Stock Market Asymmetry and It's Implications

The dirty little secret of mainstream financial gurus.

Do you consider yourself an investor? Do you have a 401K or Defined Benefit retirement plan? Many people’s retirement funds are professionally managed and invested directly in stocks and bonds. Even if you never directly participate in the markets, you are affected by them, and your money isn’t as safe as mainstream financial advisors would have you believe.

Everyone knows you can take a loss in the market, as we have all been reminded the past few years. The critical omission is that this is guaranteed to happen. This post focuses on stock market asymmetry and what it means for the average investor.

Below is a chart I created plotting expected performance of the stock market versus actual for the past 80 years. The chart shows the total number of years the market was within 1, 2, or 3 standard deviations.


 

The chart illustrates the positive bias of the stock market. More importantly for this discussion, is the “fat tail” at the -3 sigma point. Normal Distribution would predict a -3 sigma event 1.8 years out of 80. However, there were actually 3 years when the market lost over 29.9% (3 standard deviations). Statisticians would argue this sample is not statistically valid due to the small sample size, and this argument is correct. However, it has been shown (by others) that this asymmetry holds no matter the time period. It is true for daily, minute and even tick values.

The stock market more closely follows a Power Law distribution rather than a Normal Distribution, but almost all common stock market tools, including the Black-Scholes option pricing model, use the Normal Distribution, ignoring the greater-than-expected risk of substantial losses. This is the dirty little secret of mainstream financial gurus.

 

The fact that the market does not follow Normal Distribution (the Bell Curve), has several important ramifications.

1.    In order to realize something near the 10.95% mean, market timing must be correct. Retirees do not have the flexibility of waiting +/- 15-20 years to achieve optimal entry-exit timing. If the market is in a much-greater-than-expected slump at retirement time, you lose. Survivors of the 1929 crash had to wait 40 years to recover their losses. Who can  postpone retirement an extra 40 years?

2.    The stock market is MUCH more unpredictable than commonly acknowledged or accounted for.

3.    Most investment mangers lose more money for their clients than they make. This sounds incredible but is true. Managers make a little bit of profit each year until a >3 Sigma event occurs (at a rate much higher than expected due to Power Law distribution of the market) and they lose more for their clients than they have ever made for them. I will try to elaborate on this proof in a future post.

4.    When mainstream financial experts claim you can expect x% return in the stock market over time, they can only mean one of two dishonest things; either “over time” really means an unknown (and unknowable) variable amount of time, or; they are citing a mythical Normal Distribution market.

5.    It is interesting to note, that in the financial gurus make-believe ND stock market world, Black Monday 1987 and the 2008 market meltdown are statistically impossible. Their models do not allow for these events yet they happened.

6.    Day trading is generally considered risky and perhaps much like gambling. The main reason is because optimal market timing is so important and it is infamously difficult to achieve. For investment and retirement funds the time period is the only difference. The importance of timing is exactly the same. Therefore, investing in a long term equities fund is fundamentally no different than the shadowy world of day trading.

There are better ways to invest that greatly decrease risk.

The Hidden Mathematics Of Drug Testing

Imagine the following scenario:

There exists a drug test that is 99% accurate on both drug users and non-drug users; If 100 drug users are tested, 99 will test correctly positive and 1 will incorrectly test negative. If 100 non-drug users are tested, 99 will correctly test negative and 1 incorrectly positive (false positive).

Now, imagine a group of 1000 people, (workers, welfare recipients, whatever) whose rate of drug use is 0.5%. One individual from this group is chosen at random and tested. The test is positive. Most people would say that the probability of that individual being an actual drug user is 99%. The test is 99% accurate, right?

Wrong

The probability that the tested person is really a drug user is ~33%. In other words, it is more likely the person is not a drug user, even though the "99% accurate" test was positive. At first, this may sound implausible. But it's not. Why?

The absolute number of non-drug users is much larger than users. The number of false positives (0.995%) outweighs the number of true positives (0.495%).

Substituting real numbers;

1000 individuals are tested
There should be 995 non-users and 5 users.
From the 995 non-users, 0.01 × 995 ≃ 10 false positives (1% of the 995)
From the 5 users, 0.99 × 5 ≃ 5 true positives (99% of 5)
The total of all positives, 10+5 = 15. Of these 15 positive results, only 5, about 33%, are genuine.

