# 2. Risk and Financial Crises

What I want to do this time is talk about probability I don't think many of you have taken a course in probability theory I don't take that as a prerequisite for this course but I think that actually probability theory is fundamental to the way we think about

Finance so I wanted to talk about that a little bit today and I'm going to put it in a concrete context namely the crisis that the world has been through since 2007 and which we're still in at this point it's a financial crisis that's

Bigger than any since the Great Depression of the 1930s but I want to there's many different ways of thinking about a crisis like this and I wanted to focus on one way that people think about it in terms of probability models so

That's not the only way it's not necessarily my favorite way of thinking about it and but that's I think a good way of introducing our discussion of probability as it relates to finance excuse my cold I'm managing the talk I

Didn't bring any water I hope I make it through this lecture it's a little bit iffy but so let's just think about the crisis most people when they talk about financial crises they talk in terms of narrative of a historical narrative so

I'll give you a quick and easy historical narrative about the crisis the crisis began with bubbles in the stock market and the housing market and also in the commodities market bubbles are I will talk about these later but

Bubbles are events from which people get very excited about something and they drive the price up really high and it's got a break eventually and there was a pre break around 2000 when the stock market collapsed around

The world all over the world that stock markets collapsed in 2000 but then they came back again after 2003 and they were on another boom like a rollercoaster ride and then they collapsed again that's the narrative story they and then

The both the housing market stock market collapsed and then what happened is we see a bunch of institutional collapses so we see in 2007 a failures of companies that had invested in home mortgages and we see a run on a bank in

The United Kingdom that northern Northern Rock it was arrested but it looked like 1930s all over again with a bank failure we saw bank failures in the United States and then we saw International Cooperation to prevent

This from spreading like a disease and then we had governments all over the world bailing out their banks and other companies so a disaster was averted and then we had a nice rebound that's the narrative story okay and it makes it

Sound and I'm going to come back to it because I like the narrative story of the collapse but I want it today focus on something that's more in keeping with probability with the way financial theorists think about it and what

Financial theorists will think about is that actually it's not just those few big events the crisis we got into was the accumulation of a lot of little events and sometimes they accumulate according to the laws of probability

Into big events and and you're just telling stories around these accumulation of shocks that affected the economy and the stories are not by some accounts not that helpful we want to understand

The underlying probabilities and so that's that's oh thank you have a good head good assistant he knows what I need I just announced what I need he got it a bottle of water and tomorrow I may have absolutely no voice you're lucky so the

There's a couple of I'm going to talk today about probability and variance and covariance and regression and idiosyncratic risk insist systematic risk things like that which are core concepts in finance but I'm also going

To in the context of the crisis talk emphasized in this lecture breakdowns of some of the most popular assumptions that underlie financial theory and thinking particularly of two breakdowns and will will emphasize these as as

Other interpretations of the crisis one is a failure of independence I'll come back and redefine that and another one is a tendency for outliers or fat-tailed distributions so off to explain what all that means but but basically let me just

Try to elaborate on probability theory is a conceptual framework that mathematicians invented and it's become a very important way of thinking but it doesn't go back that far in time the word probability in its present meaning

Wasn't even coined until the 1600s so if you if you talk to someone before the year 1600 and say this has a probability of 05 they would have no idea what you're talking about so it's a major advance in

Human understanding to think in terms of probabilities now we do and now it's routine but it wasn't routine at all and a part of what I'm thinking about is what what the probability theorists do or in particular finance theorists like

To do is they think that the world is well let me just say it's kind of a you it's kind of a realization that the world is very complex and that it is that the outcomes that we see are the result of millions of little things and

The stories we tell are just stories but so how do we deal with the complexity of the world well we do it by by dealing with all of these little incremental shocks that affect our lives in a in a mathematical way and we think of them as

Millions of shocks how do they accumulate and they we have mathematical laws of how they cumulate and once once we once we understand those laws we can we can build mathematical models of the outcomes and then we can ask whether we

Should be surprised by the by the financial events that we've seen it's a little bit like science a real hard science so for example weather forecasters they build models that you know you see these weather forecasts

They have computer models that are built on the theory of fluid dynamics and there is a theory of all those little atoms moving around in the air and there's that there's there's too many atoms to count but there's some laws

About their cumulative movement that we understand and it actually allows us to forecast the weather and so people who are steeped in this tradition in finance think that what we're doing when we're when we're doing financial forecast is

