1 00:00:00,000 --> 00:00:05,472 So, we'll look at data now finally. And before we actually start looking at 2 00:00:05,472 --> 00:00:10,011 data, remind yourself a covariance stationary. 3 00:00:10,011 --> 00:00:14,111 So, we're going to look at data. And, when we start looking at asset return 4 00:00:14,111 --> 00:00:19,656 data, you want to think yourself that the asset returns that we observe are 5 00:00:19,656 --> 00:00:23,087 realization of a covariance stationary time series, okay? 6 00:00:23,087 --> 00:00:29,483 And so, you think of Gaussian White Noise as the benchmark and, and then you want to 7 00:00:29,483 --> 00:00:34,082 think with a critical eye of may, of, what a moving average process looks like, so 8 00:00:34,082 --> 00:00:39,154 time dependence of, you know, one period, or an autoregressive process where you 9 00:00:39,154 --> 00:00:43,470 have decaying time dependence. And you want to ask yourself, you know, 10 00:00:43,470 --> 00:00:47,192 you know, what properties do you actually see in the data? 11 00:00:47,401 --> 00:00:52,928 And if the process were covariance stationary process, what kind of process 12 00:00:52,928 --> 00:00:55,014 would it most likely be? Okay? 13 00:00:55,014 --> 00:00:58,061 So, that's the kind of game that we want to play. 14 00:00:58,061 --> 00:01:02,088 So, all of the returns have the same mean, the same variance. 15 00:01:02,088 --> 00:01:08,081 They have a correlation that depends upon how far apart they are, but on time, okay? 16 00:01:08,081 --> 00:01:12,050 There should be no trends in the data, and, and so on. 17 00:01:12,050 --> 00:01:16,045 So, that's the process. So, now let's look at some data. 18 00:01:20,064 --> 00:01:27,619 So, for this next set of lectures, I'm going to be looking well, two time series. 19 00:01:27,863 --> 00:01:34,055 I have monthly continuously compounded returns that I downloaded from Yahoo and 20 00:01:34,275 --> 00:01:40,482 one series is going to be the monthly returns on Microsoft, which is just an 21 00:01:40,482 --> 00:01:44,849 individual asset. And the other series we're going to look 22 00:01:44,849 --> 00:01:49,584 at is the, the S and P 500 index, the Standard and Poor's 500. 23 00:01:49,584 --> 00:01:55,131 The S and P 500 think of this as a portfolio of 500 assets, okay? 24 00:01:55,131 --> 00:02:01,025 So, it's a, it has, it's a portfolio of, of 500 stocks, and the weight of each 25 00:02:01,025 --> 00:02:06,071 asset in the portfolio is based on its market capitalization. 26 00:02:06,071 --> 00:02:10,880 So, for example, if Microsoft were in the S and P 500, the weight of Microsoft is 27 00:02:10,880 --> 00:02:15,011 equal to the, the market value of Microsoft which is the price of Microsoft 28 00:02:15,011 --> 00:02:19,448 times the number of shares outstanding, that's its market capitalization. 29 00:02:19,448 --> 00:02:24,241 So, if you had to go and, if you wanted to buy Microsoft as a company, what price 30 00:02:24,241 --> 00:02:27,551 would you pay? It would be price times number of shares 31 00:02:27,551 --> 00:02:31,076 outstanding, which for Microsoft is about $35 billion. 32 00:02:31,356 --> 00:02:37,463 And then, so, you take the market capitalization of Microsoft and then you 33 00:02:37,463 --> 00:02:42,220 divide it by the sum of the market capitalizations of all of the other 500 34 00:02:42,220 --> 00:02:46,093 stocks, right? So, the weights of, of all the assets in 35 00:02:46,093 --> 00:02:51,860 the S and P 500 add to one, and assets that have high market capitalizations have 36 00:02:51,860 --> 00:02:57,426 higher weight in the S and P 500. It's most of the, the 500 stocks that are 37 00:02:57,426 --> 00:03:04,494 in the S and P 500 tend to be very large capitalization stocks, so they're big 38 00:03:04,494 --> 00:03:09,430 well-established companies. And, and, and the weights, you know, of 