This is just one of many cases where mathematics is misused in public policy.  Politicians and self-righteous people can claim "We should drug test population x, the tests are 99% right! Very few people will be falsely accused!" As I have demonstrated above, this is simply not true.
The same kind of misuse of mathematics is used in DNA sampling also, with literal life and death consequences. See Misused Mathematics of DNA Sampling.

Why Dave Ramsey is Wrong (Dave Ramsey Lie #1)

Many financial "gurus" like Dave Ramsey advise their clients to "Buy and Hold"; buy investments regardless of current market conditions and hold them for a long period. My personal favorite flavor of Buy and Hold is Dollar Cost Averaging where you pretend you didn't overpay for a particular investment.
Other advisers say to invest in actively managed portfolios with the (poorly thought out) idea of beating the market. Still others recommend investing in indexes that attempt to follow one or more of the popular markets and take advantage of the market's positive bias.

They are ALL wrong

They are wrong because they are either intentionally dishonest, or more likely, willfully ignorant.

 

What does a hedge fund manager do?

 

Imagine someone has a million dollars in stock. They want to protect their million dollars. So they "hedge" their million by buying an investment whose value is inverse to their stock. They could buy a Put option on their stock, and if the stock goes down, the option goes up. They can exercise the option and gain roughly the amount they lost on the stock. But the option itself has a cost. The hedge fund manager conducts these transactions for the stockholder, and in many cases they are the "Market maker"; the one who sells the stockholder the option.

This scenario IS a Zero Sum Game. Someone gains, someone loses. It sounds like a bad deal for the stockholder (it is), but more importantly, it's a very dangerous game for the hedge fund. Sooner or later,  hedge funds collapse. 
In the real world, hedging is not limited to options, futures, and futures option. The Now Even Bigger too-big-to-fail firms create derivative instruments so complex even they can not decipher them at times.*

The Financial Sector

 

There is no doubt some people make money in the financial sector. We all hear of the success stories. But we seldom hear of the millions who lose. That's just human nature, the winners are celebrated and the losers are forgotten. But the winners all have one thing in common - they have an edge or angle. Where does the wealth of the multi-billionaire hedge fund managers come from? Does it come from growth? Added Value? No, the majority comes from the losers. That isn't to say the economy is Zero-Sum; it isn't. But economic growth can not account for all of the wealth accumulated by the top winners.

The Unpredictable Market

 

The US stock market has dropped in excess of 40% five times in the last 80 years. None of these declines were predicted mathematically. In fact, According to the models used by financial advisors, those declines were impossible. Impossible as in even one of them could not happen in billions of years. But they happened. I explore the reasons why in other posts.

As a result of this phenomenon most (all?) successful managers destroy more wealth than they have ever created.  

This wealth destruction happens because investors entrust successful managers with more money as they become more successful. The few managers who make money do so not because they are highly competent, but because they are simply lucky with favorable timing.

Why Dave Ramsey and most all other so-called experts are wrong.

  

Financial experts fail their clients by not taking into account or informing their clients of the true nature of risk. Whatever their favorite advice, they always discount risk. They happily plod along, advising their naive clients, and some of them make outstanding returns. Then something like Black Monday happens and all the gains they previously made are wiped out. But, the expert shrugs his shoulders and says that's life, no one could have seen that coming. 

No one could have seen that coming. 

And that's exactly the point. Because they can not properly quantify the risk, they ignore it. Because of this omission, investors are guaranteed to always lose unless their execution timing is exactly right. We know that is impossible. 

This exact phenomenon has been recurring over and over since markets have were invented. And it will continue.

There is a better way. 


* This was an aspect of the so-called Housing Bubble in 2008. Investment banks fraudulently created derivatives so complex they could not decipher them. The government stepped in, and they nor the contractors they brought in could not decipher them. In the end, taxpayers paid off the investment banks and their insurance companies the amount they claimed to have lost. A real-life example is a poker game, where one person wins all the money on the table. Then the government comes in and repays all the losers whatever they claimed to have lost -with no proof.

Misused Mathematics of DNA Sampling

Mathematics is the basis of modern technology. But it is also the basis of many false assertions. I have written previously about how statistics are (miss)used in stock market analysis. This article looks at the miss-application of math in DNA Sampling.

When DNA experts testify in court, they typically describe the probability of a false match as 1 in 100 trillion (1/100,000,000,000,000). That seems like a virtual certainty. But where did they get this number?