Very much like what we do when we do weather forecasts we have a statistical model we see all the shocks coming in and of course there will be hurricanes and we can only forecast them you know there's a limit to how far out we can

Forecast them so all hurricanes are surprised two weeks before they happen weather forecasters can't do that same thing with financial crises this would be the model that we we understand the probability laws there's only a certain

Time horizon before which we can forecast the financial crisis that isn't exactly my view of of the situation but I'm presenting a view this time which is very mathematical and probability theory oriented so let me get into some of the

Details and again these are going to be recovered in the review session that elan fold the one of our teaching assistants will do so I'm having I have just slides with some of graphs and equations but that's not in line fold I

Want to start out with just the concept of return which is in finance is the basic the most basic concept that and I forgot to turn my blackberry off when you invest in something you have to do it for a time interval and I'm writing

The return as one time period t is time and so it could be year or it could be months or it could be day and we're going to number these we're going to number these months let's say it's monthly return we're going to number

These months so that the the first month is number one second month is number two and so return at time T refer if T is equal to three there would be the return at month three and and we'll do price at the beginning of the month

And so what is your return to investing in something it's the increase in the price that's PT plus one minus PT all right I'm calling I'm spelling it out here price I spelled it out of the numerator

I guess I didn't do it in the denominator it's price at time t plus one minus the price at time t which is called the capital gain plus the dividend which is a check you receive if you do from the company that's you're

Investing in and that's the return we have something else called gross return which is just one plus the return returns can be positive or negative they can never be more than never be less than minus 100% in a limited liability

Economy that we live in the law says that you cannot lose more than the money you put in and that's going to be our assumption so return is between minus 100% and plus infinity and gross return is always positive it's between zero and

Infinity now what we're going to do in this is the primary thing that we want to study because we are interested in investing and in and making a return so we want to do some evaluations of the success of an investment or so I want to

Now talk about some basic statistical concepts that we can apply to returns and to other random variables as well and these are these on this slide are mostly concepts that you've already heard this is expected value this is the

Mathematical expectation of a random variable X which could be the return or the gross return or something else but we're going to substitute some thing else we're going to substitute in what they are so the expectation of X or

The mean of X mu sub X is another term for it is the weighted sum of all possible values of X weighted by their probabilities and the probabilities have to sum to one they're positive numbers that are or zero or positive numbers

Reflecting the likelihood of that random variable occurring in that value of the random variable occurring so I have here there's an infinite number of possible values for X and we have a probability for each one and the expectation of X is

Is that weighted sum of those weighted by probabilities of those possible values this is for a discrete random variable that takes on only a finite only a countable number of values if it's a continuous random variable if X

Is continuous then the expectation of X is an integral of the probability density of x times X DX I'm just writing that down for you now because it's for completeness but I'm not going to explain or elaborate that but basically

These are these two formulas here are measures of the central tendency of X ok it's the in it's essentially the average of X in the in the probability metric that we have up here but this formula is something we use to estimate the

Expected value of X and it says that and this is called the mean or average which you've learned long ago if you have n observations on a random variable X you can take the sum of the X observations the summation I equals 1 to N and then

Divide that by n that's called the average so what I want to say is that this is the average or the mean or sample mean when you have a sample of n observations which is an estimate of the expected value of x so for example if

We're evaluating a investor who has invested money you can get n observations let's say n annual returns and you can take an average of them and that's the first the most obvious metric representing the success of the

Investment if X is the return okay people are always wanting to know they're looking at someone who invests money is this person a success or not well this is the first and most obvious measure let's see what the person did on

Average you are investing for let's say N equals 10 10 years let's take the returns you made each year add them up and divide by 10 and that gives us an average I put this formula down as an alternative because it's another this is

Called the geometric mean this is the arithmetic mean this is the geometric mean and you're probably not so familiar with that because it's a different concept the geometric mean instead of adding your n observations you multiply

Them together if you form a product of them and then you instead of dividing by n you take the nth root of the product and so that's a formula that's used to estimate the average return of a portfolio of investments where we use

Gross return for X naught not just the simple return this geometric mean makes sense only when all the X's are non-negative if you put in a negative value you might get a negative product then if you took the instrument of that

It would be a magic number so let's forget that we're not going to apply this formula if there's any negative numbers but it's often used and I recommend its use in evaluating investments because if you use gross

Return it gives a better measure of the outcome of the investment so think of it this way suppose you invested money with some investment manager and the guy said I've done a wonderful job of investing your money I made 50% one

