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  • 00:00

    [MUSIC PLAYING][Business Mathematics][Module 10: Statistics, Part 2.2:Introduction to Statistics]

  • 00:08

    RICHARD WATERMAN: You can see howwe've changed in today's class, whichis about statistics, from last time's class, whichwas about probability. [Richard Waterman, Adjunct Professorof Statistics] In the probability class,it was as if we had access to complete information.We had the whole probability distributionfor our random variable.And so we were able to directly calculate these populationparameters.

  • 00:29

    RICHARD WATERMAN [continued]: In today's class, we don't have complete informationabout the population.Instead, we only have access to a sample,which is rather realistic.And so given that we only have access to a sample,we are restricted to calculating sample statistics.And then we hope to use those samplesstatistics to say something useful about the population.

  • 00:52

    RICHARD WATERMAN [continued]: That's the basic idea.And so here are our main summaries-- two measuresfor centrality, the sample mean and the sample median,and two measures for spread, the samplevariance and its close cousin, the sample standard deviation.[The Formulas]I want to briefly show you the formulaefor calculating the sample mean variance and standard

  • 01:14

    RICHARD WATERMAN [continued]: deviation.But I want to reiterate that in practice we nevercalculate these things by hand.I don't remember the last time I did oneof these manual calculations.We always use a computer to performthese calculations for us now.But by presenting the formula, youcan at least see what's going on in the background.

  • 01:37

    RICHARD WATERMAN [continued]: So one thing to note is that these formulaeinvolve that big capital sigma, the Greek letter sigma.And what that means is summation.So you sum the elements associated with the sigma sign.And so here is the definition of the sample mean.It says that x bar is obtained by adding up the observations,

  • 01:59

    RICHARD WATERMAN [continued]: the individual xi.So the xi would be going back to the table for Apple stock,they would be the values in the second column under the percentdaily returns.So you add up the observations and then divide throughby how many there are.And little n on the bottom of the formulais common notation for your sample size.

  • 02:21

    RICHARD WATERMAN [continued]: So there's the sample mean.Now the sample variance-- it's a sort of averaging processbecause we are summing up again-- the sum.See the summation sign.And we're dividing through by almost n.Technically, it's n minus 1.But what are we averaging here?It's xi minus x bar, all squared.So it's an average of the squared deviations

  • 02:43

    RICHARD WATERMAN [continued]: from the sample mean.And so there's our sample variance,our measure of the spread of the data that we're looking at.There's always the question-- whydo we divide through by n minus 1 rather than n.And I say, if you're worrying about that,you're worrying about all of the wrong thingsbecause, typically, n is going to be very large.

  • 03:04

    RICHARD WATERMAN [continued]: So let's say n is a million.So you really want to worry as to whether youare dividing through by a million or 999,999?If you're worrying about that, you'reworrying about the wrong thing.Notwithstanding that fact, I'll just quickly tell you,there's an n minus 1 down there because this summary statistics squared itself depends on another summary statistic,

  • 03:24

    RICHARD WATERMAN [continued]: x bar.And when that happens, we lose whatis called a degree of freedom, and hence, the n minus 1in the denominator.But don't worry about that.I just want to show you that the variance is essentiallyan average.It's an average of the squared deviations from the mean.Thinking about the units that that sample varianceis measured in, well, you can see that in the definition

  • 03:48

    RICHARD WATERMAN [continued]: there's some squaring going on which, for example, saysthat if x was measured in n, then the variance isgoing to be in squared n.Again, a rather confusing concept.And so we often look at the standard deviationto aid our interpretation.And the standard deviation is just the square root

  • 04:09

    RICHARD WATERMAN [continued]: of the variance.And the standard deviation would havethe same units as the raw data that we were looking at.So that's how we calculate these quantities, at least in theory.[Calculating the percentiles of the data distribution]Going back to the summaries that were presented in the box plot,those summaries are based on ordering the data,

  • 04:31

    RICHARD WATERMAN [continued]: rather than averaging the data.So the median is defined as the 50th percentile.That says that it's the observationfor which half of the data is above it, and half is below it.Now there are some technicalitieshere that I'm simply not going to get into.For example, if you had an even number of observations,

  • 04:51

    RICHARD WATERMAN [continued]: there isn't going to be one right in the middle.There's going to be two.And you would actually get the median by averaging those two.So I'm sort of neglecting those details.And they really are small details,if you have a large data set.But there's the median.It's essentially the one that is in the centerof the ordered list of observations.The lower quartile-- that was the lower end of the box

