The Gambler's Half Life

Gambling and the Half Life of Deficient Investment Strategies

Why returns are so hard to measure, especially after the fact

Mainstream economic theory relies heavily on the concept of equilibrium, especially, that interest rates are highly competitive at a given point in time, and thus assets like government securities will get discounted quickly to zero across an infinite discounting horizon. According to this viewpoint, this happens whenever the yield on government debt is lower than the market rate, and the government runs perpetual deficits. Without a competitive yield, the present value of the debt gets discounted to zero across an infinite discounting horizon, if the government runs a perpetual deficit.

The most fundamental reason why this is wrong, is because despite a continual deficit, the government is purchasing new goods and services when it runs a deficit. So despite a perpetual deficit, (indeed, paying public servants is the only way a government operates), markets and public servants create additional value when additional money is spent. This value increases both private and public wealth, which allows people to save government securities at a higher valuation, as well as pay more taxes, on more things they want to own. The second reason why a perpetual deficit does not render the debt worthless, is because discounting the value of assets further into the future, becomes more and more difficult. Mainstream theory tends to assume an infinite discounting horizon, when anticipating asset trends further than 5 or 10 years in the future becomes exponentially more difficult. The long lifecycle of government and very stable flow of tax payments(even more stable than consumer patterns) gives a huge advantage to government debt across long discounting horizons.

These reasons and others are all reasons why government debt is so valuable, and why it can support such high valuations. Of course, there are conditions where the market cannot continue to support the government debt valuation, and the result is either inflation, limiting fiscal space, or a need for increased taxation, to create that fiscal space. These reasons and more, are all things I have been trying to cover on my website(a work in progress book) "ratedisparity.com"

This essay covers a third reason why it is hard to time discount assets to a present value which reflects uniform returns. That reason is the inherent difficulty of observing historical returns, whether they reflect a trend, or a one time anamoly. There is, specifically, a technique for seeking out such irregular returns, that technique being "gambling". You put more on the line hoping to get a better result than normal.

Importantly, gambling can be fair, which means that the odds in your favor and against you are balanced. The simplest example is to have some kind of hat or a pot, into which every participant puts a certain amount of dollar bills, each with their name written on them. A bill is drawn randomly from the hat, and the winner gets the all of the money in the hat or pot. For the purposes of this essay, we will assume that all gambling is fair, and there is no house fee.

In this particular essay, I will mathematically demonstrate how the possibility of gambling, both intentionally and unintentionally, makes extrapolation of historical data a very flawed and unreliable means for determining future trends and therefore accurate asset discounting information. More than anything else, I believe this fact makes a competitive equilibrium in rates of return impractical, and instead more commonly we see an "adaptive equilibrium" in returns, which I describe in more detail again on my website and work in progress book "ratedisparity.com"

Any investment strategy which utilizes a degree of gambling, will be called a "deficient investment strategies". Notably, there is a class of betting strategies, popular in the 19th century, which seeks to do the opposite, to turn bets and random chance into a stable increase. Such strategies go by the name of "martingale betting".

In the typical martingale betting pattern, you start out with the minimum bet, and each time you lose you double your bet. The effect of this, is that it is unlikely you will lose many times in a row, so eventually you either run out of money to bet entirely, or you recover your losses.

In this essay we will be evaluating a simpler strategy, that only involves one favorable bet in every time period.

To help set up this scenario, imagine there is an unscrupuluous hedge fund manager who marches down to the casino at the end of each fiscal year and uses all his assets under management as collateral for this single bet. While he may be unscrupulous, he is not completely reckless, so he limits himself to one bet with good odds to win a small amount. As stated the house takes a neglible or zero fee, and the bet is made at fair odds.

While this deceptive and unethical practice is clearly verboten, I am arguing that because history is such a poor indicator of what we know about the world today, that people end up gambling whether they like it or not, and contrary to the common phrase "hindsight" is in fact not 20/20. In fact, it is often more difficult to judge in hindsight whether an investment was a sound call, or just blind luck. This is because all the information and narratives that survive, tend to be about adjusting to what won from the last round. So if you are trying to judge the merit of investment decisions, this is even more difficult in retrospect.

How much does this hedge fund manager try to win? Whatever that amount is, we will call the "excess return". Investing is a competitive business, and even just increasing your annual returns by a few percentage points, can give you an edge. Just hope you don't lose.

We will consider a few scenarios, where the hedge fund manager seeks different amounts of excess returns: 1%, 3%, and if he is feeling really greedy a whopping 5% extra each year.

How long will this unscrupulous manager last, in the median case for each scenario? Well it turns out that mathematically, this is exactly the same question as how long it takes you to double your money at a given rate of return. So if the manager wants to gamble for a 1% excess return, in the median case, he will last log(2)/log(1.01) = 69.66 years. If you aren't at least faking 1% excess returns, he contends, are you even trying? It is more likely not going fail to work in your lifetime.

If you are unconvinced, let me explain the math simply. We are assuming that all money gambled is a simple betting pot with perfectly fair odds. If you want to fake a 1% excess return, you simply take your entire portfolio, and you contribute 100 parts to a pot, and someone else contributes 1 part to the pot. You then have a 100 out of 101 chance of winning. Once (100/101)^t is less than or equal to 0.5, that is the median number of years to failure.

This is exactly the same as the time it takes to double a pot of money at a given return. 1.01^t >= 2, which we can see by simply inverting the inequality.

This mathematics means that if you wanted to fake 7% excess returns, you would likely get away with it for 10 years. And now we see how easy it is for ponzi schemes to get started, intentionally or unintentionally, simply by taking on unknown or unquantified risk with a positive upside. It is possible to treat the market like a casino, without even knowing it. We will leave the other cases, 3 percent and 5 percent as an exercise for the reader. They fall somewhere between 10 years and 70 years.

This informational problem, means that smart investors diversify appropriately, and stay on top of things to make sure their investments are operating, not only honestly, but with a strong sense of risk awareness. Furthermore, you should not try to compete directly with others on returns, but with your own history, and only very long term historically verified results, across large populations. Returns will vary much more across space, than they do across time, and the entire notion of "equilibrium" returns, indeed the entire premise of controlling the "price of money" with policy rates, is a flawed endeavor, much more likely to prove costly for a country that seeks higher real rates(because that applies to their treasury securities).