Year I made 30% another year oh and by the way I made one I had one bad year what was – 100 % okay so what do you think of this investor what do you think about it if he made 50% one year and then 30% another year and then he lost

Everything you know that dominates everything right if you have a – 100 percent simple return your gross return is zero okay so if I plug in if I put in a 0 here – any of the X's right the GM it that this product will be zero

Anything times zero is zero and I take the nth root of zero and what's that it's zero so if there's ever a year in which the return is minus a hundred percent then the gross the geometric mean is zero that's a good discipline

This obviously doesn't make sense as a way to evaluate investments accesory are you following me with me on this because you care a lot if the guy wipes you out whatever else is done after that doesn't matter so that's why we want to use the

Geometric return so we have different these are all measures of central tendency that is what is the central result sometimes the investor had a good year if none to the investor had a bad year but what was the typical or central

Value so these are a couple of measures of of them but we care more than just about central tendency when evaluating risk we have to do other things as well and so we want to talk about and this is very fundamental to finance we have to

Talk about risk what could be more fundamental than risk for finance so what we have here now is a measure of variability and the upper equation here is something called variance and it's equal to the weighted average of the X

Random variables squared deviation from the mean weighted by probabilities okay all it is is the mean squared the expectation of the square of the deviation from the mean the mean is the center value and the

Deviations from the mean are if whether they're positive or negative you square them they become positive numbers and so that's called variance so for example if X tends to be if they return tends to be plus or minus 1% from the mean returns

They don't say the mean return for an investor is is 8% a year and it's plus or minus 1% then you would see a lot of ones when you squared the deviation from the mean and the variance would probably be one and the standard deviation which

Is the standard deviation is the square root of the variance and it would also be one okay this is a very simple concept it's just the average squared deviation from the mean the estimate of the variance or the sample variance is

Given by this equation and it's s squared of X is just the sample mean take the deviations of the of the variable from its sample mean you have n observations say someone has invested money for 10 years you take the average

Return for the 10 years and that's x-bar and then you take all 10 deviations from the mean and square them and then divide by n some people divide by n minus 1 but I'm just trying to be very basic and simple here so I'm not going to get

Into these ideas then the next thing is covariance and we're getting through these concepts they're very basic concepts covariance is a measure of how two different random variables move together so I have two different random

Variables x and y so X is the return on let's say the IBM corporation and Y is the return on General Motors Corporation and I want to know when when IBM goes up does General Motors go up or not so a measure does a measure of the co

Movement of the two would be to take the deviation of X from its mean and times the deviation of Y from its mean and take the average product of those and that's called covariance it's a positive number if when X is high relative to its

Mean Y is high relative to its mean also and it's a negative number if they tend to go in opposite directions if GM tends to do well when IBM does poorly then we have a negative covariance because if one is above its mean and the other is

Below its mean the product is going to be a negative number and if we get a lot of negative products like that it means that they tend to move opposite each other and if they are unrelated to each other then the covariance tends to be

Zero and this is a the core concept that I was talking about some idea of unrelated nasai is a lot of our thinking in risk so if x and y are independent they're generated suppose ibm's business has just nothing at all to do with GM's

Business they're so different then I'd say the covariance is probably zero and then there we can we can use that we can use that as a principle which will underlie our later analysis correlation is a

Scaled covariance and it's a measure of how much two variables move together but it's scaled so that it varies only over the range of minus 1 to +1 so the correlation between two random variables is their covariance divided by the

Product of their standard deviations and you can show that that always ranges between minus 1 and plus 1 so if two variables I have a plus 1 correlation that means they move exactly together the when one moves up 5% the other one

Moves up 5% exactly if they have a correlation of minus 1 it means they move exactly opposite each other these things don't happen very often in finance but in theory that's what happened if they have a zero correlation

That means there's no tendency for them to move together at all if two variables are independent then their correlation should be zero okay so these are the variance of the sum of two random variables is the variance of the first

Random variable plus the variance of the second random variable plus twice the covariance of the random variables so if the two random variables are independent of each other then their covariance is 0 and then the variance of the sum is the

Sum of the variances but that is not necessary that say that that's true if the random variables are independent but we're going to see that breakdown of independence is the story of this lecture right now I don't

We want to think about independence as as mattering a lot and it's a it's a model or a core idea but when do we know that things are independent so okay this is a plot I was telling you earlier about the VC okay let me just hold off

