Numbers that Play Games
Numbers that Play Games
This is a fairly surface-level overview of the ways numbers can play tricks on us, particularly when they relate to money and betting. These tricks can leave us feeling cheated or simply out of pocket. While some examples may be familiar, I’ve also included a mistake that I personally made on the job when trying to develop trading strategies, just to show that nobody is immune!
1. Winning coin flips
Let’s start with the basics. I am going to give you $100 and flip a coin many times. You can bet as much or as little as you want. To make things even better, you have an advantage, which is that you know that there is a 60% probability that the coin will land heads. However, this would be too easy! I will also charge a 1.5% fee on the sum of money you have after every turn. See if you can make a million dollars!
It is surprisingly difficult to know how much to bet. The fee makes it so that we certainly cannot bet too little, and a poor run of coin flips would leave us bankrupt very quickly with overly aggressive bet sizes. Despite this game having a positive expected value—1that is to say, your expected return is positive when you bet heads with a sufficiently large amount—you would find that it is all too easy to go broke! Indeed, we can take this lesson into our own lives too, which is to say that since we only live once, we shouldn’t ever chance losses we cannot weather regardless of whether the bets have positive expected value. For example, would you ever play the “reverse lottery”, where you get \$10 for free but have a one-in-a-million chance of losing your family home (once you lose the home, you cannot play again)? Sure, if your family home is worth less than \$10M, this would seem like a good bet, but play this enough times and your luck might catch up to you…
2. Volatile stocks
For our next game, you will be shown 12 random stocks and a history of their prices. To make things a little easier, each stock is independent of all other stocks—that is, the prices of the stocks do not influence one another. Furthermore, each stock has a positive expected return of about 1% per time step. How much can you make?
Once again, it seems almost inconceivable that you could reliably lose money on a game with positive expected value. Truth is, given enough time, every stock in this game is almost certain to decline towards zero. Yet, the expected value of holding any one of these stocks is certainly positive. How is this possible? The crucial property of those stocks is that they were very volatile. In this game, most outcomes for these highly volatile price paths are modest losses, while a small number of exceptionally large gains pull the mean upward.
A related concern appeared during the recent market frenzy in Korea. During a semiconductor bull run led primarily by SK Hynix and Samsung, retail investors purchased leveraged ETFs. A modest drawdown in otherwise solid stocks could then be amplified by leverage and trigger forced liquidation. This is related to volatility drag, in which daily compounding causes a leveraged ETF’s longer-term performance to diverge from the multiple of the underlying asset’s longer-term return. In the game more generally, these highly volatile returns can have an extreme upward skew: despite having a positive expected return, they have a large probability mass substantially below the expected return and will therefore fall below it more often than not.
3. Multiple winning coin flips
Let’s make our first game more exciting! In this game, you know that each coin lands heads 70% of the time, and you get to flip three coins at once! The catch? A 17.3% fee after every turn. Can you make it to a million dollars?
There are some interesting mechanics at play here, not the least of which is something I didn’t tell you! Ordinarily, betting a large portion of your total worth would maximize the game’s long-term logarithmic growth, but it has a hidden quirk: if you bet at least 75% of your total worth, then one time in five a sudden disturbance in the force causes all the coins to land on the same face. While the probability of each coin being heads in this circumstance is still 70%, the correlation substantially increases the risk of ruin.
4. Predictable markets
Every turn, a predictive model would give you a noisy but informative indication of where the stock is going to go. The stock’s next moves are correlated with the model’s predictions. Can you get rich here?
On a quiet Friday afternoon in the office, I trained a new model for the firm’s HFT operation. It used a slightly modified dataset and was trained in a somewhat novel way, mostly for my own research and entertainment. After it had finished training, I excitedly opened up the model’s predictions and did some preliminary analysis, which looked promising.
Putting the model through the trading simulator, the results were extremely positive. Encouraged, I also plotted some interesting behavior from the model. I plotted times when it would take on large positions in the market, and something very curious showed itself. Despite the model being trained for the HFT application, it seemed to be taking large positions at the end of the trading day, before holding them overnight and making a large profit. This was a very exciting development, and one that I had not anticipated because I totally forgot that there would be multiple days of data stuck together in the dataset. My model was attempting to predict moves on the following trading day when it reached the end of the previous trading day.
Nevertheless, I found that the model would predict these large moves minutes before the close and figured that this was a discovery worth sharing with the team. Indeed, its predictions of overnight moves were very accurate and I couldn’t shake the feeling that it was perhaps simply too good to be true. And indeed it was. The next day (yes, I often worked on weekends), I did a little more data analysis and realised that all of its predictive power was from predicting stocks that were in the limit-up and limit-down states2. Specifically, it would predict that limit-up stocks would go up a lot and that limit-down stocks would go down a lot, which is useless because hardly anybody would sell you a stock that is limit-up nor buy a stock from you that was limit-down. Filtering out those examples, it had more or less no overnight predictive alpha.
Embarrassed, I came in on the Monday morning and explained to my research partner why what I had told him was rubbish. Thankfully, he was really nice about it and said that the follow-up research I had done to expose the flaw in my “discovery” was really good analysis!