  • 05:13

    RICHARD WATERMAN [continued]: in the box plot that has 25% of the data beneath itand 75% above it.And the upper quartile-- that has 75% of the data below itand 25% above it, once you've sorted the data.And so those are summaries that are oftenused to get a sense of the median,

  • 05:34

    RICHARD WATERMAN [continued]: telling you where the center of the distribution isand the distance between the lower and the upper quartilesis another measure of the spread of the distribution,the length of the box in the box plot that we had just seen.So that's how you would go ahead and calculatethese numerical summaries.

  • 05:55

    RICHARD WATERMAN [continued]: Again, I hope you don't ever have to do that by hand.I know I'm not going to do it by hand.We're always going to rely on software to do this for us.[Calculating the summary statistics]So here are the summaries from the daily returns on Apple.And obviously, you can see these werecalculated through software.There's lots of packages that you can use

  • 06:15

    RICHARD WATERMAN [continued]: to create summaries like this.Excel will do it for you, as willdedicated statistics packages.So let's just have a look at these summaries.Under summary statistics, you can see the mean reported hereat 0.1338.So that's the average percent daily return in Apple stock.

  • 06:35

    RICHARD WATERMAN [continued]: So on average, Apple has been going up,at least over this time period, by 0.13 of a percent.But data-- it's got positive.The standard deviation is 2.34.That's a measure of the spread and the variance-- 5.497.So those are our numerical summariesbased on the distribution of Apple stock.

  • 06:57

    RICHARD WATERMAN [continued]: And the quantiles presented here--you can see the median at the 50th percent.So that's equal to 0.13075.Notice that's pretty close to the mean.And the reason for that is that the distributionis essentially symmetric.We're also presenting here the lower

  • 07:18

    RICHARD WATERMAN [continued]: and the upper quartiles, the 25th percentileand the 75th percentile.And the 25th is minus 1.05, roughly a 1%.So 25% of days, you lose about 1% on Apple stock.And the 75th percentile is 1.378.

  • 07:41

    RICHARD WATERMAN [continued]: And that says for about a quarterof days, 25% of the days, you're going to make more than 1.38.Doing a bit of rounding percent on Apple stock.So those are the numerical summaries.They're telling us where the center of the distribution isand how spread out it is.[The Normal Distribution]Now that we've seen the numerical summaries

  • 08:01

    RICHARD WATERMAN [continued]: and the main graphical summaries,I want to introduce one of the key ideas thatsits behind statistics.And that's a particular distribution for data.And it's known as the normal distributionor-- colloquially termed-- the bell curve.And on this slide, I'm really not going to say much about it,except to say, look, it's a bell curve.

  • 08:23

    RICHARD WATERMAN [continued]: So this is what we're about to discuss.So in a little bit more detail now.So the normal distribution is probablythe most ubiquitous of all of the statistical distributions.There are in fact many statistics distributions.And I'm certainly not going to go into them all here.But of all the different distributionsfor data that we come across, if there was only one

  • 08:44

    RICHARD WATERMAN [continued]: that one could learn about, it wouldbe the normal distribution.So that's what we're going to do.We see normal distributions in many real life data sets.It's not the case that every distribution you look atis going to be approximately normal.But it is often the case that it providesa reasonable approximation to what's going on.

  • 09:04

    RICHARD WATERMAN [continued]: And as I said, you might have heardthis particular distribution term-- the bell curve.Now many of the statistical techniques that get usedrely at some level on an assumption of normality.Now it's not always an assumptionof normality of the raw data.Sometimes, it's a feature of the data

  • 09:26

    RICHARD WATERMAN [continued]: that we're looking at that is assumedto have a normal distribution.But when I was talking about assumptions,a common assumption behind statisticsis normality somewhere in the background.Now what makes the normal distribution important?One of the things that makes a normal distribution importantis that it's characterized by knowing the mean-- that we

  • 09:48

    RICHARD WATERMAN [continued]: write as mu and the standard deviation sigma-- alone.If you know those two features, then you essentiallyknow everything about the distribution.And so it's the distribution that'scharacterized by its mean and its standard deviation.Now in practice, we rarely know what the population meanand standard deviations are.We don't get to see the whole population.