And had a minute the that well I'll tell you what that was that was a plot of the of the stock market since from 2000 to 2010 in the US and I'm going to come back to that these are the crises I was telling you about

This is the decline in the stock market from 2000 to 2002 or 3 and this is the more recent decline from 2007 to 2009 those are the accumulate accumulative effect of a lot of little shocks that didn't happen all at once it happened

Over years and we want to think about the probability of those shocks occurring and that's that's where I am going for but what I want to talk about is the core concept of independence leading to the some basic principles of

Risk management the crisis that we've seen here in the stock market is the cumulation of you see all these ups and downs in the stock market and then all these ups and downs on the way up there were there were relatively more downs in

The view from 2000 to 2002 and there were relatively more ups from the period 2003 to 2006 but how do we understand the accumulative effect of that which is what matters so we have to have some

Kind of probability model the question immediately is are these shocks that affected the stock market are they independent or are they or are they are they somehow related to each other and that that is the core question that made

Us so difficult for us to understand how to deal with the potential of such a crisis and why so many people got and got in trouble dealing with this crisis so we had a we had a big financial crisis in the United States in 1987 when

There was a stock market crash that was bigger than any before in one day we'll be talking about that but the after the 1987 crash companies started to compute a a measure of the risk to their company which is called

Value at Risk I'll write it up like that I capitalized the first in the last letter so you'll know that I'm not this is not the same thing as variance this is value at risk and what companies would do after 1987

To try to measure the risk of their activities is to compute a number something like this they would say there's a 5% probability that we will lose 10 million dollars in a year that's that's the kind of bottom line that

Value at Risk calculations would make and so you need a probability model to make these calculations and so you need probability theory in order to do that many companies had calculated value at risk numbers like this and told their

Investors we can't do too badly because there's no way that we could lose the probability is only 5% that we could lose 10 million and they have other numbers like this but they were implicitly making assumptions about

Independence or at least relative independence and that's where the that's the concept I'm trying to emphasize here it's a core concept in finance and it's not one that is easy to to be precise about we have an intuitive idea that the

You know we see the ups and downs of the stock market and we've noticed them and they all average out to something not too bad the problem is that brought us this crisis is that the Value at Risk calculations were too optimistic

Companies all over all over the world were estimating very small numbers here relative to what actually happened and and that's that's a problem I wanted to emphasize core concepts here intuitive concepts that you probably

Already have but one of these concepts is something we'll call the law of large numbers okay and the law of large numbers says that there's many different ways of formulating it but putting it in its simplest form that if I have a lot

Of independent shocks and averaged them out it's on average there's not going to be much uncertainty if I if I flip a coin once and let's say I'm making a bet plus or minus if it comes up heads I'll win a dollar if it comes up tails I'll

Lose a dollar well I have a risk I mean I have a standard deviation of one dollar in my outcome for that but if I do it a hundred times an average the result there's not going to be much risk at all and that's the law of large

Numbers it says that the variance of the average of n random variables that are all independent and identically distributed goes to zero as the number of elements and the average goes to infinity and so that's a fundamental a

Fundamental concept in underlies both finance and insurance the idea that tossing a coin are throwing a die in a small number of yeah as uncertainty in a small number of observations but the uncertainty vanishes in a large number

Of observations goes back to the ancient world Aristotle made this observation but he didn't have probability theory and he couldn't carry it further so the idea of insurance the fundamental concept of

Insurance is is is relies on this intuitive idea and the idea was intuitive enough that insurance was known and practiced in ancient times but the insurance concept depends on independence and so the independence is

Something that apparently breaks down at times like these like these big down prices that we've seen in the stock market in the two episodes in the beginning of the 20th century so the law of large numbers has to do with the idea

That if I have a large number of random variables what is the variance the variance of X 1 plus X 2 plus X 3 up to X n is if all if there's if they're all independent then all the covariances are 0 so it equals the variance of X 1 plus

The variance of X 2 plus the variance of X n there's n terms I'm not showing them all ok so if they're all if they all have the same variance then the variance of the sum of n of them is n times their variance okay and that means the

Standard deviation which is the square root of the variance is equal to the square root of n times the standard deviation of one of them the mean is divided by n so that means that the standard deviation of the mean is is

Equal to the standard deviation of one of the X's divided by the square root of n so as n goes large you can see that this standard deviation of the mean goes to zero and that's the law of large numbers

