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Thrust of argument: Robert Shiller tells us <<

Jeremy Siegel says that in finance we should be using geometric and not arithmetic averages. Why is that? Well I'll tell you in very simple terms, I think. Suppose someone is investing your money and he announces, I have had very good returns. I have invested and I've produced 20% a year for nine out of the last ten years. You think that's great, but what about the last year. The guy says, "Oh I lost 100% in that year." You might say, "Alright, that's good." I would add up 20% a year for nine years and than put in a zero-no, 120 because it's gross return for nine years--and put in a zero for one year. Maybe that doesn't look bad, right? But think about it, if you were investing your money with someone like that, what did you end up with? You ended up with nothing. If they have one year when they lose everything, it doesn't matter how much they made in the other years. Jeremy says in the text that the geometric return is always lower than the arithmetic return unless all the numbers are the same. It's a less optimistic version. So, we should use that, but people in finance resist using that because it's a lower number and when you're advertising your return you want to make it look as big as possible.

We also need some measure of--We've been talking here about measures of central tendency only and in finance we need, as well, measures of dispersion, which is how much something varies. Central tendency is a measure of the center of a probability distribution of the--Central tendency is a measure--Variance is a measure of how much things change from one observation to another. We have variance and it's often represented by σ², that's the Greek letter sigma, lower case, squared. Or, especially when talking about estimates of the variance, we sometimes say S² or we say standard deviation². The standard deviation is the square root of the variance. For population variance, the variance of some random variable x is defined as the summation i = 1 to infinity of the Prob (x = xi) times (xi - μx)². So mu is the mean--we just defined it of x--that's the expectation of x or also E(x), so it's the probability weighted average of the squared deviations from the mean. If it moves a lot--either way from the mean--then this number squared is a big number. The more x moves, the bigger the variance is.

There's also another variance measure, which we use in the sample--or also Var is used sometimes--and this is ∑². There's also another variance measure, which is for the sample. When we have n observations it's just the summation i = 1 to n of (x - x bar)²/n. That is the sample variance. Some people will divide by n-1. I suppose I would accept either answer. I'm just keeping it simple here. They divide by n-1 to make it an unbiased estimator of the population variance; but I'm just going to show it in a simple way here. So you see what it is--it's a measure of how much x deviates from the mean; but it's squared. It weights big deviations a lot because the square of a big number is really big. So, that's the variance.

So, that completes central tendency and dispersion. We're going to be talking about these in finance in regards to returns because--generally the idea here is that we want high returns. We want a high expected value of returns, but we don't like variance. Expected value is good and variance is bad because that's risk; that's uncertainty. That's what this whole theory is about: how to get a lot of expected return without getting a lot of risk.

Another concept that's very basic here is covariance. Covariance is a measure of how much two variables move together. Covariance is--we'll call it--now we have two random variables, so I'll just talk about it in a sample term. It's the summation i = 1 to n of [(x - x-bar) times (y - y-bar)]/n. So x is the deviation for the i-subscript, meaning we have a separate xi and yi for each observation. So we're talking about an experiment when you generate--Each experiment generates both an x and a y observation and we know when x is high, y also tends to be high, or whether it's the other way around. If they tend to move together, when x is high and y is high together at the same time, then the covariance will tend to be a positive number. If when x is low, y also tends to be low, then this will be negative number and so will this, so their product is positive. A positive covariance means that the two move together. A negative covariance means that they tend to move opposite each other. If x is high relative to x-bar--this is positive--then y tends to be low relative to its mean y-bar and this is negative. So the product would be negative. If you get a lot of negative products, that makes the covariance negative.

Then I want to move to correlation. So this is a measure--it's a scaled covariance. We tend to use the Greek letter rho. If you were to use Excel, it would be correl or sometimes I say corr. That's the correlation. This number always lies between -1 and +1. It is defined as rho= [cov(xiyi)/SxSy] That's the correlation coefficient. That has kind of almost entered the English language in the sense that you'll see it quoted occasionally in newspapers. I don't know how much you're used to it--Where would you see that? They would say there is a low correlation between SAT scores and grade point averages in college, or maybe it's a high correlation. Does anyone know what it is? But you could estimate the corr--it's probably positive. I bet it's way below one, but it has some correlation, so maybe it's .3. That would mean that people who have high SAT scores tend to get higher grades. If it were negative--it's very unlikely that it's negative--it couldn't be negative. It couldn't be that people who have high SAT scores tend to do poorly in college. If you quantify how much they relate, then you could look at the correlation.

