Putting an errorbar in your money

I, and also an increasingly larger population of the world, have concerned myself and dedicated countless hours to deliberating about the unfortunate fact of life that as we grow older, we eventually might not be able to provide for ourselves due to the natural degradation of our bodies. The capitalist society, however, provides us with the choice of converting our human capital (i.e., ability to work) into assets (money, stock, real estate) that can be used in the future to keep us going even when we lose the ability to work.

Therefore, it is important (and heavily underappreciated) to put aside a portion of your hard-earned capital for when those hard times come. Apparently, however, human psychology does not align very well with this rational argument. We naturally find ourselves jeopardizing these long-term goals by enjoying ourselves too much when we’re young and active, to the point of entering debt to buy the latest gadget.

You see, the consumerist culture of capitalism and the necessity for saving for the future are not mutually-exclusive behaviors and a reasonably intelligent and disciplined person should be able to consume goods and promote the advancement of society through the fostering of competition and the funding of technological research that is one of the greatest achievements of consumerist capitalism. It is a matter of fact, though, that this consumerist nature tempts the less rational part of ourselves to all sorts of dubious behaviors from the financial planning perspective. Thus, many different countries institutionalized retirement savings as mandatory through social security. The optimality of this solution is questionable, but under the terms of a “greater good” goal function it definitely is a sensible decision.

The collection of such a large pool of retirement capital under the management of a single countrywide institution has benefits and caveats that are rather important to be discussed. High levels of money management specialization should be expected from such an institution, given that the best in their field can be afforded to manage such a large asset. On the other hand, effectiveness of highly specialized money managers working with institutional capital against buying and holding the market has been demonstrated not to be corroborated by the data [as very well explained by Benjamin Felix, references in the video]. It is also reasonable that dilution of many behavioral and idiosyncratic market risks is possible with a larger pool of capital, averaging out the effects of spurious market movements. In third-world countries like the one I come from [Brazil], however, there is lesser trust in the effectiveness of the management of these funds as the transparency of social security data is low and a lot of room is available for dishonest behavior. I personally see social security in such environments as another “tax”, which does pay back in the long term but is prone to mismanagement and corruption.

Don’t retire early

With my stance in the argument set, I believe that regardless of social security one should save personal funds for discretionary retirement. You see, differently from people of the FIRE movement [Financial Independence, Retire Early], I believe careful selection of your professional career path during your 20’s should be enough to provide sufficient personal satisfaction from your job such that you wouldn’t need – or want – to retire early. If one’s job is fulfilling and provides them with a sense of contribution to society, why would they trade it off for “enjoying life” by doing absolutely nothing useful? Obviously, enjoying vacation trips now and then is important for a healthy life balance, but I’d say that would become boring rather fast if that was the only thing you did for a couple decades.

Granted you chose a fulfilling career, it is sensible to keep contributing to society for as long as we physically and mentally can. If you did not, consider changing while you can. Even if it is financially less rewarding, in the long run you’ll keep it up for much longer. And the fact that you enjoy what you do usually makes you willing to spend the time to do the “extra mile”, which is key to becoming respected in your area.

Index funds

Nobel laureate Eugene Fama showed through his research the evidence that, in an efficient market, actively managing your money gives you no statistical edge against an investor that simply buys and holds the market. Fama also shows that there are some specific factors that have reasonable theoretical foundations and explain the gains of the market as a whole. Their description of the “Three-factor model” shows that regression fitting of historical stock pricing data can explain the performance of asset portfolios by three factors: The market factor would be a “premium” for investing in the higher-risk market; The size factor would be a premium for investing in higher-risk smaller capitalization stocks; and the value factor would be a premium for investing in companies that possess a higher book-to-market ratio. I confess I don’t fully understand the theoretical justification for increased average returns for the higher risk stocks. One thing that I find reasonable, in my personal ignorance of the financial market, is that the market factor justifies itself as long as we have large positive macroeconomic movements (i.e., as long as population grows, total amount of goods produced increases through more technology, etc.). I think it is a rather important limitation of Fama’s model that we need these macroeconomic movements to occur in order to have our stocks grow long term, and that major macroeconomic downturns are not impossible in the future if catastrophic events occur. Due to the unlikelihood of these events and the hopelessness of safeguarding from them even if they do occur, I still believe investing is a reasonable strategy.