Okay but the problem is so you know you can look at a financial firm and they have returns for a number of years and those returns can be accumulated to give some sense of their total outcome but does the total outcome really behave

Properly as does it become certain over a longer interval of time well apparently not because of the possibility that we are that the observations are not independent so you want to move from analysis of variance

To something that's more I told you that var came in in 1987 or thereabout after the stock market crash of 87 there's a new idea coming up now after the recent crisis and it's called Kovar and this is a concept emphasized by Professor Bruner

Meyer at Princeton and some of his colleagues that we have to change analysis of variance to recognize I've said we have to change Value at Risk to recognize that portfolios can sometimes Co vary more than we thought that there

Might be episodes when everything goes wrong at the same time so suddenly the covariance goes up so Kovar is a is an alternative to very to Value at Risk that does different kinds of calculations in the present environment

I think we recognize the need for that so this is the the aggregate stock market and let me go to another plot which shows both the same aggregate stock market that's this blue line down here and one stock the one

Stock I have shown is Apple the computer company and this is from the year 2000 this is just the first decade of the 20th century can you see this is my podium in the way for some of you the you might be surprised to say wait a

Minute did I hear you right is this blue line the same line that we just saw but you know I'll go back it is the same line it's just that I rescale it there it is it's a blue line this is the aggregate

This looks scary doesn't it the stock market lost something like almost half of its value we've dropped 40% between 2000 and 2002 Wow then it went all the way back up and then it dropped almost 50% these are

Scary numbers right but when I put Apple on the same plot the computer had this because Apple did such amazing things it had to compress and that's the same curve that you were just looking at it's just compressed so that I can plot it

Together I put both of them at 100 in the year 2000 so what I'm saying here is that somehow Apple did rather differently than the estimate this is the S&P 500 as a measure of the whole stock market and the the the Apple

Computer is that one of the breakout cases of dramatic dramatic success in investing it went up 25 times this incidentally is the adjusted price for Apple because in 2005 Apple did a two-for-one split you know what that

Means when when is my tradition in the United States stock should be worth about $30 per share and there's no reason why they should be $30 per share but a lot of companies when they when

The price hits $60 or something like that they say well let's just split all the shares in two so that they're back to $30 well Apple went up more than double but they only did one split in this period so we've corrected for the

Otherwise you can see a big apparent drop in their stock price on the day of the split are you with me on this but they it just it really doesn't matter it's just a units thing but you can see that an investment in Apple went up 25

Times whereas an investment in the other 8,500 went up only only it didn't go off actually to it it's down so now now this is a plot showing the monthly returns on Apple now it's only the capital gain returns I didn't include dividend but it

Is essentially the return on these two on the S&P 500 and on Apple now this is the same data that you were just looking at but it looks really different now doesn't it it looks really different they're unrecognizable as the same thing

You can't tell from this plot that Apple went up 25 fold that matters a lot to an investor and maybe you can if you've got very good eyes there's more up ones and there are down one up months and downwards there's a huge number of

Enormous variability in the month but I'd like to look at the picture like this because it conveys to me the incredible complexity of the story what was driving Apple up and down so many times it's only a pretty simple picture

By Apple and your money will go up 25 fold incidentally if you were a precocious teenager and you told your parents 10 years ago okay were you were you into this then but just imagine you say mom let's take

Out a four hundred thousand dollar mortgage on the house and put it all in Apple stock okay okay your parents would thank you today if you told them to do that your parents could do that they could

Take they've probably paid off their mortgages right they could go and get a second mortgage easily come up with four hundred thousand dollars most of you houses would be worth it so what would it be worth today maybe ten million

Dollars your father would be your mother would be saying you know I've been working all ten years and your little advice just got me 10 million is more than I made and all those much more than I made in all those years so these kind

Of stories attract attention but you know it wasn't an even ride that's that story seems too good to be true doesn't it I mean 25 fold the reason why it's not so obvious is that the write the ride was your as you're observing this

Happen every month it goes opposite it's just so big swings you make 30 percent in one month do those 30 percent in another month it's a scary ride and you can't see it happening unless you look at your portfolio and see what you can't

Tell it there's just so much randomness from one month to the other but incidentally I I had I was a dinner speaker last night for Yale alumni dinner in New York City and I wrote in with Peter Salovey who was Provost of

Yale and on the ride back he reminded me of a story that I think I heard but it took me a while to I remember this do but I'll tell you that it's an important Yale story and that is that in 1979 the the Yale class of 1954 had a 25th