I want to move to regression. This is another concept that is very basic to statistics, but it has particular use in finance, so I'll give you a financial example. The concept of regression goes back to the mathematician Gauss, who talked about fitting a line through a scatter of points. Let's draw a line through a scatter of points here. I want to put down on this axis the return on the stock market and on this axis I want to put the return on one company, let's say Microsoft. I'm going to have each observation as a year. I shouldn't put down a name of a company because I can't reproduce this diagram for Microsoft. Let's not say Microsoft, let's say Shiller, Inc. There's no such company, so I can be completely hypothetical. Let's put zero here because these are not gross returns these are returns, so they're often negative. Suppose that in a given year--and say this is minus five and this is plus five, this is minus five and this is plus five--Suppose that in the first year in our sample, the company Shiller, Inc. and the market both did 5%. That puts a point right there at five and five. In another year, however, the stock market lost 5% and Shiller, Inc. lost 7%. We would have a point, say, down here at five and seven. This could be 1979, this could be 1980, and we keep adding points so we have a whole scatter of points. It's probably upward sloping, right? Probably when the overall stock market does well so does Shiller, Inc.

What Gauss did was said, let's fit a line through the point--the scatter of points--and that's called the regression line. He chose the line so that--this is Gauss--he chose the line to minimize the sum of squared distances of the points from the lines. So these distances are the lengths of these line segments. To get the best fitting line, you find the line that minimizes the sum of squared distances. That's called the regression line and the intercept is called alpha--there's alpha. And the slope is called beta. That may be a familiar enough concept to you, but in the context of finance this is a major concept. The way I've written it, the beta of Shiller, Inc. is the slope of this line. The alpha is just the intercept of this curve. We can also do this with excess returns. I will get to this later, where I have the return minus the interest rate on this axis and the market return minus the interest rate on this axis. In that case, alpha is a measure of how much Shiller, Inc. outperforms. We'll come back to this, but beta of the stock is a measure of how much it moves with the market and the alpha of a stock is how much it outperforms the market. We'll have to come back to that--these are basic concepts.

I want to--another concept--I guess I've just been implicit in what I have--There's a distribution called the normal distribution and that is--I'm sure you've heard of this, right? If you have a distribution that looks like this--it's bell-shaped--this is x and--I have to make it look symmetric which I may not be able to do that well--this is f(x), the normal distribution. f(x) = [1/(√ (2π)σ)] times e to minus [(x-μ)2 / 2σ]. It's a famous formula, which is due to Gauss again. We often assume in finance that random variables, such as returns, are normally distributed. This is called the normal distribution or the Gaussian distribution--it's a continuous distribution. I think you've heard of this, right? This is high school raw material. But I want to emphasize that there are also other bell-shaped curves. This is the most famous bell-shaped curve, but there are other ones with different mathematics.

A particular interest in finance is fat-tailed alternatives. It could be that a random distribution--I don't have colored chalk here I don't think, so I will use a dash line to represent the fat-tailed distribution. Suppose the distribution looks like this. Then I have to try to do that on the other side, as symmetrically as I can. These are the tails of the distribution; this is the right tail and this is the left tail. You can see that the dash distribution I drew has more out in the tails, so we call it fat-tailed. This refers to random variables that have fat-tailed distributions--random variables that occasionally give you really big outcomes. You have a chance of being way out here with a fat-tailed distribution. It's a very important observation in finance that returns on a lot of speculative assets have fat-tailed distributions. That means that you can go through twenty years of a career on Wall Street and all you've observed is observations in the central region. So you feel that you know pretty well how things behave; but then, all of a sudden, there's something way out here. This would be good luck if you were long and now suddenly you got a huge return that you would not have thought was possible since you've never seen it before. But you can also have an incredibly bad return. This complicates finance because it means that you never know. You never have enough experience to get through all these things. It's a big complication in finance.

