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!

Small Fields And Exotic Convex Bets (Part 1)

Small Fields And Exotic Convex Bets

In horse-racing, Convex betting is predicated on frequent small losses and infrequent big wins. But in order to take advantage of this strategy, we require markets with fields of 11+ runners to generate live outsider prices.

Unfortunately, in the last couple of years, British racing has begun to noticeably decline in terms of competitive fields in both Flat and Jump codes with the consequential drop in price ranges! We, sports traders, cannot fix the current problems facing British racing so we must either adapt or die!

One possible solution for us is to look for better prices in the exotic (e.g. Betfair's 'Forecast' and 'Reverse FC') markets.

To that end, using an Excel spreadsheet, we can set up a table as follows to generate either 'Forecast' or 'Reverse FC' tickets subject to the following conditions:

• Runner saddle numbers as both row and column labels,
• Populate the table with 1s and 0s subject to the following constraints:
  • Row totals must equal two.
  • Column totals must equal two.
• Use Solver to generate a possible solution. Otherwise, use trial and error.
• Read off the row labels of the columns containing 1s to generate a subset of all possible tickets.
• There are multiple valid combinations of tickets.
• No matter what is the winning combination of the race, the generated set will contain two tickets with at least one of the winning numbers.

This is an example of generating a 'Reverse FC' solution for an eight-runner field.

and this is an example of generating a 'Forecast' solution for a five-runner field.

Enjoy!

Bellman Bets Meets Shannon's Demon

Bellman Bets Meets Shannon's Demon

As 'Royal Ascot' week is upon us with a splendid array of graded stakes races, it would be nice to enjoy some convex betting opportunities without risking too much capital and ending the week with either a small loss or a small gain.

To that end and with a certain amount of tongue-in-cheek attitude, we present our 'Bellman Bets meets Shannon's Demon' (or Session Handicapping meets Market Rebalancing) strategy. The ideal conditions for using this approach include:

• Markets should be volatile (i.e. range of short-, medium-, and long-priced winners),
• Markets should be negatively correlated (i.e. Local Track and Royal Ascot) or uncorrelated (e.g. cash), and
• Rebalancing costs should be very low or zero.

Let us assume that we have the following parameters for both markets:

• Number of Races = 7
• Current Bankroll = 117
• Target Bankroll = 525
• Win Probability (Avg) = 0.15
• Decimal Odds (Avg) = 7.00
• Current Race = 1

In order to put the protocol through its paces, we simulate the first four Bellman bets (assuming three losses and one win):

[Local Track; 13:45]
java.exe GamblersRuin 7 117 525 0.15 7.00 1
Success = 0.23789
Stake   = 13.0

[Local Track; 14:20]
java.exe GamblersRuin 7 112 525 0.15 7.00 2
Success = 0.22352
Stake   = 29.0

[Royal Ascot; 14:30]
java.exe GamblersRuin 7 103 525 0.15 7.00 1
Success = 0.21001
Stake   = 36.0

[Local Track; 14:55]
java.exe GamblersRuin 7 131 525 0.15 7.00 3
Success = 0.24309
Stake   = 19.0

leaving us with the a small profit. Note that the winning bet at Royal Ascot assumes it was on one of multiple selections in that particular market.

Ideally, we should consider quitting for the day if we either win or lose 50% of the initial bankroll. If the former outcome occurs, then we should also revisit our estimates for average price and average win percentage. Either way, our 'portfolio' is automatically rebalanced for the next day.

Mathematically, Bellman maximizes our probability of reaching a specific target using a limited number of events and Shannon reduces the impact of volatility drag on our portfolio by rebalancing after every event.

Tread carefully, do your own research, and enjoy!