Fama’s research sparked the composition of potentially one of the greatest tools for financial investing: Index funds. Though index funds have been around for several decades now, index plus other factor-related funds have popped into existence with incredibly small fees and the liquidity of a stock through exchange-trading. This allows small, individual investors, to decide for themselves their investment strategy and their risk tolerance in a DIY approach. If you’ve seen any other post from this blog, you know I love DIY!

the largest casino in the world

When I was 18 years old I had my first time experience with the stock market. After playing with it for a few months, I concluded – in my naive view of the world of then – that the stock market is just institutionalized gambling. The emotions you feel when your money fluctuates in the market are rather bewildering and I honestly experienced some real adrenaline pump while binge-watching my long positions fluctuating with the Market’s tide. It all seemed random, though. I tried looking for patterns, learned technical analysis and applied it as a guide to my investments. But after getting deeply acquainted with it, I felt like I was just finding patterns in randomness as we do when we see faces in clouds or stick figures made from stars in the sky. These patterns appeared to have the same predictive power of flipping a quarter. After that experience, I decided that I would not touch stocks ever again in my life.

Academic research really helped me to have a more sober view of the market. The outreach work by Ben Felix also helped me see through the bullshit of financial channels and blogs in the internet. I felt, after what was pretty much a decade, more prepared to give it another shot. The knowledge of statistics, scientific bias, data analysis and just plain critical thinking developed through higher education were instrumental in the establishment of my current, totally non-expert opinion of the financial market. So I decided to write this and share some of my humble data analysis results in the hopes that other people might find it “dumbed down” enough to give it a go. I still confess that some of the papers by Fama and French are still over my head due to sheer academic jargon and encoding.

As I glanced before, it is worth the exercise to ask ourselves why in the first place it makes sense to invest in the market. Why does the stock market seem to grow ever higher in value? Where is the wealth being generated? Is the market a zero-sum game? If so, who is losing money?

These questions still linger in my head, to be honest. We need to address what is a zero sum game, I think, to get started. A zero-sum game is just description for some systems where the total amount of a token is conserved such that only transfers of that token are possible between the players. This means, no “token” is being created out of nowhere. All games in casinos are zero-sum, for example. They involve the players putting their money in a pot, and the results of the game determine how much of that pot is distributed to the winners/losers. Usually in a casino, the game is such that the “house” has a slight statistical edge and will, over thousands of rounds, accumulate wealth. Since the game is zero-sum, that wealth must come from the players of the game. We have then, a very good distinction between “investing” and “gambling”. While both endeavors are risky and statistical in nature, “gambling” is a zero-sum game. “Investing”, on the other hand, is a positive-sum game.

But how is this even possible? How can one create money out of thin air? Well, surely the Federal Reserve in the US (and their equivalent in other countries) do, right? That makes the game positive-sum because now money has been created out of thin air. Well, not really. Though the total numerical amount of money might be larger due to “materialization” of money, no actual wealth was created by doing so.

This brings us to an important point in investing. What does money mean? What is the nature of wealth? Well, I don’t pretend to know the answer of these questions. My readings lead me to believe that money is a token that is institutionalized in our governments through thousands of years of iterations. It seems to be a natural manifestation of society. Instead of trading goods directly, we use the money token as a convenience. It by itself only stores value because everyone agrees it has value. Without diverting too much on why money has value, one can meditate that a way one can earn money, and therefore generate value, is through work. Careful application of one’s time and expertise to transform raw materials into more useful devices, goods or other consumables is a reasonable means of earning money. Let’s take the example of a material good, say, a chair. A chair stores value within itself because it is a useful device that allows humans to comfortably sit while they’re doing less involved activities or just enjoying themselves. It retains its value over time, because it keeps on accomplishing that task for a relatively long period until it finally decays to the point of becoming undesirable.