My friend Nassim Talib has just written a book about it called--maybe I'll talk about that--called The Black Swan. It's about how so many plans in finance are messed up by rare events that suddenly appear out of nowhere. He called it The Black Swan because if you look at swans, they're always white. You've never seen a black swan. So, you end up going through life assuming that there are no black swans. But, in fact, there are and you might finally see one. You don't want to predicate making complicated gambles under the assumption that they don't exist. Talib, who's a Wall Street professional, talks about these black swans as being the real story of finance. >>
Direction of resistance / implied resistance: Money is a problematic and outmoded concept, an extension of jungle law and I absolutely advocate for its eradication and for a utopian society where our resources are managed in ways which match our capacity for extreme advancement which, collectively, is very high right now, as Chomsky has advised us, even though ironically Feynman to some extent warned of the opposite (nothing had changed, he said, since before the war - the existence of atomic weaponry, merely the existence, spelled the end for all humanity). We can do whatever we want, we can make a great society, Chomsky, in among his many lectures, has strongly advised us and still does, repeatedly, to this day on which I am writing this and possibly, depending on the details, when you, reader, are reading this.

Nonetheless, it exists and financial systems exist and, though some may well criticise my choice of example, George Soros is out there trying to use money to help take us out of a society which requires the existence of money. He had to set the ball rolling, others will have to take it much further than he. Apparently he started poor and became rich. This may explain his, in my view, lack of fear of a post-money future. Plus he's very very old and won't have to face it himself, only via his 'descendants'.

 

 

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Removal of resistance: And so accepting that hopefully we will soon see the absolute and enlightened (and peaceful) end to money, the economic systems and financial 'trading' / 'speculation' (it can be both, either or neither and still win or lose, consistently, depending on you and your maths, not on anything else - more proof that it is essentially a valueless entity, not benefiting humanity or any other life form, when it comes right down to it, still part of our past jutting out through the present and into too much of our future) - if one, nonetheless, wanted to master financial markets, face them in any way and derive short term benefit from them, eg in order to rebalance the distribution of power which itself may or may not block the development of a post-money society, that stretch of Shiller is extremely important. Unification: My own methods work only because they absolutely do not fall down when tested against the requirements set out there by Shiller, which is no surprise as I listened to him give that lecture many times over before I'd even begun to try and figure out the making of currency and stock trading algorithms which 'beat the market'. (Some equations shown above not properly represented on this page, view original in references for exact articulation).
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Beat the devil, beat the market, but you can never beat the maths.

Robert Shiller tells us <<

Jeremy Siegel says that in finance we should be using geometric and not arithmetic averages. Why is that? Well I'll tell you in very simple terms, I think. Suppose someone is investing your money and he announces, I have had very good returns. I have invested and I've produced 20% a year for nine out of the last ten years. You think that's great, but what about the last year. The guy says, "Oh I lost 100% in that year." You might say, "Alright, that's good." I would add up 20% a year for nine years and than put in a zero-no, 120 because it's gross return for nine years--and put in a zero for one year. Maybe that doesn't look bad, right? But think about it, if you were investing your money with someone like that, what did you end up with? You ended up with nothing. If they have one year when they lose everything, it doesn't matter how much they made in the other years. Jeremy says in the text that the geometric return is always lower than the arithmetic return unless all the numbers are the same. It's a less optimistic version. So, we should use that, but people in finance resist using that because it's a lower number and when you're advertising your return you want to make it look as big as possible.

We also need some measure of--We've been talking here about measures of central tendency only and in finance we need, as well, measures of dispersion, which is how much something varies. Central tendency is a measure of the center of a probability distribution of the--Central tendency is a measure--Variance is a measure of how much things change from one observation to another. We have variance and it's often represented by σ², that's the Greek letter sigma, lower case, squared. Or, especially when talking about estimates of the variance, we sometimes say S² or we say standard deviation². The standard deviation is the square root of the variance. For population variance, the variance of some random variable x is defined as the summation i = 1 to infinity of the Prob (x = xi) times (xi - μx)². So mu is the mean--we just defined it of x--that's the expectation of x or also E(x), so it's the probability weighted average of the squared deviations from the mean. If it moves a lot--either way from the mean--then this number squared is a big number. The more x moves, the bigger the variance is.

There's also another variance measure, which we use in the sample--or also Var is used sometimes--and this is ∑². There's also another variance measure, which is for the sample. When we have n observations it's just the summation i = 1 to n of (x - x bar)²/n. That is the sample variance. Some people will divide by n-1. I suppose I would accept either answer. I'm just keeping it simple here. They divide by n-1 to make it an unbiased estimator of the population variance; but I'm just going to show it in a simple way here. So you see what it is--it's a measure of how much x deviates from the mean; but it's squared. It weights big deviations a lot because the square of a big number is really big. So, that's the variance.