  • 10:09

    RICHARD WATERMAN [continued]: We don't have access to a perfect probability model.What we have as analysts is access to data.And so what we use are the sample analogsof the population mean and standard deviation.In other words, x bar and s, we use those two summariesto tell us about the normal distribution

  • 10:29

    RICHARD WATERMAN [continued]: that we think we are observing-- the factsabout the normal distribution.[The Empirical Rule]I'm now going to introduce a rule thatapplies when your raw data is approximatelynormally distributed.And that rule is called the empirical rule.And it's one of those ideas that has a huge amount of legs

  • 10:52

    RICHARD WATERMAN [continued]: associated with it.You see this idea, the empirical rule, used all over the placein terms of decision making, in the presence of uncertainty.And so I always say if there's one thing you'regoing to learn about statistics, Iwould say learn the empirical rule.So let us get some ideas out there so we can discusswhat the empirical rule is.

  • 11:13

    RICHARD WATERMAN [continued]: So first of all, when data is bell-shaped and symmetric,then we are going to call it approximately normal.And you'll see this word approximate in a lotof the slides coming up.And the reason why we use that is because in real lifewe never know whether or not something is absolutely true.Typically, we just don't have access to that.And what we can do is hopefully say

  • 11:35

    RICHARD WATERMAN [continued]: that something is at least approximately true and closeenough to be useful is the way that I think about it.So when data is bell-shaped and symmetric,then we're going to say it's approximately normal.Under these circumstances, then the meanand the standard deviation summarizedthe data efficiently.That's the idea of the normal distribution

  • 11:55

    RICHARD WATERMAN [continued]: being characterized by its mean and standard deviation.So those are the two key numbers that you need to have.And you know how to calculate thembased on the previous slide.Or at least you're going to get them from software.Now the empirical rule only applieswhen the data is approximately normal.So how do you know the data is approximately normal?Well, that's why you draw the histogram to start off with.

  • 12:18

    RICHARD WATERMAN [continued]: You have a look at the data and you make a call onwhether or not it's reasonably normally distributed.Now there are some techniques thatwill be discussed in the statistics classesas to how to do this a little bit more formally.But for now, we'll just note that you

  • 12:38

    RICHARD WATERMAN [continued]: get to apply the empirical rule when you have approximatelynormally distributed data.So what is this rule do for you?Well, the answer is it provides a rule of thumband it ties together our two key summaries, the meanand the standard deviation.Mu and sigma, if you had them.X bar and s, if you have to estimate those features.

  • 12:59

    RICHARD WATERMAN [continued]: And it ties them into a rule that establishes wheremost of the data should lie.And one of the benefits of knowing where most of the datashould lie is it tells you that if you'reoutside of where you're expected to be,then you are a little bit atypical.And so many of you would have heard about Six Sigma

  • 13:24

    RICHARD WATERMAN [continued]: in quality control.People sometimes throw those words around-- Six Sigma.So what are we trying to say by that?So Six Sigma is meant to be intimatingto you that you're observing somethingthat's very, very unlikely.So we use the empirical rule essentiallyto establish the probability of events happening.And if we know where events are likely to be

  • 13:47

    RICHARD WATERMAN [continued]: and we observe an event outside of that likely area,then we have a basis to say that that's unusual or atypical.So the empirical rule is going to tie our two summariestogether, mean and standard deviation,into a rule telling us where we expectcertain percentages of the data to lie.Don't forget, empirical rule only

  • 14:07

    RICHARD WATERMAN [continued]: can be applied when you have approximately normallydistributed raw data.Now you say, what does it mean by the raw data.I'm talking about when we were discussing the Applestock, the numbers that were sittingin the last column of the table, the daily returns on the Applestock.That's what I would term the raw data.So here is the empirical rule.

  • 14:29

    RICHARD WATERMAN [continued]: It says that given the raw data as approximatelynormally distributed, then you can expect 68% of the datato lie within one standard deviation of the mean.95% of the data is going to lie within two standard deviationsof the mean.And essentially all or a little bit more precisely 99.7%

  • 14:52

    RICHARD WATERMAN [continued]: of the data will lie within three standard deviationsof the mean.Again, you can see I wrote here the rule definedby sigma, which is what I would use if I had access to sigma.But typically in practice we're goingto have to replace sigma with s when we apply the rule.Now we often round things to make