In the case of the chair, the people involved in the process of harvesting the naturally-ocurring materials to build it, cutting them into shapes that embody the function of the chair, and finally putting it together, need to be compensated for their time in doing so. Furthermore, the people involved in auxiliary services such as delivery, selling, handling, managing and others also will have spent a small fraction of their time in the particular chair you’re sitting while reading this, for which they also need to be compensated. Their time, therefore, is stored in the value of the chair. And you, when making its purchase, is willing to pay your earned money to have it. Of course, your function in society also does produce tangible or intangible goods in some sense, and your time is compensated such that you can afford to pay for it.

Through this reasoning I believe we can establish that goods and services store value and the production of such goods and services is how wealth (and therefore, money) is created. Some goods will last for longer, thus storing wealth for a longer time. Others, however, will last for very little time before spoiling (i.e., foods) or destroying themselves, thereby retaining their value for less time. This means that wealth is also destroyed over time, and in order to have a net positive wealth generation people need to be producing more value than the value that is naturally destroyed over time. I would say that a key requirement for this to happen is that populations keep on growing, because that would increase the overall demand for goods and services.

My current understanding based on this argument is that money is just an agreed-upon representation of people’s productive time. This representation is also useful to quantify the impact of one’s relative productiveness, since some people earn more money for the same amount of time invested in contributing for society. I’m not claiming that this is a fair representation, but the dynamics of market offer/demand should to at least some extent dictate the relative usefulness of people’s contributions. The efficiency of the job market is a point that I myself haven’t researched too much into, however. But in a sense, this is why it is somewhat accepted that there is some positive correlation between individual wealth production and their relative contribution to society (i.e., the dichotomy between highly regarded jobs such as doctors, engineers, etc.; versus lower-waged jobs like the exploited workers of fast food restaurants and supermarkets). But I think this is a highly controversial topic to be discussed here, because I don’t believe that people deliberately want to be useless in society.

So, HOW MUCH IS the errorbar?

Ok, this was a lot of meditation about capitalism. For personal financial decision-making, I’m sure none of that is necessary. What I really wanted to share, though, is my underwhelming observations of historical data. You see, if one believes index fund investing is a viable alternative for not only keeping their money value but also increasing it over time, then the evidence should point out to a mean effective growth of value over time, net of inflation effects, right? Well, though that has already been proven from numerous papers, I wanted to also give it a go. So let’s take the historical S&P500 index data as a benchmark for data analysis. However, the S&P500 index does not account for inflation. So the first step is to remove inflationary effects. If we do that, we get the following chart:

Inflation-adjusted S&P 500 index (computed on 12/05/2019). Data sources: S&P500Inflation

Interestingly, the chart indicates about 6 times growth in the index over the course of 90 years. As of the time of this writing, the US markets are regarded to be in a “bull run”, which obviously needs to be taken into account. But I’d say that everyone agrees that, on average, there is indeed an overall trend of growth even after inflation correction. For comparison, the first data point of the series in December of 1927 shows an index value of 17.66 before correction and 262.3 after corrected to 2019 money.

So there’s a mean growth. But when we are buying into stock, we generally do not know where exactly we are sitting in this curve. Maybe now we’re at a peak? Maybe not, maybe we are still on the rise and the next crash will be way past 4000 points. The point is, we don’t know a priori and we can’t know. Especially for us peasants that are not involved in finance, it is a waste of our valuable human asset and skill in other fields of knowledge to attempt to predict that. The practical question for non-specialists is more about whether, statistically speaking, there is an expected return (which seems to be the case from the figure above) but also, what is the amplitude of the other outcomes (i.e., good and bad). This is what I mean by putting an errorbar in your money. Every time you look at the stock market, the nominal value of your holdings is volatile – there’s some fluctuation, or noise, to it. The question I want to personally answer by analyzing the data here is, how much is that noise?