So, that completes central tendency and dispersion. We're going to be talking about these in finance in regards to returns because--generally the idea here is that we want high returns. We want a high expected value of returns, but we don't like variance. Expected value is good and variance is bad because that's risk; that's uncertainty. That's what this whole theory is about: how to get a lot of expected return without getting a lot of risk.

Another concept that's very basic here is covariance. Covariance is a measure of how much two variables move together. Covariance is--we'll call it--now we have two random variables, so I'll just talk about it in a sample term. It's the summation i = 1 to n of [(x - x-bar) times (y - y-bar)]/n. So x is the deviation for the i-subscript, meaning we have a separate xi and yi for each observation. So we're talking about an experiment when you generate--Each experiment generates both an x and a y observation and we know when x is high, y also tends to be high, or whether it's the other way around. If they tend to move together, when x is high and y is high together at the same time, then the covariance will tend to be a positive number. If when x is low, y also tends to be low, then this will be negative number and so will this, so their product is positive. A positive covariance means that the two move together. A negative covariance means that they tend to move opposite each other. If x is high relative to x-bar--this is positive--then y tends to be low relative to its mean y-bar and this is negative. So the product would be negative. If you get a lot of negative products, that makes the covariance negative.

Then I want to move to correlation. So this is a measure--it's a scaled covariance. We tend to use the Greek letter rho. If you were to use Excel, it would be correl or sometimes I say corr. That's the correlation. This number always lies between -1 and +1. It is defined as rho= [cov(xiyi)/SxSy] That's the correlation coefficient. That has kind of almost entered the English language in the sense that you'll see it quoted occasionally in newspapers. I don't know how much you're used to it--Where would you see that? They would say there is a low correlation between SAT scores and grade point averages in college, or maybe it's a high correlation. Does anyone know what it is? But you could estimate the corr--it's probably positive. I bet it's way below one, but it has some correlation, so maybe it's .3. That would mean that people who have high SAT scores tend to get higher grades. If it were negative--it's very unlikely that it's negative--it couldn't be negative. It couldn't be that people who have high SAT scores tend to do poorly in college. If you quantify how much they relate, then you could look at the correlation.

I want to move to regression. This is another concept that is very basic to statistics, but it has particular use in finance, so I'll give you a financial example. The concept of regression goes back to the mathematician Gauss, who talked about fitting a line through a scatter of points. Let's draw a line through a scatter of points here. I want to put down on this axis the return on the stock market and on this axis I want to put the return on one company, let's say Microsoft. I'm going to have each observation as a year. I shouldn't put down a name of a company because I can't reproduce this diagram for Microsoft. Let's not say Microsoft, let's say Shiller, Inc. There's no such company, so I can be completely hypothetical. Let's put zero here because these are not gross returns these are returns, so they're often negative. Suppose that in a given year--and say this is minus five and this is plus five, this is minus five and this is plus five--Suppose that in the first year in our sample, the company Shiller, Inc. and the market both did 5%. That puts a point right there at five and five. In another year, however, the stock market lost 5% and Shiller, Inc. lost 7%. We would have a point, say, down here at five and seven. This could be 1979, this could be 1980, and we keep adding points so we have a whole scatter of points. It's probably upward sloping, right? Probably when the overall stock market does well so does Shiller, Inc.

What Gauss did was said, let's fit a line through the point--the scatter of points--and that's called the regression line. He chose the line so that--this is Gauss--he chose the line to minimize the sum of squared distances of the points from the lines. So these distances are the lengths of these line segments. To get the best fitting line, you find the line that minimizes the sum of squared distances. That's called the regression line and the intercept is called alpha--there's alpha. And the slope is called beta. That may be a familiar enough concept to you, but in the context of finance this is a major concept. The way I've written it, the beta of Shiller, Inc. is the slope of this line. The alpha is just the intercept of this curve. We can also do this with excess returns. I will get to this later, where I have the return minus the interest rate on this axis and the market return minus the interest rate on this axis. In that case, alpha is a measure of how much Shiller, Inc. outperforms. We'll come back to this, but beta of the stock is a measure of how much it moves with the market and the alpha of a stock is how much it outperforms the market. We'll have to come back to that--these are basic concepts.