  • 15:13

    RICHARD WATERMAN [continued]: statements a little bit easier.And I might say 68%-- and that's about two-thirds--two-thirds of the data lies within one standard deviationof the mean, 95% within two, and 99.7% within three.Just looking at the third statement, if 99.7% of the datalies within three standard deviations of the mean,

  • 15:34

    RICHARD WATERMAN [continued]: then that says only point 0.3%, Or 3 out of 1,000 of the datapoints would be expected to lie morethan three standard deviations away from the mean.So that's a way of saying that a surprising event has happened,if you were more than three standard deviations awayfrom the mean.So those are the most common values

  • 15:56

    RICHARD WATERMAN [continued]: of sigma that we look at, one, two, and three.But I wanted to introduce a special onebecause I'm going to use this when I do some riskcalculations in a minute.It is the case that if you go out 1.645 standard deviationsfrom the mean, then 90% of the datawill lie within 1.645 standard deviations of the mean.

  • 16:19

    RICHARD WATERMAN [continued]: If 90% lies within that range, then 10%lies outside of the range.And given that a normal distribution is symmetric,it has that mirror symmetry line down the middle,you could split that 10% up and say there'sgoing to be 5% in each side, which we sometimes call

  • 16:40

    RICHARD WATERMAN [continued]: the tail of the distribution.So 1.645 sigma is the correct distanceto go from the mean to capture exactly 90% of the data.So that's what the empirical rule says for us.And we're going to be able to apply it when our data isapproximately normal.So let's do a couple applications.

  • 17:01

    RICHARD WATERMAN [continued]: [Using the Empirical Rule to calculate probabilities]So I'm going to return now to the Apple stock that,on reviewing the histogram, is approximately normallydistributed and apply the empirical rule to itto answer some questions.And I'm thinking about making statements about what's

  • 17:22

    RICHARD WATERMAN [continued]: going to happen tomorrow.So let's call the return on Apple stock tomorrow x.It's a random variable because tomorrow hasn't happened yet.It's an unknown, uncertain future event.But we'll assume that x tomorrow comesfrom the same distribution as we have observedover the last three years.

  • 17:42

    RICHARD WATERMAN [continued]: And again, that's an assumption.And one would, if you were going to use that assumption,one should be willing to stand behind itto make some argument as to why you believe that it'sa reasonable assumption.So call tomorrow's return x.I'm going to find a couple probabilities to illustrate

  • 18:02

    RICHARD WATERMAN [continued]: the empirical rule.First of all, we'll find the probabilitythat x is greater than 2.47%.So what's approximately the probability that Apple goes upby more than 2 and 1/2%?And then I'll find another probability-- part B here--what's the probability that the return isgreater than minus 4.55% and less than 4.81%?

  • 18:25

    RICHARD WATERMAN [continued]: So those are a couple probabilitiesthat I would like to calculate.Now based on my observation that this data is approximatelynormally distributed, I'm going to use the empirical ruleto help on these questions.So unfortunately, I don't know mu and sigma.And I am going to estimate those quantities as x bar and s.

  • 18:51

    RICHARD WATERMAN [continued]: And based on the numerical summariesthat I provided before, here are those estimates of x bar.And based on those numerical summariesthat I provided before, here's x bar and s. x bar is 0.13%and s is 2.34%.So those are the numbers that I'mgoing to plug into the empirical rule.

  • 19:13

    RICHARD WATERMAN [continued]: So given that I'm going to use the empirical rule to answerthis question, what I have to do is figure out howmany standard deviations these eventsx correspond to in terms of deviations from the mean.And to do that, I introduce what is called the z-score.And the z-score, for any particular value x,

  • 19:35

    RICHARD WATERMAN [continued]: is just defined as x minus x bar over s.And what that is doing for you is countinghow many standard deviations s the xis away from the mean x bar.So the z-score is a standard deviation counter--how many standard deviations are you away from the mean.So when I look at question A, what's the probability that x

  • 19:59

    RICHARD WATERMAN [continued]: is greater than 2.47%, what I need to dois figure out how many standard deviations 2.47%is away from the mean.And then given I know how many standard deviations it's awayfrom the mean, I can then apply the empirical rule to itbecause the empirical rule gives youa specific probability for events associated

  • 20:21

    RICHARD WATERMAN [continued]: with being a given distance from the meanwhen measured in terms of standard deviations.So the z-score is going to be the keyto answering these questions-- a standard deviation counter.[Solutions]So I'll work on, not surprisingly, part A first.And it's a question about an observation-- tomorrow's

  • 20:44

    RICHARD WATERMAN [continued]: return-- greater than 2.47.So I note that 2.47 turns out to beexactly one standard deviation above the mean.How do I know it's one standard deviation above the mean?Because I'm using the z-score formula.I'm doing 2.47-- that's the value of xI'm interested in-- minus the mean, 0.13, divided by 2.34.

