Stochastic Models

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Stochastic Models

Postby blairfix » Wed Feb 03, 2016 11:46 pm

Using R, I've created a simulation of stock market gains/losses. The video can be viewed here:

https://youtu.be/XPzcmEesAM4

It's a simulation of 100,000 investors. During every iteration, each investor earns a random rate of return chosen from a normal distribution with mean 0 and standard deviation 8.5%. All investors begin with $300. Most investors lose over the long term (due to the fact that a loss of x% is not recuperated by a gain of x%).

The left panel shows the evolution of the distribution of net worth, from very equal, to very unequal. The Gini index quantifies the growing inequality. The right panel shows the net worth of every individual. Note how a few investors win big.

The take home lesson is that differential returns will always lead to increasing inequality, in the absence of some stabilizing force.
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Re: Stochastic Models

Postby Jonathan Nitzan » Thu Feb 04, 2016 12:57 pm

Thank you for the very neat simulation.

1. I find the result of growing inequality out of randomness puzzling. Is there an intuitive explanation for this outcome?

2. How is the model bounded? If I understand you correctly, you start with a total market value of $3,000,000 distributed equally between 10,000 investors, each owning $300. You then assume that each investor earns returns drawn from a normal distribution with a mean of 0% and a standard deviation of 8.5%. My question is how this distribution squares with the total market value of $3,000,000: (1) Is this total contracting/expanding depending on the average growth rate in each period? Or (2) is the total market value fixed throughout, which means that the average (geometric?) growth rate across the 10,000 investors has to be 0% in every period? Is this difference important for your results?

3. Have you run this simulation using different parameters/assumptions? For example, would it make a difference if the average growth rate is positive rather than zero, which would be the case if the overall market is rising (I guess, my question is whether this system is linear or nonlinear)? What would be the result if the returns are not distributed normally? Etc.
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Re: Stochastic Models

Postby blairfix » Thu Feb 04, 2016 5:15 pm

I'll answer your questions in order:

1. I find the result of growing inequality out of randomness puzzling. Is there an intuitive explanation for this outcome?


My response required some math, so I've attached a pdf.
Multiplicative Randomness.pdf
(73.11 KiB) Downloaded 115 times


The short answer is no ... I think this stuff is unintuitive. But the mathematics is straightforward.

2. How is the model bounded?


The only bound is the distribution of growth rates. Growth rates come from an exogenous distribution, determined by me. The total market value of all investors will vary dynamically, depending on the parameters of the growth rate distribution.

3. Have you run this simulation using different parameters/assumptions?


Yes. I've made more videos.

Here is a model where market gains have a mean of 0 and a standard deviation of 1%: https://www.youtube.com/watch?v=u9HO-eBVVII

Inequality grows very slowly. The standard deviation controls differential growth which is what ultimately causes inequality.

Here is a model where market gains have a mean of 1% and a standard deviation of 2%:
https://www.youtube.com/watch?v=FJtTD4p62A8

Average net worth increases with time (as we might expect), but inequality also grows. The distribution propagates through "space".

Here is a model where market gains have a mean of 3% and a standard deviation of 8.5%:
https://www.youtube.com/watch?v=0OnDcpzk1Zo

Both average net worth and inequality increase rapidly.

So to summarize, the growth rate mean determines how quickly (if at all) the distribution will move through "space", while the growth rate standard deviation determines the rate of increase of income inequality.
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Re: Stochastic Models

Postby leedoran » Thu Feb 04, 2016 6:00 pm

I too find it elegant and fun and totally not intuitive, Blair. Many thanks...

Are there any parallels or models from natural (or social) systems that could help inform our (lack of) intuition for this simulation?

best,

L.
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Re: Stochastic Models

Postby blairfix » Thu Feb 04, 2016 9:22 pm

Hi Lee,

The basic problem with multiplicative randomness is that it is unstable. It is a basic property of differential exponential growth. By definition, all systems undergoing exponential growth are unstable.

In nature, all exponential growth is bounded by resource constraints, so all systems must evolve to a state of dynamic equilibrium. So we don't find examples of pure multiplicative randomness.

This was a problem for Gibrat, who proposed that income distribution could be explained in terms of a series of random, multiplicative "shocks" to individual income. Kalecki pointed out that this process was intrinsically unstable.

For instance, the distribution of income from dividends is quite stable over time. How can this be, given that stock markets are inherently a multiplicative process. One answer, first suggested by Kalecki, is that
rates of return are not independent of one's capital stock. There is good evidence to support this hypothesis.

Many studies have found that the standard deviation of rates of return declines with corporate size. This is intuitive. Penny stocks are highly unstable ... while blue chip stocks are very stable.