Reunion okay this is history do you know the story do know where I'm heading so somebody said you know we're here at this reunion there's a lot of us here that's all as an experiment let's chip in some money and ask an investor to

Take a risky portfolio investment for Yale and let's give it to Yale on our 50th anniversary all right sounds like fun so they got a portfolio manager his name was Joe McNay and they said and they put together it was three

Hundred and seventy-five thousand dollars well it's like one house you know for the whole class of 1954 no big deal so they gave Joe McNeil McNay a three hundred and seventy five thousand dollar start and they said just have fun

With this we you know we're not conservative if you lose the whole thing go ahead but just go for maximum return on this so Joe McNay decided to invest in Home Depot Walmart and internet stock okay and under 50th reunion that was

2004 they presented Yale University with ninety million dollars that's that's an amazing stories and but I'm sure it was the same sort of thing same kind of rollercoaster ride the whole time it and now that we're trying to decide is joe

Mcnally Gnaeus what do you think is he a genius the I think maybe he is but the other side of it is I've just told you what to do in just a few words it's Walmart Home Depot and internet stocks and the other thing is they sold he

Started liquidating in 2000 right at the peak of the market so it must be partly luck but the the thing is should he have how did he know that Walmart was a good investment in 1954 I don't know it's sort of he took he took the risks

Maybe that's right I'm just digressing a little bit to think about the way things go in history it seems that I talked about the Forbes 400 people and I was presenting that I mentioned last lecture about Andrew

Carnegie's gospel of wealth and he says that some people are just very talented and they make it really big we should let them then give their money away it's kind of the American idea that we let talented people prove themselves in the

Real marketplace and then they end up becoming philanthropists and guiding our society but maybe they're just lucky and it's just no one could have known that Walmart was going to be such a success and I think that history is like that

The people you read about in history these great men and women of history are often just phenomenal risk takers like Joe McNay and for every one of them that you read about there's a thousand of them that got squashed and I was reading

The history of Julius Caesar as written by Plutarch it's a wonderful story and I was reading this I thought this guy is a real risk taker you know read all the details of his life he just went for it every time and he ended up emperor of

Rome but you know what happened to him he got assassinated and so it was you know it turned out not entirely a happy story so maybe it's all those poor all those ordinary people who live in a little house four hundred thousand

Dollar house they don't risk it maybe there are the smart ones you just don't ever hear of them well these are issues for finance but you wonder what are all of these things here all of these big movements

I tried to get this is the worst one here where it lost about a third of its value in one month and I researched it what was it you anyone know what caused it in 2008 well I'll tell you what caused Apple to lose a third of its

Value in one month Steve Jobs who is the founder of Apple and genius behind the and he gave a or is that an annual meeting or press conference and people said he doesn't look well and so they recall that he had

Pancreatic cancer in 2004 but the doctors then said it's curable no problem so the stock didn't do anything but reporters called Apple and said is he okay and the company spokesman wouldn't say anything so it started a

Rumor mill that Steve Jobs was dying of cancer it quickly rebounded because he wasn't that's how crazy these things are his market movements so now the next plot and this is important for our concept here I can pull out the same

Data in different ways and this shows a different sort of complexity now let me just review what we've seen here we started out with Apple stock this is the stock price normalized to one hundred and two thousand okay and it goes up to

Twenty five then the next thing I did is I did capital gains as a percent the percentage increase in price for each month it looks totally different and it shows such complexity that I can't tell a simple narrative or I've

Just told you about one blip here but there were so many of these blips on the way and they all have some story about the success of some Apple product or people aren't buying some product every month looks different but now what I

Want to do and I'm here the blue line is the return on the S&P 500 not what I want to do is plot a different sort of plot it's a scatter plot I'm going to plot the return on the Apple against the return on S&P 500 okay do you know what

I'm referring to here so this is this is a scatter plot on the vertical axis I have the return it's actually the capital gain on Apple and on the horizontal axis I have the capital gain on the whole stock market

Okay and each point represents one of the points that we saw on the on the mark actually I think it was I was telling you the second-lowest story Steve Jobs I'm not sure which point it was one of these points in 2008 was when

Steve Jobs looked sick but so each point is a month and there I have the whole decade of 2000 of the beginning of the 2000s plotted so the the best success was in December January of 2001 where the stock price went up 50% in one month

I try to figure out what that was about why did it go up 50% in one month it turns out that the preceding two months it had gone down a lot there down here somewhere there were these big drops and people