  • 21:10

    RICHARD WATERMAN [continued]: And when I do that, it tells me that 2.47 is exactlyone standard deviation above the mean.Now I have to acknowledge that I created these questions.So that when we do the standard deviation counting,the numbers come out conveniently.

  • 21:31

    RICHARD WATERMAN [continued]: So it's not coincidental that's a whole number there.But it's going to allow me to apply the empirical rule.And I'll talk a little bit about whatone would do in practice if that wasn'tone of these whole numbers associatedwith the empirical rule.But for now, bear with me, and notethat 2.47 is exactly one standard deviationabove the mean.Now the empirical rule says to usthat there's a 68% chance of being within one

  • 21:54

    RICHARD WATERMAN [continued]: standard deviation of the mean.Given that the distribution is symmetric,we can split this 68% up into two pieces of 34%.And in the figure on the next slide,I have highlighted the area of interestthat corresponds to getting a value of the random variable--

  • 22:15

    RICHARD WATERMAN [continued]: tomorrow's return on Apple stock-- greater than 2.47%.That's shaded in light green.[Question (A) illustrated]So let's have a quick look at an illustration of the areathat we're trying to calculate.Here it is.You can see I've drawn the normal distribution centeredaround its mean, 0.13.

  • 22:36

    RICHARD WATERMAN [continued]: And I have identified that 2.47 is exactlyone standard deviation above the mean.And I've put in the respective probabilities of the pieces.The area to the left of the mean,to the left of 0.13-- that has a 50% areabecause the distribution is symmetric.

  • 22:56

    RICHARD WATERMAN [continued]: And based on the empirical rule, because they're68% within one standard deviation of the mean,I can split that 68% up into two 34% pieces.And when I do that, I can see thatthe light green area, which is my area of interest,it corresponds to returns greater than 2.47%.

  • 23:16

    RICHARD WATERMAN [continued]: That must have 16% of the area in it.And so that's the probability that I'm interested in,the number that is illustrated inside the light green areaof this curve.[Solutions]So going back to the actual calculationitself, using the fact that the area under the entire curve

  • 23:37

    RICHARD WATERMAN [continued]: must be 100%, because something has got to happen,and by the 50% symmetry argument and the breaking upof the 68% into two pieces-- 34 and 34--I see that the probability that x is greater than 2.47%is 0.16.And so there is a probability calculation

  • 23:58

    RICHARD WATERMAN [continued]: that has come from implying the empirical rule.And the key thing to note here wasthat the answer was obtained by countinghow many standard deviations that 2.47 was awayfrom the mean, the 0.13, 3 and thenusing the empirical rule, given I knew how many standard

  • 24:19

    RICHARD WATERMAN [continued]: deviations the event of interest was away from the mean.[Empirical rule, question (B)]Let's have a look at the second question, part B, now.And that was to find the probabilitythat x, our daily return on Apple stock, the valueto be observed tomorrow, lies between minus 4.55%

  • 24:41

    RICHARD WATERMAN [continued]: and plus 4.81%.This time around, we're going to have to find the z-scores,or count standard deviations for eachof the endpoints of this interval.And it's always useful to be able to identifyon the distribution the area thatcorresponds to the event whose probability I'm

  • 25:03

    RICHARD WATERMAN [continued]: trying to calculate.And so I'm going to quickly jump to the next slide, whereI illustrate with the area that I'm interested in.And once again, it's that light green areathat I'm trying to calculate, the area of the probability.And notice that the boundaries of thatare minus 4.55 and 4.81.So what's the area between those two numbers?