Now, no one gets rich by investing only in blue chip stocks. Instead, millionaires and billionaires are usually made at the fringes, investing in obscure stocks. But for every millionaire made, there are hundreds of people who lost their shirt investing in this way.

On the flip side, billionaires have relatively limited avenues for investment. Bill Gates has doesn't mess with penny stocks ... he would have to invest in hundreds of thousands of different companies. Instead, he invests in big corporations. As a result, his net worth is fairly stable, but unlikely to beat the average by a significant rate.

I've run simulations using rates of return from Compustat. Assuming all investors have access to all rates of return, we get wildly increasing income inequality over a very short period of time.

If we add the assumption that investors have a lower capital threshold below which they do not invest -- so they don't invest in companies with capitalization less than say 10% of the investor's net worth, then we get a stable distribution over time.

This is because we no longer have pure multiplicative randomness. Now, the volatility in rates of return is dependent on one's net worth.
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Re: Stochastic Models

Postby blairfix » Thu Feb 04, 2016 9:23 pm

Hi Lee,

The basic problem with multiplicative randomness is that it is unstable. It is a basic property of differential exponential growth. By definition, all systems undergoing exponential growth are unstable.

In nature, all exponential growth is bounded by resource constraints, so all systems must evolve to a state of dynamic equilibrium. So we don't find examples of pure multiplicative randomness.

This was a problem for Gibrat, who proposed that income distribution could be explained in terms of a series of random, multiplicative "shocks" to individual income. Kalecki pointed out that this process was intrinsically unstable.

For instance, the distribution of income from dividends is quite stable over time. How can this be, given that stock markets are inherently a multiplicative process. One answer, first suggested by Kalecki, is that
rates of return are not independent of one's capital stock. There is good evidence to support this hypothesis.

Many studies have found that the standard deviation of rates of return declines with corporate size. This is intuitive. Penny stocks are highly unstable ... while blue chip stocks are very stable.

Now, no one gets rich by investing only in blue chip stocks. Instead, millionaires and billionaires are usually made at the fringes, investing in obscure stocks. But for every millionaire made, there are hundreds of people who lost their shirt investing in this way.

On the flip side, billionaires have relatively limited avenues for investment. Bill Gates has doesn't mess with penny stocks ... he would have to invest in hundreds of thousands of different companies. Instead, he invests in big corporations. As a result, his net worth is fairly stable, but unlikely to beat the average by a significant rate.

I've run simulations using rates of return from Compustat. Assuming all investors have access to all rates of return, we get wildly increasing income inequality over a very short period of time.

If we add the assumption that investors have a lower capital threshold below which they do not invest -- so they don't invest in companies with capitalization less than say 10% of the investor's net worth, then we get a stable distribution over time.

This is because we no longer have pure multiplicative randomness. Now, the volatility in rates of return is dependent on one's net worth.
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Re: Stochastic Models

Postby blairfix » Sat Feb 20, 2016 5:54 pm

Why the Mind Abhors Pure Randomness

Over the past few weeks I've been thinking about why inequality generated by randomness is not intuitive. I've come to the conclusion that it is the result of the following:

1. Pure random processes have no "memory".
2. Humans have a memory, and it informs our expectations about the future. These expectations are biased towards fairness.

It is precisely the lack of memory that causes purely random process to generate dispersion and inequality. Let us illustrate with an example. Suppose a large group of people are given $100 at a casino.

Each person then gambles $1 on the outcome of a coin toss: heads they win, tails they lose. So after each flip, the individual either gains or loses a dollar (from the casino).

This is a random process with no memory. It is guaranteed to generated inequality. Why? It generates dispersion precisely because it is memoryless: the past has no effect on the future. A (fair) coin toss is always 50/50, no matter what the past outcomes have been.

Suppose that after 10 tosses, one lucky individual receives 10 heads, thus winning $10. They now have $110. In the next coin toss, they are still just as likely to win as they are to lose. Having won/lost more in the past has no bearing on winning/losing in the future. Our memory is "wiped" after every iteration. This cause a "random walk" to occur. Every new position becomes a new "home base", from which positive/negative deviations are equally likely.

But humans have a memory, and we have an intuition (instinct) for what ought to happen given our recollection of the past. That is, we expect the past to affect the future. In the above scenario, we expect that the initial conditions should be the home base. Past deviations away from home base "ought" to yield future deviations towards home base. This expectation, I believe, is the result of our instinctive desire for fairness. We think that winners "ought" to lose more in the future, and losers "ought" to win more. This keeps things fair.

