To Infinity and Back

Out of the blue, my daughter says to me: “Infinity is when you keep counting forever.” 

That’s not something you hear every day from a 4 year old.  We have been working on simple addition, so intrigued, I ask her: “What is infinity plus 1?”

My daughter looked at me confused: “Daddy, counting to infinity is impossible. Don’t you know that?” 

While I would love to believe I have a little mathematical savant on my hands, I’m pretty sure she heard this on the TV show Peg + Cat. 

Nevertheless, infinity is a fairly deep philosophical concept that underpins much of high level mathematics, and in some ways, financial investment strategies.  Let’s explore through the lens of trend following.

Trend Following and Infinity

Trend following is an investment strategy that tries to ride trends in the market.  It believes that stocks that are trending will continue to trend in the same direction, very similar to momentum. The main difference between the two is while momentum strategies entirely change their holdings every so often and start over with a new portfolio, trend followers hold a stock until the the trend ends.

There are many ways people try and implement trend following strategies, but they all boil down to a general rule:  Sell an asset after it incurs losses.

Most trend following strategies use trailing moving averages to define the level of loss required to trigger selling.  Those averages lag behind the current price.  If the price falls back down below the trailing moving average, the asset is sold in order to limit losses.  Others use a trailing moving stop, i.e, when the price falls 10% from its peak, they sell it. 

Some people use a tight number and sell very quickly on a loss, others use looser limits and wait for larger pullbacks before selling.  Universally all trend following strategies sell because a loss crosses some threshold.

For this analysis, I’m only going to discuss trend following on individual assets, i.e. single stocks or single commodities.  Trend following on indexes is different because portfolios of assets behave differently than individual assets.¹

Modeling with Randomness

If you read my blog, you know I don’t believe individual assets trend or show momentum.  But, trend following is still a very viable strategy, with some very unique attributes.  Let’s model how trend following would look with random coin flips.  The game is as follows:

Bet $100.  Flip a coin, heads you win 100% of the bet (receive $200 back, for a gain of $100), tails you lose 50% of your money (get $50 back, for a loss of $50). 

Applying the trend following concept of selling upon losses to these random coin flips creates a game where you sell the asset after a certain number of tails.   You could pick any number of consecutive tails to end the game, just as you can the size of the trailing moving average, but for our example to keep it simple, we’re going to end the trade at a single occurrence of tails.

For our model, let’s say you can only play one total game each day.  You will start the day with a coin flip and keep going until you get a tails.  Once you get a tails, you will be done for the day. The total return for that day is what we will focus on.  The next day you can play again.

Calculating the returns from this game are not straightforward since theoretically the game can go on forever. So to evaluate the game, we’re going to set an artificial cap on the # of flips, evaluate the result, and then examine where the values head as this cap increases. The distribution of winnings start out like the following:

Half the time you lose right away.  A quarter of the time you win once and then lose.  Still not a great result over all.  But in the other 25% you will make money by winning two rounds.  Lets push the table out further.

The First Infinity

The winnings keep growing exponentially.  The total payout becomes extremely “skewed to the right”, which means the loses occur often. However, the winners, which are rare, are enormous. 

If you limit the game to only a few flips, the average return is positive. The very few times you end up with multiple heads, you make plenty of money to overcome the many repeated losses you receive. 

But the game isn’t limited to only a few flips.  It can keep on going.  And as we keep on going, you can see that the average return keeps on going with it. 

The game ultimately has an arithmetic return of infinity.  It is undefined.  My daughter would say it is “impossible”.  In a way, I would agree. 

The St. Petersburg Paradox

This game is nearly identical to one invented in the 1700’s by Daniel Bernoulli, called the St. Petersburg Paradox.  It is a paradox because Bernoulli asked, “how much should you pay to play a game with an infinite return”?  The answer is clearly not infinity itself, although you could argue it should be. 

But what is the correct answer?  Bernoulli “solved” this problem by saying each person has their own risk aversion “utility” which discounts this infinite value to a much lower number that they are willing to pay.  This concept has underpinned much of economic theory and in some ways finance ever since.  It’s the foundation of Expected Utility Theory. 

I don’t want to digress from the trend following concept just yet, so I’ll expand on this concept in another post.  But realize this concept of infinite returns has been confounding people for centuries.

Trend Following and Infinity

Considering investors appreciate high returns, “infinity returns” are quite attractive.  It’s not hard to understand how a portfolio similar infinite “games” can produce a very fine investment return. 

Now you may point out that I picked an extreme example. I did that on purpose. Any less than this example will not produce an infinite return. However, the trend following methods still “expand” returns, even when they are not infinite.

Expanding Returns

To show how, let’s look at the example from the investing games post. Let’s put a 50% gain on heads, 40% loss on tails, through the same “trend following method”.

As shown on the right, the arithmetic return converges to a 20% return as the cap on the # of flips increases. This is quite impressive as the underlying game’s return is 5%. Trend following effectively expands (increases) the arithmetic return of the game.

This lead to the conclusion that trend following individual assets may work because it:

• Combines a group of low correlated, return expanding, “infinity” games together
• Uses the natural tendency of an investment fail quickly to partially rebalance the games,

This system works, even if the underlying stocks behave entirely randomly, never actually trending.

However, there is another infinity lurking.

The Second Infinity

So random coin flip games set up to simulate trend following rules produce great returns.  But what about the variance and standard deviation of those returns?

Turns out the +50%/heads, -40%/tails game produces infinite standard deviation as well.  As the winnings expand with a large number of consecutive heads, the standard deviation grows even faster, shooting upward to infinity, just as the arithmetic return did in the first example. 

Ok this is great. We now have games with infinite standard deviation, on top of games of infinite return.²  Infinite return on its own is enough of a paradox.  How do you make reasonable conclusion on how to invest in games with infinite volatility and uncertainty?

Enter the Geometric Average

Lets go back to the first example with infinite return. Just like the second example, it too has infinite standard deviation. But what about the geometric return?

Following the same process, you can see that as the number of flips increases, the geometric mean converges toward a defined number.  It’s converging toward the square of the underlying geometric return.

Why?

The entire game is built from a single foundational game that is compounded. Compounding distorts arithmetic returns, and standard deviation, to the point of driving them to infinity. But since the geometric return is an average built for compounding, it never gets distorted.³

The geometric return is the “truth” return when it comes to multiplicative games. You can’t hide from it and it will always be there. Investing games are always multiplicative games.

Further Proof Why the Geometric Return Rules All

I harp on the geometric return constantly on this blog, and this is a big reason why.  Even when the arithmetic return is undefended, and the variance is undefined, the geometric return can still very much be defined.  The geometric return is actually more tangible, more real, than any other measure of a compounded random processes.

There’s nothing impossible about it.

1-There are reasons to believe portfolios of random assets can “trend”, even though individual stocks don’t and we will study why in a later blog post. 

2-Some may point out that if I reduced the game’s return from 50% to 41.4%, the standard deviation would no longer be infinite. That’s true. But what’s the fun in that example?

However, skew would still be infinite until return gets down to 26%. And then kurtosis will be infinite until 18.9%. All the while each statistic, when not infinite, will still become very distorted.

3- It goes to the square of the return because the most likely number of flips is 2.