  • 25:25

    RICHARD WATERMAN [continued]: Well, the approach is to take the z-scoreof both ends of the interval.And if I do that, I'll find that 4.81 is exactlytwo standard deviations above the mean, minus 4.55,is exactly two standard deviations beneath the mean.And so this event of interest, the onethat was shaded in light green on figure 2,

  • 25:47

    RICHARD WATERMAN [continued]: is an event corresponding to beingwithin two standard deviations of the mean.So this is exactly the question--what's the probability that the random variable lieswithin two standard deviations of the mean.And assuming approximate normality-- the distribution--and applying the empirical rule, I just

  • 26:08

    RICHARD WATERMAN [continued]: read off directly from what the empirical rule states.And it tells me that there's a 95% chanceof lying within this particular two standard deviationsof the mean.So the answer to the question is 0.95.And that's what's illustrated on this slide figure,showing figure 2 with a 95%.

  • 26:33

    RICHARD WATERMAN [continued]: Now I did note that I created these questionsso that when we did the standard deviation counting,the numbers turned out to be whole numbers,so we could apply the empirical rule directly.Now in practice, that isn't going to be the case.So what are you going to do?Well, in practice, we use software

  • 26:53

    RICHARD WATERMAN [continued]: to answer these questions.And so you would get a software package--Excel can do this for you-- that will tell youthe probabilities associated withany particular standard deviationdistance from the mean.It doesn't just have to be 1, 2, or 3, or 1.6, or 5.The software will tell you the answer to any distancethat you choose to look at from the mean.

  • 27:15

    RICHARD WATERMAN [continued]: So the idea certainly generalizes.We just have to use software to implement it, typically.But the point here is to get a senseof the sorts of statements that we'reable to make based on the empirical ruleand to get a good understanding of howthe standard deviation relates to where we expect data to lie.

  • 27:37

    RICHARD WATERMAN [continued]: If I go out one standard deviation,I'm going to see about two-thirdsof the data in that distance.If I go out two standard deviations, about 95%of the data.And if I go all the way three standard deviationsof the mean, then I'm going to see almost allof the data, or 99.7%, according to the empirical rule.So it gives us a sense of where most of the data

  • 27:59

    RICHARD WATERMAN [continued]: is going to lie.[Turning the question around]So what I'm going to do now is turn the question aroundthat we've been looking at.So the way that the questions have been articulated,the questions A and B that we've just done,is that there was a particular valuefor x that was of interest.And we worked out the probabilityof being greater than that x or being

  • 28:21

    RICHARD WATERMAN [continued]: within some particular distance.So the point was that you were given a value of x and youworked out the probability.Now you can turn the question around where somebody gives youthe probability, and you figure out what the associated x is.And you might say, well, that sounds rather abstract.

  • 28:43

    RICHARD WATERMAN [continued]: Why am I doing that?Well, it turns out that when we turn this empirical rulequestion around, somebody comes to youwith a probability of an event, you figure outhow far away from the mean you areto be associated with that event,then that is exactly the idea behind something that'scalled Value at Risk, or VaR

  • 29:04

    RICHARD WATERMAN [continued]: And VaR was one of the early methodologies.I's still around in various formsthat was used to measure risk.It was an application of the empirical rule.And I'm going to illustrate a VaR calculationfor daily returns under the assumptionof approximate normality for the distribution of returns-- here,

  • 29:25

    RICHARD WATERMAN [continued]: approximate normality of the Apple stock.OK, to illustrate what I mean by turning the question around,here's how we're going to articulate it.We want to find the percent return, which we'll terman adverse market move, for whichthere's only a 5% chance of seeing something

  • 29:46

    RICHARD WATERMAN [continued]: this bad or more extreme.From a mathematical point-of-view,we're trying to find a particular value-- call itx with a little 0, so that's going to be our adverse marketmove-- such that the probability that the random variable isless than this value, or less than or equal to this value,is exactly 0.05.

  • 30:07

    RICHARD WATERMAN [continued]: So you can see you're being given the probability here,the 0.05, and you've got to figure outwhat the little x0 is.So it's really just applying the empirical rule the other wayaround.[An illustration of the VaR calculation]And in terms of a graph that shows youexactly what we're trying to do here, here is the idea.So we've got a normal distribution for returns.

