For a player who gambles his entire bankroll each round, it appears to be a wash. No matter the order of returns, if there are an equal number of heads and tails, the player ends up having exactly as much as he did at the start.

For example...

- Starting bankroll: $100
- Round 1 (heads): $200
- Round 2 (tails): $100
- Round 3 (tails) $50
- Round 4 (heads): $100

However, for a sophisticated investor this game would represent an incredible profit opportunity. Claude Shannon illustrated this by proposing that an intelligent player would wager only half of their bankroll each round. This seemingly small differences turns the game into a winner.

- Starting bankroll: $100
- Round 1 (heads): $150
- Round 2 (tails): $112.5
- Round 3 (tails) $84.375
- Round 4 (heads): $126.5625

Converting this game into investment-speak, Claude Shannon proposed a portfolio of 50% coin flip and 50% cash. This portfolio was rebalanced at the beginning of each round. The results of this game are quite profound. It shows that risk reduction has the ability to increase returns by bringing your realized portfolio geometric return closer to the weighted arithmetic returns of the portfolio's components.

Taking the bankroll management strategy in a slightly different direction, we can observe how a diversified portfolio of coin flip games that is rebalanced each round offers a return that is far greater than the weighted performance of the individual games.

(This chart will update automatically every 10 minutes or so, offering a new randomized experiment each time.)

Internalizing the benefits of diversification takes time, but I can’t imagine going back to a concentrated portfolio. Diversification combined with rebalancing offers the opportunity to increase returns while simultaneously decreasing risk. Although the real world is far more complicated than a simple coin flip operation, I think the game can help investors understand how looking at assets in isolation never makes sense. Sophisticated investors only care about what an asset will do to a diversified and periodically rebalanced portfolio. If you enjoyed the topic of this post, try Googling phrases such as “volatility pumping”, “volatility harvesting”, “Shannon’s Demon”, and “Kelly criterion.” Additionally a well done paper on the subject can be found here.