It is reasonable to expect this noise would be changing over time. Fluctuations on the daily basis should be small, but larger excursions should be expected over time, both for bull runs and bear runs. So this question only makes sense to me under a specified time horizon. We can analyze the historical data with different time horizons, then. If we look at, say, a week time horizon, we can look at any arbitrary pair of dates 7 days apart and see the return % distribution. Then we look at the return time series and average the returns, to get a mean return over a week. We can also look at statistical properties like standard deviation and percentile values, which would give us the size of that “errorbar”. So I’ve done that. The results over overlapping periods between 7 days and 40 years look like this:

Average and deviations of returns of the S&P500 index over time windows of various lengths.

Interesting observations can be made over the long term with the chart above: A mean trend of positive returns is expected over the course of 40 years. You should expect to triple your money (200% return), inflation corrected, over 40 years. Not as much as I hoped for, to be honest, but also not that bad. It gives, to me, a very good sense of how much saving money now will be worth when I retire. This also gives grounds for decision making, which is awesome!

Furthermore, observe in the chart above that in about 30 years some positive return is not only expected, but 99% guaranteed. It takes that long of a wait. This gives a good sense of the investment horizons we are talking about here.

Unfortunately a logarithmic scale can only be used in the time axis, as negative returns cannot be plotted in logarithmic scales. Therefore, the returns over periods less than an year are rather difficult to observe. So the chart below shows a zoomed version of the data, from 7 days to 1 year. We can see the growth of the “errorbar size”. Within a week, the standard deviation is 2.8% and the 1-99 percentile encompasses returns between -8.2% and +7.5%. In a month, the standard deviation grows to 5.8% and the 1-99 percentile now encompasses returns between -16% and +13%. Within an year, the 1-99 percentile grows to between -43% and +55%. Even though the mean of the returns is always positive (over an year it’s +4.54%) it is interesting to see that the distribution has a slightly larger negative tail. This shows how emotional responses tend to affect negative movements of the market in the short term more strongly.

Average and deviations of returns of the S&P500 index over time – zoom in the short term.

Another interesting observation I made with this data is displayed in the animated GIF below. It is interesting to see how the probability distribution is pretty much a normal distribution for periods less than 4 months, losing its character as the periods grow longer. For an year the distribution is more triangle-shaped, and for over 3 years it starts to morph into a long positive tail. The fact that the distribution looks exactly like a random normal distribution for the short period (i.e., less than a quarter) demonstrates the amount of time that companies need to realize gains. It also delineates the change between stock market gambling over the short term versus actual generation of wealth over the long run.

Probability distribution of gains of the S&P500 for increasingly long time windows


This personal analysis for me is very captivating evidence that the stock market is a positive-sum game. I know this is limited to the U.S. market and that the political hegemony of the U.S. is probably biasing the results to a positive conclusion, which might not be true in the long run. Nevertheless, I believe for my short little lifespan it might still be of somewhat valid, empirical application. The statistical distribution of gains makes me more resilient to market downturns now, since I know what kinds of movements to expect in the short term, which unfortunately are rather large.

As a matter of practical mnemonic, I’d say 2 standard deviations is enough to capture the expected movement of the short-term market. This would mean weekly movements are expected to fall within about ±5%, monthly movements should be within ±10%, quarterly movements about ±20% and yearly movements about ±40%. It’s a large errorbar to put in your money, but one that has to be done if any positive expected return, inflation corrected, is desired.

I hope you also got inspired to look at the data yourself. If that’s the case, have a look at my code in Github. It is a simple code and I’ve done some simplifications for the sake of analysis. Nevertheless I think the conclusions are quite valid. Hope you’ve learned something!

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