17:49,496 --> 00:17:53,651 if, even if you have to integrate something, we can do it in r numerically. 201 00:17:53,651 --> 00:17:58,085 So, we have the function, one in front of the area under the curve, we can write a 202 00:17:58,085 --> 00:18:03,504 function in r to represent the probability curve and then, we can use the function 203 00:18:03,504 --> 00:18:08,506 called Integrate to numerically calculate the area under the curve for us. So, we're 204 00:18:08,506 --> 00:18:13,726 going to be using tools that will do Calculus for us and, but we need to know 205 00:18:13,726 --> 00:18:21,144 the concept of, of what's going on, alright. A very simple example of a 206 00:18:21,144 --> 00:18:27,590 continuous random variable in a distribution is, is the so-called uniform 207 00:18:27,590 --> 00:18:34,510 distribution over the interval ab. So we say, x is distributed uniform. So, this is 208 00:18:34,510 --> 00:18:40,903 a bit of notation in a probability theory. X represents a random variable. This 209 00:18:40,903 --> 00:18:47,622 little squiggle character, character represent, is to be read as is distributed 210 00:18:47,622 --> 00:18:54,000 a. So, x is distributed as uniform over the interval ab, okay? The probability 211 00:18:54,000 --> 00:19:01,624 curve of the uniform distribution is a rectangle. So, if we want the think about 212 00:19:01,624 --> 00:19:10,459 the uniform distribution, we have some interval a to b, and we know that the area 213 00:19:10,459 --> 00:19:18,006 under the probability curve has to equal a 100%. And the idea of a uniform 214 00:19:18,006 --> 00:19:23,634 distribution is that, you know, probability over any interval of the same 215 00:19:23,634 --> 00:19:28,347 length is the same, okay? So, it's a, it's a way of thinking of like, equal 216 00:19:28,347 --> 00:19:34,054 probability for events of the same size, so to say. So, if this total area has to 217 00:19:34,054 --> 00:19:39,946 be 100%, then we know that length times width is equal to one, so the height of 218 00:19:39,946 --> 00:19:49,219 the probability curve is one / b - a. So, that represents the probability curve for 219 00:19:49,219 --> 00:19:56,795 a uniform end variable. And we know that this probability curve is greater or equal 220 00:19:56,795 --> 00:20:03,001 to zero provided the, the right end point is bigger than the left end point. And we 221 00:20:03,001 --> 00:20:08,071 know that the total area under this curve, length times width, if we do the 222 00:20:08,071 --> 00:20:11,001 integration it's equal to one.