Are getting really pessimistic because Apple products weren't going well they had introduced some new products and they MobileMe I think you forget about these products that don't work it didn't work very well and then somehow people

Decided what really wasn't so bad and so we have plus 50 almost 50% return in one month now the reason why it looks kind of compressed on this way is because the stock market doesn't move as much as Apple so basically Apple return is the

Sum of of two components which is the overall market return and the idiosyncratic return okay so the return for a stock for the I stock is equal to the market return which is represented here by the S&P 500 which is the pretty

Much the whole stock market plus video syncretic return okay and if they're independent of each other the variance of the sum is the sum of the variances so the variance of the stock return is the variance of the

Variance of the Apple return is the sum of their market return and their idiosyncratic return well let me be clear about that let's add a regression line to the scatter point okay it's the same scatter that you saw is it clear

Everyone clear what we're doing here I've got s and P on this axis and Apple on this axis and now I've added a line which is a least square fit which minimizes the sum of squared deviations from the line it tries to get through

The scatter of points as much as it can and the the line has a slope of one point four five and we call that the beta all right these are these are concepts that well I'm asking

Elon to elaborate for you in the review session but it's a simple idea here what it means is that it seems like Apple shows a magnified response to the stock market it goes up and down approximately one and a half times as much as the

Stock market does on any day so the market return here is equal to the beta times the return on the ESPY that you see here so I wonder why that is why does Apple respond more than one for one with the

Stock market I guess it's because the aggregate economy matters right if you think that maybe because Apple is kind of a vulnerable company that if the economy tanks Apple will tank even more than the economy then the aggregate

Economy because they're such a volatile dangerous strategy company and if the if the market goes up then the it's even better news for Apple but and so but even so that the idiosyncratic risk just dominates look at these observations way

Up and way down here Apple has a lot of idiosyncratic risk and I mentioned one example it's it's Steve Jobs health the the Steve Jobs story is remarkable he founded Apple and Apple prospered and then he kind of had a falling out with

The management and got kind of kicked out of his own company and he this all right I'll start my own computer company my second I'll do it again so he found it next computer but meanwhile Apple started to really tank

This was in the 90s and they finally realized they needed Steve Jobs so they they brought him back so it's the company's ups and downs the idiosyncratic risk has a lot to do with Steve Jobs and what he does the mistakes

He made those are what causes these big these big movements this line I thought would he have an even higher beta but I think it's this point which is bringing the beta down and this is I think this is the point the month after it turns

Out that Steve Jobs really wasn't sick okay and it turned out to be the same month that the Lehman Brothers collapse occurred so you see this point here is between September and October of 2008 and that's the point it was September 15

That we had the most significant bankruptcy in US history Lehman Brothers the investment bank went bankrupt and through the whole world in chaos so the stock market in S&P 500 stock market return was minus 16% in one month

Horrible drop but for Apple it really was only about minus 5% because they were getting over the news of Steve Jobs so that's the way things that's the way things work now I wanted to move on now to the next topic which

Is outliers and talk about another assumption that is made in finance traditionally that turned out to be wrong in this episode and the assumption is that random shocks to the financial economy are normally distributed now you

Must have heard of the normal distribution this is the I have Annabelle shake the famous bell-shaped curve that was discovered by the mathematician Gauss over a hundred years ago the the bell-shaped curve is thought

To be this particular bell-shaped curve which is the in the log of this curve is a parabola it's a particular mathematical function the the curve is thought by statisticians to recur in nature many different ways it has a

Certain probability law so I have plotted two normal distributions and I have them for two different standard deviations one of one of them the black line is the standard deviation of three and the other one the pink line is the

Standard deviation of one but they both look the same it just scaled differently and these distributions have the property that the area under the curve is equal to one and the area between any two points

We say between minus 5 and minus 10 the area under this curve is the probability that the random variable falls between minus 5 and minus 10 so a lot of probability theory works on the assumption that variables are random are

Normally distributed but random variables have a habit of not behaving that way especially in finance it seems and so we had a mathematician here in the Yale math department then while Mandelbrot who was really the discoverer

Of this concept and I think the most important figure in it he said that in nature the normal distribution is not the only distribution that occurs and that especially in certain kinds of circumstances we have more fat-tailed

Distributions so this blue line is the normal distribution and the pink line that I've shown isn't is a fat-tailed distribution that Mandelbrot talked about called the Cauchy distribution and you see how it differs the pink line

