Time Bankroll And Kalman Filters

Time Bankroll And Kalman Filters

In a number of prior posts, we have referred to the differences between mean and median outcomes of a series of trades (bets) in sports events to varying degrees.

• Time-Value Vs Expected-Value Or Likely Poverty Vs Average Riches - John Allen Paulos,
• Volatility Drag - Aaron Brown, and
• Ensemble Averages Vs Time Averages - Ole Peters.

On this occasion, we would like to explore the idea of a Time Bankroll.

Time Bankroll: Bankroll metric that reflects trader's unique time-printed sequence of trades - generated using a Kalman Filter.

Expected value summarizes the average performance across all traders and is of critical importance to the bookmaker whereas time value best reflects the most likely, individual outcome and is of paramount importance to the trader!

We can use a Kalman Filter to track our bankroll through a sequence of trades thereby mapping our unique journey through time in a single number.

First, starting with a low bankroll (1700) and ending with a somewhat higher final bankroll (2300) gives us a time bankroll (2122) which reflects that particular journey.

Second, starting with a high bankroll (2900) and ending with a somewhat lower final bankroll (2300) gives us a time bankroll (2500) which reflects a different time-print.

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Note that the specific bankroll values are randomly generated and are not meant to accurately reflect relative changes from trade to trade.

In sum, given that the time bankroll better reflects our individual time-printed journey through a sequence of trades, should it form the basis for calculating future stakes?

Enjoy!

Proebstings Paradox - Price Is Right If Marked-To-Market

Proebsting's Paradox refers to a situation in which a sports trader makes successively increasing bets on the same selection in a single market, ostensibly using the Kelly Criterion to calculate the stakes while taking advantage of better and better odds, only to ultimately face ruin.                 

 

The difficulty arises because the sequence of bets appears to cost more in total bankroll percentage than the Kelly Criterion would recommend as a standalone bet at the highest odds in the sequence.

In the Todd Proebsting example, the sports trader initially bets 25% of his bankroll on a 2/1 selection with a 50% win probability. Some time later, he is offered 5/1 on the same selection and calculates his Kelly stake at 22.5% leading to a total wager of (250 + 225) = 475. The problem with this result is that the Kelly stake for a single bet at 5/1 (assuming 50% win probability) is only 40% of bankroll (400) - leading to the theoretical possibility of ruin from betting at successively more attractive odds on the same selection in a single market.
To resolve this paradox, both Ed Thorp and Aaron Brown recommend that the sports trader should "mark to market" his bankroll after the initial 2/1 bet (reducing it by 12.5% from 1000 to 875) and use this updated position to calculate the 5/1 stake. In fact, to stay within the upper limit (400) defined by the standalone 5/1 bet, the trader also needs to reduce his estimated win probability to 42.5%! Using the above example, this would lead a sports trader to bet at most 146.25 = (16.71% * 875) at 5/1 giving a total wager of (250 + 146.25) = 396.25, which is roughly equivalent to the Kelly standalone stake (400) at 5/1 but with a lower upside!