39 00:03:09,430 --> 00:03:14,018 these assets, it's, I don't want to say it's equally weighted but it's fairly 40 00:03:14,018 --> 00:03:17,205 equally weighted. So, there isn't a, a huge concentration on 41 00:03:17,205 --> 00:03:20,513 one or two stocks. It's, it's a fairly diversified spread-out 42 00:03:20,513 --> 00:03:24,980 portfolio. So, the key thing to keep in mind is, this 43 00:03:24,980 --> 00:03:30,194 is an individual stock. This is a highly diversified portfolio of 44 00:03:30,194 --> 00:03:33,271 500 stocks. And what we want to do is we wanna say, 45 00:03:33,271 --> 00:03:37,709 what are the, the stylized facts about the returns on an individual security and a 46 00:03:37,709 --> 00:03:42,070 highly diversified portfolio. And we're going to see that they're very 47 00:03:42,070 --> 00:03:47,023 distinct differences in the properties of the returns of these two series, okay? 48 00:03:47,230 --> 00:03:52,000 Now, here are the plots of the continuously compounded return, and the 49 00:03:52,000 --> 00:03:56,040 data goes from 1998 to 2008. So, we don't have the financial crisis in 50 00:03:56,040 --> 00:04:01,052 this period and so we'll look, we'll do a homework assignment you know, you'll be 51 00:04:01,052 --> 00:04:04,767 looking at more updated data. And so, when you put the financial crisis 52 00:04:04,767 --> 00:04:07,066 in here, you'll see a big negative returns, okay? 53 00:04:07,066 --> 00:04:10,066 So, that's you know, that's something to be considered. 54 00:04:10,066 --> 00:04:15,746 Now, when we look at these two graphs, you want to ask yourself, what properties do 55 00:04:15,746 --> 00:04:20,000 you see, and what similarities do you see between the two series? 56 00:04:20,000 --> 00:04:22,487 Okay? So, first of all, do they look like 57 00:04:22,487 --> 00:04:27,053 Gaussian White Noise? Notice in, in the, from 1998 to 2002, the 58 00:04:27,053 --> 00:04:33,013 fluctuations of Microsoft were pretty big, so we call that the volatility. 59 00:04:33,013 --> 00:04:38,058 But, after about 2002, notice the magnitude of the fluctuations decrease, 60 00:04:38,058 --> 00:04:41,088 right? So, is that a property of a covariance 61 00:04:41,088 --> 00:04:44,042 stationary time series? Yes or no? 62 00:04:44,042 --> 00:04:47,015 No. It's because the variance appears to 63 00:04:47,015 --> 00:04:51,006 depend upon time. During this period the variance was big. 64 00:04:51,006 --> 00:04:54,055 During the, this, this period the variance was small. 65 00:04:54,055 --> 00:04:59,083 So, that's an indication, perhaps, that the volatility is, is changing over time, 66 00:04:59,083 --> 00:05:02,084 okay? So, that's one, one future to see. 67 00:05:02,084 --> 00:05:08,032 What about the mean in the series? There doesn't appear to be any obvious 68 00:05:08,032 --> 00:05:11,041 trend, right? So, the, the data appears to fluctuate 69 00:05:11,041 --> 00:05:16,039 pretty close to zero in, in this plot. And so, so if we think about covariance 70 00:05:16,039 --> 00:05:21,158 stationary in terms of the mean being constant, we don't see like the mean is up 71 00:05:21,158 --> 00:05:26,237 here, and then the mean is down below, and then the mean is up here, and, and things 72 00:05:26,237 --> 00:05:30,068 like that. So, now one of the things about, and, and 73 00:05:30,068 --> 00:05:36,052 so we, I, I want to follow up on this, this volatility comment that was made 74 00:05:36,052 --> 00:05:40,020 before. Microsoft has high volatility here than 75 00:05:40,020 --> 00:05:44,017 low volatility, we also that in the S and P 500 index. 