I want to--another concept--I guess I've just been implicit in what I have--There's a distribution called the normal distribution and that is--I'm sure you've heard of this, right? If you have a distribution that looks like this--it's bell-shaped--this is x and--I have to make it look symmetric which I may not be able to do that well--this is f(x), the normal distribution. f(x) = [1/(√ (2π)σ)] times e to minus [(x-μ)2 / 2σ]. It's a famous formula, which is due to Gauss again. We often assume in finance that random variables, such as returns, are normally distributed. This is called the normal distribution or the Gaussian distribution--it's a continuous distribution. I think you've heard of this, right? This is high school raw material. But I want to emphasize that there are also other bell-shaped curves. This is the most famous bell-shaped curve, but there are other ones with different mathematics.

A particular interest in finance is fat-tailed alternatives. It could be that a random distribution--I don't have colored chalk here I don't think, so I will use a dash line to represent the fat-tailed distribution. Suppose the distribution looks like this. Then I have to try to do that on the other side, as symmetrically as I can. These are the tails of the distribution; this is the right tail and this is the left tail. You can see that the dash distribution I drew has more out in the tails, so we call it fat-tailed. This refers to random variables that have fat-tailed distributions--random variables that occasionally give you really big outcomes. You have a chance of being way out here with a fat-tailed distribution. It's a very important observation in finance that returns on a lot of speculative assets have fat-tailed distributions. That means that you can go through twenty years of a career on Wall Street and all you've observed is observations in the central region. So you feel that you know pretty well how things behave; but then, all of a sudden, there's something way out here. This would be good luck if you were long and now suddenly you got a huge return that you would not have thought was possible since you've never seen it before. But you can also have an incredibly bad return. This complicates finance because it means that you never know. You never have enough experience to get through all these things. It's a big complication in finance.

My friend Nassim Talib has just written a book about it called--maybe I'll talk about that--called The Black Swan. It's about how so many plans in finance are messed up by rare events that suddenly appear out of nowhere. He called it The Black Swan because if you look at swans, they're always white. You've never seen a black swan. So, you end up going through life assuming that there are no black swans. But, in fact, there are and you might finally see one. You don't want to predicate making complicated gambles under the assumption that they don't exist. Talib, who's a Wall Street professional, talks about these black swans as being the real story of finance. >>

Money is a problematic and outmoded concept, an extension of jungle law and I absolutely advocate for its eradication and for a utopian society where our resources are managed in ways which match our capacity for extreme advancement which, collectively, is very high right now, as Chomsky has advised us, even though ironically Feynman to some extent warned of the opposite (nothing had changed, he said, since before the war - the existence of atomic weaponry, merely the existence, spelled the end for all humanity). We can do whatever we want, we can make a great society, Chomsky, in among his many lectures, has strongly advised us and still does, repeatedly, to this day on which I am writing this and possibly, depending on the details, when you, reader, are reading this.

Nonetheless, it exists and financial systems exist and, though some may well criticise my choice of example, George Soros is out there trying to use money to help take us out of a society which requires the existence of money. He had to set the ball rolling, others will have to take it much further than he. Apparently he started poor and became rich. This may explain his, in my view, lack of fear of a post-money future. Plus he's very very old and won't have to face it himself, only via his 'descendants'.

And so accepting that hopefully we will soon see the absolute and enlightened (and peaceful) end to money, the economic systems and financial 'trading' / 'speculation' (it can be both, either or neither and still win or lose, consistently, depending on you and your maths, not on anything else - more proof that it is essentially a valueless entity, not benefiting humanity or any other life form, when it comes right down to it, still part of our past jutting out through the present and into too much of our future) - if one, nonetheless, wanted to master financial markets, face them in any way and derive short term benefit from them, eg in order to rebalance the distribution of power which itself may or may not block the development of a post-money society, that stretch of Shiller is extremely important.

My own methods work only because they absolutely do not fall down when tested against the requirements set out there by Shiller, which is no surprise as I listened to him give that lecture many times over before I'd even begun to try and figure out the making of currency and stock trading algorithms which 'beat the market'. (Some equations shown above not properly represented on this page, view original in references for exact articulation).



http://openmedia.yale.edu/projects/iphone/departments/econ/econ252/transcript02.html
https://www.youtube.com/watch?v=WMkD8HKJQCM