Distribution moments

hello David Harper of the banach turtle here with a brief tutorial on the moments of a distribution moments are measures that tell us about a distribution they capture key qualities about the distribution in numerical form so I'll use the normal as an example here but keep in mind moments don't need to refer to the normal distribution the normal is just a sort of benchmark for us moments tell us about qualities of any distribution generically we can say the KF moment is given here and generally is a function of this caithe power where X is the observation and a is going to be equal zero or the mean so this is the generic form of the moment or the caithe moment I won't worry too much about that because I'll just show you the first moment which is the mean or specifically the arithmetic mean and so this is the easy one to understand if we still think about this normal distribution here it peaks at the arithmetic mean of the distribution that is the first moment and that mean is given by it's the summation of the X values divided by the number of values and so you can see it fits into our generic our general formula if a equals zero and we're raising to the K of power so the first moment is the mean it's the measure of central tendency it's also sometimes called the location because if you think about it it sort of anchors the distribution and says well it starts to be located here the second moment is the variance and this is also called the shape of the distribution it's also called them a measure of scatter but it tells us about the dispersion of the distribution and I would also say because it's in unit squared you will recall we oftentimes take the square root to standardize it and convert the variance into a standard deviation so variance and standard deviation can almost be used synonymously but the variance as a measure of dispersion or shape is the second moment and it also fits into our general form for general formula for the moment of the distribution because it's the distance between observations and the mean now in this case a equals the mean squared so we're raising to the second power and it's really the expected value of these distances squared so it's divided by n that's the second moment and the normal distribution that I still highlight in blue behind my other boxes is it happens to be fully described by both that first and second moment if we know the mean and we know the variance then we fully characterized the normal distribution but that's just the normal of course we can go to the third moment and talk about the skew and if you imagine these normal could be pulled to the left or sort of yanked over here Inc to the right legging to the left it could be skewed and we would say it's not symmetrical so skew is a measure of symmetry or asymmetry also fits into our general formula for the moment but it's rate it's a function of the expected value raised to a distance between the observation and the mean raised to the third power so here's a measure of skewness the normal because it's fully described by the mean and variance has a skew of zero so anytime if the skew is over here the right to be positively skewed over the left it would be negatively skewed and the fourth moment is kurtosis this turns out to be very relevant in quantitative finance you can see it also fits into the general were Matt for the moment we're talking about the fourth moment and we it's a function of the fourth power now the normal has a kurtosis of three such that we sometimes say we refer to excess kurtosis as kurtosis that is greater than 3 so kurtosis of 3 is sort of normal so to speak excess kurtosis would be greater than 3 if we did have a kurtosis greater than 3 we would have what is technically called lepto kurtosis and that refers to fat tails and so really there's really two parts to that the fat tails in part are a function of a higher peakedness so if you'd imagine if the mean was up a little higher but there was less density here at the top we would have higher peakedness and some of that mass would fall into the fat tails that would be fat tailed distribution and many financial returns are fat tailed and they're for exhibit lepto kurtosis that's the fourth moment okay so I'll quickly go just go to excel and show you how to use skew and kurtosis in Excel so briefly in Excel all I did here was I went to Yahoo Finance and pulled up the daily price closes for Yahoo's own stock so that's daily closed as of yesterday stock closed at $19 86 cents here the day before looks like it or the last trading day before closed at $20 and 78 cents and so on so I have daily price closes here in this column and then I just calculate a daily periodic return so that's the natural log of the ratio so we've got daily periodic returns and then I simply put them into frequency bins so that I could plot a histogram over here and I'd actually pulled for 10 years because it's a pretty easy to do so what I have here is a histogram or frequency plot of Yahoo's daily returns over the last 10 years okay so you can see this is 400 daily returns up here somewhere near the mean which is going to be pretty close to zero and so is it normal well it's hard for me to say that what I wanted to show you is that in Excel there are two built-in functions for the test of that third and fourth moments that I just showed you so here the first one is skew s ke W that's a test of that skewness and it's very simple it just takes the series as the input here so I said equals skew all of those daily returns and I got point 0 5 and what that tells me is 0 would be no skew or symmetry so I've got some positive skewness in Yahoo's daily returns and then then then I computed the fourth moment which is equals K u R T is a parameter it only takes a single range or array so it's also a pretty easy function and I've just got as its input the series of daily periodic turns I hit enter and for kurtosis I get three point six nine and you will call kurtosis of three is normal and any greater than three means excess kurtosis or fat tails so for Yahoo I do in fact have fat tails or excess kurtosis which is typical as turn as far as stock price returns go so that's about all you need to do in Excel to test for the data set to see if you've got symmetry and fat tails thanks for your time this is David Harper of the Bionic turtle

Channel: Bionic Turtle

Views: 41,435

Rating: 4.702857 out of 5

Keywords: Finance, Quant, Excel

Id: SZ3T1cSXP7w

Length: 8min 45sec (525 seconds)

Published: Wed Jan 23 2008