  • 30:27

    RICHARD WATERMAN [continued]: It's centered around 0.13, the mean.And what we want to do-- have a lookat the left-hand side of this graph--we want to find the value.And I've just put in question marks for this valueright now because I don't know what it is.But it's to be defined as the value for which there

  • 30:49

    RICHARD WATERMAN [continued]: is 5% area to the left of it.And so that cut-off can be interpreted in various ways.I mean, formally, it is the valuewith 5% area to the left of it.But you could articulate it by saying something like,ruling out the worst 5% of days, what'sthe worst we're likely to see, given I've

  • 31:11

    RICHARD WATERMAN [continued]: ruled out the worst 5% of days.So different ways of articulating that cut-off.But that cut-off with the question markshere is what is going to be termed our adverse market move.And what I would like to do is figure outthe value of those question marks,and then use that value as a measure of the riskthat I have in some particular position,

  • 31:32

    RICHARD WATERMAN [continued]: in this case, Apple stock.So that's the idea here.Can you figure out what the question marks are going to be?In order to figure out what the question marks are,which boils down to one of these Value at Risk calculations,I'm going to make a quick assumption.And the assumption is-- and this follows the VaR, or Value

  • 31:55

    RICHARD WATERMAN [continued]: at Risk methodology-- I'm going to assume herenow that the mean is 0.Now I note that when I did the calculation,the sample mean was a little bit different from 0,but in terms of this formal calculation of VaR,we do it under the assumption that the mean is 0.And there are reasonable argumentsfor why we should do that.For now, we'll just take it as the case.So let's say now that the mean is 0.

  • 32:18

    RICHARD WATERMAN [continued]: And what I want to do is figure out under the empirical ruleassumption how far away I need to be from the meanin order to only have 5% area to the left of me.Now going back to how I introduced the empirical rule,

  • 32:39

    RICHARD WATERMAN [continued]: remember I gave that very special 1.645 event.It said that 90% of the data lieswithin 1.645 standard deviations of the mean.If 90% is within 1.645 standard deviations, then 10%is outside.And given symmetry, there would be exactly 5% in each tail.

  • 33:01

    RICHARD WATERMAN [continued]: So it turns out that the answer to this question, the valueof the return that has exactly 5% area to the left of it,is defined as being exactly 1.645 standard deviationsbeneath the mean.And how far is 1.645 standard deviations

  • 33:24

    RICHARD WATERMAN [continued]: in this particular case, looking at the Apple stock?Well, the standard deviation was times 2.34 is 3.849.So under the assumption of a zero mean here,the adverse market move will be defined

  • 33:44

    RICHARD WATERMAN [continued]: as being 3.849% beneath the mean.And in terms of how much money I have at risk,then what I do to calculate the VaRis to take the position size that I have.So let's assume we have $1,000 in Apple stock.

  • 34:05

    RICHARD WATERMAN [continued]: And I say, well, how much of thatam I going to lose when this adverse market move occurs?And the adverse market move was losing 3.849%.So 3.849% of $1,000 is $38.49.So given a position size of $1,000, then my VaR for that,

  • 34:27

    RICHARD WATERMAN [continued]: or Value at Risk, is $38.49.And you can see that's a way of quantifying the risk associatedwith that position.It is purely an application of the empirical rule.Now I wanted to make a couple comments before I move off

  • 34:48

    RICHARD WATERMAN [continued]: of this topic.And that is, I've assumed normality here.And over time, as people look more carefully at these returnsdistributions, they concluded that market returns arenot quite normally distributed.There are other distributions that can betterdescribe the shape of returns.

  • 35:10

    RICHARD WATERMAN [continued]: And this VaR calculation is oftenaugmented with more complicated distributionsare the normal now.But nonetheless, the concept still holds true in the sensethat the VaR is being calculated as an adverse market move.You might just calculate what is an adverse market

  • 35:30

    RICHARD WATERMAN [continued]: move under different scenarios, under different distributions.And--[Music: Repeater by Moby, courtesy of][Business Mathematics][Richard Waterman]

Video Info

Series Name: Business Statistics

Episode: 12

Publisher: Wharton

Publication Year: 2014

Video Type:Tutorial

Methods: Descriptive statistics, Central tendency, Sampling, Standard deviations, Normal distribution

Keywords: mathematical applications; mathematics; Software; value at risk

Segment Info

Segment Num.: 1

Persons Discussed:

Events Discussed:



In part 2.2 of his series on business mathematics, Professor Richard Waterman provides an introduction to statistics. Statistics are used when complete information is only available for a sample, rather than the full population. Waterman discusses statistical formulas, the empirical rule, and value at risk.

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Introduction to Statistics, 2.2

In part 2.2 of his series on business mathematics, Professor Richard Waterman provides an introduction to statistics. Statistics are used when complete information is only available for a sample, rather than the full population. Waterman discusses statistical formulas, the empirical rule, and value at risk.

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