When we apply this intuition to true randomness, it deceives us. Randomness is fair in the moment, but unfair over time. However, we expect fairness in the moment to correspond to fairness over time.

Mathematically speaking, the only way that randomness can be fair over time is if we introduce a memory to the process. In the above example, this could be accomplished by introducing coin bias that depends upon one's past winnings. Again, heads = win, tails = lose. But now we give a tails biased coin to those who have a winning record, and a heads biased coin to those who have a losing record.

Notice that now we do not have fairness in the moment. Each player has a different probability of receiving heads/tails. As a result of this unfairness in the moment, we get fairness over time, and a stable distribution of income. This fairness is the result of active (non-random) redistribution.

If a pure, memoryless random process is to occur in a society, the natural instinct for fairness must somehow be suppressed. I believe that the stock market is the perfect example. We must be convinced that stock market winners "deserve" their winnings. Our natural instinct for fairness (and non-random redistribution) must be suppressed. This must come from some rationalizing ideology.
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Re: Stochastic Models

Postby leedoran » Mon Feb 22, 2016 8:56 am

Many thanks Blair.

This all makes good sense to me … I like it a lot.

To try and summarize: our human morality (fairness guilt) that remembers and learns from the past defeats the forces of randomness so we get winners and losers. But the bottom line is that the rich get richer, the poor get poorer, and the middle class melts downward when this ‘natural’ process is given its head, i.e. free reign.

But we also know from the four decades of managed capitalism known as Keynesianism from the 1940’s through the 1970’s that government regulation works. To flatten the distribution, democratize the outputs and provide the middle class its measure of security stabilizes the system against its open-ended rush for the top by the top of the top (0.1%).

The lesson for today’s emerging capitalist crisis is to bring back such substantive regulation (updated), especially for the banks and financiers at the institutional apex of the pyramid who set the tone and devise the means for the lesser lights’ own scramble up the slippery sides of the pyramid.

My own ‘take’ on this is that the unfettered drive at the very top for the very top is fundamentally psychopathic. These are institutions without human morality (see Antifragile by Nassim Nicholas Taleb and The Corporation and many others, now) driven most often by individual ♂ psychopathic humans without the constraint of fairness guilt that ordinary human morality brings to most members of the species.

This is Ayn Rand capitalism that speaks to many aspiring ♂♂ still, daily. They come by it naturally for the T that fuels them works not only to acquire ♀ resources and attention (and servicing) directly for their personal reproductive needs but also drives them to seek out resources of all kinds that attract ♀♀. Hence their competition with their fellow ♂♂ because 20% of the total will never reproduce at all and 20% of them get 80% of the (limited) sex that is available. No wonder the middle 60% fight so hard and so diligently for the remaining 20%!

Keep up the good work!

Best,

L.

PS As ever, mine is based on scientific evidence, as referenced in the piece plus a dollop of standard textbook Evolutionary Psychology. For the morality link, here’s a snippet from my Curating Sex, Briefly on that:

We find that that very scientific method has turned our human understanding of human understanding exactly on its head. We borrow one of Jonathan Haidt’s metaphors to tell this part of the story. It is from his book The Righteous Mind. He speaks of the elephant and its rider, who one would expect to be in charge. Rider directs, elephant responds. Well, now it turns out the elephant is calling the shots. The rider is following. The rider is in the service of the brute.

That’s right, the Enlightenment reason that was in charge of the project is now said to be working at the behest of the elephant. To deconstruct: the elephant’s rider is reason, logical thinking, our conscious, scientific understanding of the world. The elephant is emotion, unconscious thinking, fast (vs slow) thinking, the autonomic nervous system.

Previously, the elephant had been largely unknown and mostly, unknowable. The philosophers, forever, and the psychologist-psychiatrists, more recently, had speculated and proposed and debated.

Now, the science of the elephant has arrived. Reason has discovered that it is the handmaiden of the unconscious, the emotional.

What directs the elephant, then? If it’s not the rider, then who -- or what -- is in charge, by this telling?
It appears to be a group of six – a sextet of principles – identified by Professor Haidt, as well. They are the foundations of morality, or moral systems. Together, they constitute a human universal, one common basis for all human behaviors in all human cultures. Our morality is built on a singular foundation in six parts.

They come in pairs:
1. Care vs harm --> Tend & befriend; Fight or flight
2. Fairness vs cheating --> Justice (Proportionality); Not
3. Liberty vs oppression --> Freedom; Tyranny (Violence)
4. Loyalty vs betrayal --> In-group; Out-group
5. Authority vs subversion --> Respect (Hierarchy); Not
6. Sanctity vs degradation --> Avoidance of contamination (Purity); Not
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