Why Parrondo’s Paradox can’t be exploited in Asset Management

Modern financial markets are incredibly fast. Algorithms identify and close out traditional arbitrage opportunities even before they can arise! It is for this reason that fund managers and traders constantly have to find new, creative ways to ensure satisfying returns for their clients. From time to time, one comes across people talking about Parrondo’s Paradox. It is said they make a lot of money by combining cheap losing assets to emerge outstanding winners. But does this really work?

Well, Spanish physicist Juan Parrondo caused some sensation in 1996, when his studies of the Brownian ratchet led him to discover that it is indeed possible to combine two or more losing strategies into a winning one. The so-called Parrondo’s Paradox became fairly famous in game theory and soon caught the attention of scientists and practitioners across the globe.

It has been 20 years since and scientists surely did a tremendous amount of research, so shouldn’t we all be rich by combining our junk to treasures? Well, unfortunately, this is not how Parrondo’s Paradox works.

How Parrondo’s Paradox works, the classic example:

The classic example of Parrondo’s Paradox include two games, which are combined to create a third one:

Game A is a simple coin tossing game: Head we win one dollar, tail we lose one dollar. If assuming that the coin is biased and showing tail in 50.1% of the tosses, it is easy to see that we will slowly lose all our money (no matter how much money we might have), if we keep playing for a sufficiently long period of time – Game A is a losing game.

Game B involves a slightly more complex coin toss depending on your current wealth. If your wealth is a multiple of three, you toss a coin with winning odds of 99:901 (winning change 9.9%). Otherwise, 3 you toss a coin with winning odds of 749:251 (winning change 74.9%). The odds while having wealth that is a multiple of 3 are so bad that they easily offset the favorable odds while not having wealth that is multiple of 3 – game B is also a losing game.

Parrondo’s Paradox occurs now, if these two losing games are combined to create a game ‘C’ by playing them in an alternating order, let’s say AABBAABB… and so on. This game C, though the combination of two losing games, turns out to be a winning game. A rigorous mathematical proof is compliment (Harmer and Abbott (G. P. Harmer and D. Abbott did some work on it in “Losing strategies can win by Parrondo’s paradox”, Nature 402 [1999], 864), but it is easy to see through the simulations.

Why Parrondo’s Paradox doesn’t work in Asset Management

I know what you must be thinking now. It sounds tempting but NO, you are unlikely to get rich by using Parrondo’s Paradox for your portfolio of stocks. The reason is that Parrondo’s Paradox is created by combining two independent losing games A and B in a way that disrupts their independency. This is not only impossible with common stocks (you would need an investment bank to design you a very niche security) but even worse, once you fully understand this concept, you can eliminate the influence of chance and will be able to create easy examples like the following:

In game A, you lose $1 every time you play. In game B, you count how much money you have left. If it is an even number, you win $3. Otherwise, you lose $5. It is obvious that both, game A and game B are losing games. If you combine these games to C as ABABABABAB and you start with any uneven number you will be turning these games into a winning game – basically also turning Parrondo’s Paradox into a simple sleight of hand – making money with a trick like this sounds more difficult to me than choosing a good portfolio of winning stocks in the first place.

A simple thought experiment

If you are still not convinced consider the following situation: Let’s assume you found an investment bank, which was willing to sell you such an exotic security that can be used to create Parrondo’s Paradox. Let’s also assume you would go ahead and build a complicated portfolio of long instruments, which starts to increase its value while the values of your stocks and underlying companies actually decrease at the same time.

At one point you might be rich by owning shares of worthless companies. Something just doesn’t add up here, right? Yes, this bubble would inevitably have to burst eventually.

Moreover, knowing which stocks are going to depreciate is exactly as difficult as to know which stocks will rise. Therefore, my advice is to forget magic tricks and Parrondo’s Paradox and find a fund manager with solid expertise and a good track record!

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Dustin Lichey
Dustin Lichey
About the Author

Dustin Lichey is the world's only PRM in Strategic Marketing. He started his career as an Intern in Risk Controlling at Allianz Deutschland AG before he shifted to Deutsche Postbank AG and afterward returned to the Investment Management of Allianz SE, where he participated in managing a portfolio worth EUR 420 bn.

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