Looks pretty much the same they're both bell-shaped curves right but the pink line has tremendously large probability of being far out these are the tails of the distribution so if you observe a random variable that looks you observe

It for a while maybe get a hundred observations you probably can't tell it apart very well from a normal distribution whether it's Koshi your normal they look about the same the way you find out that they're not the same

Is that an extremely rare circumstances there will be a sudden major jump in the variable that you might have thought couldn't happen so I have here a a plot of a histogram of stock price moon from 1928 everyday I've taken every day

Since 1928 and I've shown what the S&P composite index I it didn't have 500 stocks in 1928 so I can't call it the S&P 500 for the whole period but this is a essentially the S&P 500 and I have everyday there's something like 40,000

Days and what this line here shows is that the stock return the percentage change in the stock price in one day was between zero and one percent over nine thousand times and it was between zero and minus one percent around eight

Thousand times okay so that's typical day you know it's less than less than one percent up or down but occasionally we'll have a 2% day well this is between one and two percent that occurred about two thousand times and about two

Thousand times we had between minus 1 and minus 2% and then you can see that we've had you can see these outliers here these are look like outliers they're not extreme outliers so you if you look at a small number of data you

Would get an impression that well you know the stock market goes up between plus or minus two percent usually not so much and that's the way it is and after here they don't seem to be anything which means that it looks like you never

See anything more than up or down 5% or 6% it just doesn't happen well because it's so few days that it does those extreme can you see these little where it's that's between 5 & 6 there were there were maybe like 20 days

I can't read off the chart when it did this since 1928 you can go through 10 years on Wall Street and never see a drop of that magnitude so eventually you get kind of assured can't happen and some of what about it

An 8% drop well I look at this I said I've never seen that you know I've been watching this now I've seen thousands of days and I've never seen that but I have here the two extremes here the stock market went up 125 3% on October 30th

1929 that's the biggest one-day increase that's way off the charts and if you compute the normal distribution this what's the probability of that if it if a normal distribution that fits this central portion it would say zero a

Virtually zero it couldn't happen anyone any idea what happened on October 30th 1929 it's it's obvious to me but it's not obvious to you I'm asking you to I won't ask what happened then I told anyone know what happened in October

1929 yeah you're close you're right but someone is yeah absolutely it was the rebound after the crash the the stock market crash of 1929 had two consecutive days boy is that probability independence doesn't seem right it went

Down about 12% on October 28th and then the next day it did it again what's going on here we were down like 24 percent in two days if people got up on the 30th and say oh my god is he going to do that again but did just the

Opposite it was going totally wild so we don't know whether covariance broke down or not I guess it didn't because it rebounded that was the biggest one-day increase ever but that is if that weren't enough however let's go back to

October 19 1987 it went down twenty point four seven percent in one day it even went down even more on the DA some people say it went down more than that didn't it but on the S&P that's how much it went down so I figured what if this

Were normally distributed with the standard deviation suggested by this what's the probability of a decline that's that negative it's 10 to the minus 71 power 1 over 10 so you take one and you divide

That beat by one followed by 71 zeroes that's an awfully small number if you believe in normality October 19 1987 couldn't happen but there it is it happened and in fact I was I told you I've been teaching this course for 25

Years I was giving a lecture in this not in this room and nearby here and I was talking about something else and a student had a transistor radio remember transistor radios and he was holding it up I'm listening doing okay then he

Raised his hand and said do you know what's happening and he said the stock market is totally falling apart it just came as a complete surprise to me so after class I didn't go back to my office I went downstairs went downtown

The Merrill Lynch and I walked up so a story I like to tell it's not that anyway I walked up when I talked to a stockbroker there and I said I was about to say something but he didn't let me

Talk he said don't panic he he thought that I had shown up as a someone who was losing everything his life savings all in one day and he said don't worry it's not going to it's going to rebound so it didn't rebound because I showed up at

Lunchtime and it kept going down so so anyway there was something wrong with independence those let me just recap the two themes are the independence these two the law of large numbers it leads to some sort of stability either

Independence through time or independence across stocks so if you diversify through time or you diversify across stock you're supposed to be safe but that's not what happened in this crisis and that's the big question

And then it's fat tails which is kind of related but it's it that that distributions fool you you get big in credible shocks that you thought couldn't happen and they just come up with a certain low probability but with

A certain regularity in finance alright I'll stop there and see you on next Wednesday