76 00:05:44,017 --> 00:05:48,695 So, very often, the volatility of the S and P 500 index is called market 77 00:05:48,695 --> 00:05:53,327 volatility because the S and P 500 is thought to be a proxy for the overall 78 00:05:53,327 --> 00:05:57,522 movements in the stock market. And so here, market volatility was high, 79 00:05:57,522 --> 00:06:00,357 and then down here market volatility was low. 80 00:06:00,357 --> 00:06:05,810 So, notice that Microsoft's volatility is tracking the market volatility as well. 81 00:06:05,810 --> 00:06:10,658 So, we see some common behavior between the series and in this regard. 82 00:06:10,658 --> 00:06:15,347 Now, these two graphs are a little bit misleading because the scales are 83 00:06:15,347 --> 00:06:18,651 different. Notice that here, this, this is -40%, this 84 00:06:18,651 --> 00:06:23,262 is twenty percent for Microsoft. For the S and P 500, this is -fifteen%, 85 00:06:23,262 --> 00:06:26,728 this is six%. So, the volatility of the market is much, 86 00:06:26,728 --> 00:06:31,439 much lower than the volatility of Microsoft, but that's not apparent because 87 00:06:31,439 --> 00:06:37,097 of the way that the graphs are drawn. If we put these two asset, assets on the 88 00:06:37,097 --> 00:06:41,862 same scale. So, here, I have Microsoft in red, and the 89 00:06:41,862 --> 00:06:47,032 S and P 500 in blue. Then, we can see I think, more clearly, 90 00:06:47,032 --> 00:06:51,040 some similarities and differences between the series, okay? 91 00:06:51,040 --> 00:06:56,027 So, very clearly, the volatility of Microsoft is much greater than the 92 00:06:56,027 --> 00:07:00,085 volatility in the S and P 500, okay? So, this one fact about assets. 93 00:07:00,085 --> 00:07:06,006 When you have a highly diversified portfolio like the S and P 500, you, you 94 00:07:06,006 --> 00:07:10,349 do, you have a smaller volatility than an individual security, okay? 95 00:07:10,349 --> 00:07:14,406 And remember, volatility is very often used as a proxy for risk. 96 00:07:14,406 --> 00:07:17,810 Volatility is telling you uncertainty about the mean. 97 00:07:17,810 --> 00:07:22,713 So, the uncertainty about the mean for the S and P 500 is a lot smaller than the 98 00:07:22,713 --> 00:07:25,585 uncertainty associated with, with Microsoft. 99 00:07:25,585 --> 00:07:30,551 So, we might think there's less risk in the market than there is in Microsoft by 100 00:07:30,551 --> 00:07:35,054 itself, okay? Now, what's another feature that we can 101 00:07:35,054 --> 00:07:42,041 see, or, or another commonality between Microsoft and the S and P 500, that we can 102 00:07:42,041 --> 00:07:48,050 see in this graph, that we couldn't see very clearly in the other graph? 103 00:07:48,050 --> 00:07:52,044 Are Microsoft and the S and P 500 correlated with each other? 104 00:07:52,044 --> 00:07:57,092 So, Microsoft and the S and P 500 are kind of moving up and down together at the same 105 00:07:57,092 --> 00:08:00,050 time. Well, that means they're positively 106 00:08:00,050 --> 00:08:03,019 correlated, right? When one, when the market goes up, 107 00:08:03,019 --> 00:08:07,226 Microsoft goes up. When the market goes down, the Microsoft 108 00:08:07,226 --> 00:08:10,411 goes down. Now, I'm not saying that there is a causal 109 00:08:10,411 --> 00:08:14,461 relationship between the two. They're, they're just moving at the same 110 00:08:14,461 --> 00:08:15,612 time, okay? Right. 111 00:08:15,612 --> 00:08:21,306 So, those are some important properties and these are the kinds of things that we 112 00:08:21,306 --> 00:08:24,515 want to do when we look at data. We want to see, what do we see? 113 00:08:24,515 --> 00:08:26,746 What are commonalities? What are features? 114 00:08:26,965 --> 00:08:29,514 Is it covariance stationary? If so, why? 115 00:08:29,514 --> 00:08:31,701 If not, why not? Those kinds of things. 