St. Petersburg Paradox and VXX Runs

February 5. 2012, 2:45 PM

Trading NYSE VXX stock and hoping to get in on the first day of runs is a bit like the St. Petersburg Paradox invented by Nicolaus Bernoulli in 1713, an old riddle still relevant to investing today described recently in the WSJ:

Proposed in the 18th century, the paradox works like this: I will toss a coin until it comes up heads, at which point you get paid and the game ends. You get $1 if the coin comes up heads on the first toss, $2 if the coin comes up heads on the second, $4 if it is heads on the third, $8 on the fourth and so on. Your prospective payoff doubles with each successive flip until the coin finally lands on heads and the game is over.

How much would you pay to play? The paradox is that the potential payoff is infinite; after the first flip, you have a 50% chance of winning $2, plus a 25% chance of winning $4, plus a 12.5% chance of winning $8 and so forth. You will win $537 million if you get heads on the 30th toss; on the 50th, $563 trillion.

The measurement of risk has been generally agreed upon since mathematicians first began to study risk. Expected values are computed by multiplying each possible gain by the number of ways it can occur, then dividing this sum of products by the total number of possible occurrences. (Source)

We would like to apply the St. Petersburg Paradox in a slightly different way for trading or investing in VXX.

Whenever VXX closes up: 
  1. You buy VXX, and 

  2. Every time VXX closes up you can receive a multiple of your initial investment of the total # of times VXX closed up since your initial investment, and you quit, or 

  3. Game over, and you lose it all first time VXX closes down after your initial investment.
What's the most you could gain investing #100,000 any time VXX closes up, and what's the probability of doing so? 

You can use our calculated probabilities in Facts to post your answer in Comments — Let us know if you need more information. 

We will withhold posting our answer until readers have a chance to answer.

Interesting followup to the St. Petersburg Paradox at

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