First a little background on DNA Sampling: Technicians extract DNA, use enzymes to cut it into pieces, process it then compare the different segments, or loci as they call it. To be admissible in court, there must be matches on 9 loci, or segments. DNA analysts typically use 13 loci, and empirical evidence suggests a random match occurs about 1 in 10 times for one loci. These two pieces of information is where the above number comes from;

(1/10)¹³

This is pure mathematics. The probability of two DNA samples matching exactly is 1 in 100 trillion. But there is problem – “empirical evidence suggests a random match occurs about 1 in 10 times for one loci”. This is one of innumerable cases of getting subjective probability mixed up with frequentist probability. The former measures knowledge of an event, the latter measures mathematical probability. The problem with this particular mixup is that we don’t know for a fact that random matches occur at an exact frequency of 1 in 10. There is missing information, specifically, do the random matches always occur at this rate, or are there circumstances we haven’t encountered where this is not the case?

The Empirical Case

A study was done on the Arizona CODIS DNA database that found 1 in every 228 profiles in the database matched another profile in the database at nine or more loci. This in a database containing only 65,493 entries.

Conclusion

So the miss-applied mathematical probability of false DNA matches is claimed to be 1 in trillions (depending on number of loci). Real world practice reveals a probability of 1 in 228 (or less, with a larger dataset). Which is correct? Which number should be used in court? Definitely not the mathematical one, because it is falsely applied.

Addendum

What is the frequentist probability of finding an exact false match in a database containing 10 million entries? You may be surprised to find out the odds are 51%. 

The Misapplication of Gaussian Math in the Financial Sector

Business is about mathematics. Macro-economics is not, it is about people and reality. This is a fundamental flaw in economic thinking. And one that causes economic disasters one after another.

This is the formula for the Gaussian Distribution, better known as the “bell curve”. Most financial model up until very recently used this model. Many still do. It is used in the ubiquitous Black-Scholes option pricing model. 

However, because it is a Gaussian distribution, it does not properly account for risk (events) outside of 5σ, therefore pricing for far-out-of-the-money options are mis-priced.

In 2000, David Li published his “Li’s Function”, a type of Gaussian copula function.

Pr – Probability.. as in default. This is the variable that is solved for. In other words, The rest of the equation gives the answer for Pr
T – Time between now and when default is expected.
= - Equality used to eliminate uncertainty
F – Probability of survival 
Φ – Used to sum the probabilities of A and B 
γ – Used to reduce correlation.

Li’s Function was quickly adapted for use in pricing Collateralized Debt Obligations (CDOs) because it allowed investors to quantify risk. Unfortunately, it suffered from the same type of flaw as the Gaussian Distribution – it failed to account for unlikely events1.

Whatever other factors involved in the financial meltdown of 2008, the use of this formula was the singular major cause of investor losses. Because it failed to assign proper risk values, the actual risks were far greater than expected.

Even after the fiasco of 2008, modified versions of Li’s Function and other Gaussian Copula functions are still being used in financial circles to price multi-instrument products such as currency swaps.

Economists and financial advisers are still attempting to modify Gaussian mathematics and Game Theory to fit economic reality. But all attempts suffer the same weakness. 

It is impossible to account for all possibilities since the bounds of all possible things that could happen are infinite. 

This isn’t the case with Game Theory. The roll of a roulette Wheel or the deal of a card hand may be random, but they are bounded. There are a known number of slots on the roulette wheel, and a known number of possible card hands. The bounds of possibility in the real world, and by implication the financial world, are infinite and unknown. We do not know what possibly can happen to affect financial markets in the future. Therefore, if no one knows, how can risk be calculated? It can’t. And so the next big financial catastrophe is just around the corner.

Mainstream financial “gurus” (like Dave Ramsey and most financial advisers) mislead their clients by not taking into account the huge losses that happen regularly, but are not predicted or accounted for in their models. The fact that a lot of retirement accounts and other wealth were wiped out in 2008-12 is NOT unusual or dramatic. It is guaranteed to happen. Putting money into non-cash, non-good based instruments is reckless; no difference in essence than gambling.


Business is about mathematics. Macro-economics is not, it is about people and reality.



1. In fairness to Mr. Li, his function was theoretical, and may have never meant to be applied to real-world trades. We do not know, since Mr. Li returned to his native China and refuses to discuss the matter.