116 00:08:31,701 --> 00:08:36,576 Is there a relationship between the volatility of the series and the expected 117 00:08:36,576 --> 00:08:38,818 return? So, if you invest, if you take a stock 118 00:08:38,818 --> 00:08:43,220 that has a high volatility, do you tend to get a high average return as well? 119 00:08:43,220 --> 00:08:47,141 And the answer is, is the stylized fact is, is generally yes. 120 00:08:47,141 --> 00:08:51,345 When we look at stocks with high volatility, they tend to have higher 121 00:08:51,345 --> 00:08:54,438 expected returns than lower volatility stocks. 122 00:08:54,438 --> 00:08:58,340 Not always. This is the idea that risk and return go 123 00:08:58,340 --> 00:09:01,173 together. So, and we'll be, we'll be doing some 124 00:09:01,173 --> 00:09:05,668 plots later on while we're say, put expected average return or expected return 125 00:09:05,668 --> 00:09:11,373 on this axis and volatility on this axis, and we'll see that you know, things tend 126 00:09:11,373 --> 00:09:16,618 to move up in a direction like this and it's, it's tends to be a somewhat stylized 127 00:09:16,618 --> 00:09:22,580 fact but it's not always true. Okay. 128 00:09:22,580 --> 00:09:29,035 So, our observed sample is a realization of this covariance stationary time series. 129 00:09:29,035 --> 00:09:34,078 And, now, what we want to do is we want to compute some of descriptive statistics, 130 00:09:34,078 --> 00:09:36,950 okay? Now, in, in a very simple term, you know, 131 00:09:36,950 --> 00:09:42,092 a statistic is just a data summary. So, we compute descriptive statistics, 132 00:09:42,092 --> 00:09:48,091 that means we're creating data summaries and we're trying to capture particular 133 00:09:48,091 --> 00:09:52,001 features. Now, when we studied random variables and 134 00:09:52,001 --> 00:09:56,010 looked at probability distributions, I said there were certain shape 135 00:09:56,010 --> 00:09:58,088 characteristics of a probability distribution. 136 00:09:58,088 --> 00:10:03,034 There was the center of the distribution, which was the expected value, the 137 00:10:03,034 --> 00:10:07,008 variance, the spread, the skewness, asymmetry, kurtosis, fat tails. 138 00:10:07,008 --> 00:10:11,054 Well, all of these properties that we've talked about for random variables, we can 139 00:10:11,054 --> 00:10:14,025 create a descriptive statistics from the sample. 140 00:10:14,025 --> 00:10:18,055 So, if we want to know what is the probability distribution look like? 141 00:10:18,055 --> 00:10:22,878 We can compute a histogram which is a graphical descript, description of the of 142 00:10:22,878 --> 00:10:27,094 the distribution of the data. And then, of that histogram, we can think 143 00:10:27,094 --> 00:10:31,098 where's the histogram centered? You know, what's the spread in the 144 00:10:31,098 --> 00:10:34,065 histogram? What's it's skewness, or kurtosis? 145 00:10:34,065 --> 00:10:39,007 So, for all of the probability model concepts, there's going to be a sample 146 00:10:39,007 --> 00:10:43,250 analogue, a descriptive statistics that gives us, you know, the same kind of 147 00:10:43,250 --> 00:10:46,362 information, okay? So, we're going to create data summaries 148 00:10:46,362 --> 00:10:51,433 to describe certain features of the data, to learn about the unknown probability 149 00:10:51,433 --> 00:10:55,773 distribution that we think is actually generating the data. 150 00:10:55,773 --> 00:11:01,535 So, the, the thing about, you know, probability modeling and statistics in the 151 00:11:01,535 --> 00:11:07,005 real world is, we don't know what the true model is that generated the data. 152 00:11:07,005 --> 00:11:09,073 I mean, we'll never know what it is, right? 153 00:11:09,073 --> 00:11:14,306 And so, what we're trying to do is we're trying to kinda guess at what we think is 154 00:11:14,306 --> 00:11:19,013 the underlying model. And, we're going to use these descriptive 155 00:11:19,013 --> 00:11:24,611 statistics to try to help us pick good models for describing the data versus you 156 00:11:24,611 --> 00:11:30,084 know, bad models, and, and so on. Alright.