Chapter 28 · Worked solutions
Chapter 28 workings: Simulate Before You Speculate
Chapter twenty-eight worked: the chance of a losing season with a real edge, at 6/4 and at evens, plus a free season simulator for the third exercise.
The first two exercises use the shortcut from the chapter: work out the mean and spread of a single one-unit bet, scale both to the whole season, and read the chance of finishing behind off the normal curve. That is accurate enough at these sample sizes.
The third exercise needs a simulation. The book promises a template on this site; it is the season simulator, which runs thousands of seasons in your browser and reports the 5th-percentile path, level stakes or fractional Kelly.
The exercises
500 bets at 6/4 with a 3 per cent edge
A season of 500 level one-unit bets, all at 6/4 (2.50), with a true 3 per cent edge on each. Work out the win probability, the per-bet mean and spread, and how often the season finishes behind.
Win probability from the edge
An edge e at decimal odds O means p × O = 1 + e, so p = 1.03 ÷ 2.50 = 41.2 per cent. Break-even is 1 ÷ 2.50 = 40.0 per cent.
Mean per one-unit bet
p × (2.50 − 1) − (1 − p) = +0.03 units: the edge, as it should be.
Spread per bet
A winner pays 1.50, a loser costs 1. The variance is p × 1.50² + (1 − p) − mean² = 1.5141, so the standard deviation is 1.230 units.
Scale to the season
Means add and variances add: expected +15 units, standard deviation 1.230 × √500 = 27.5 units.
Chance of still being behind
The expected profit sits 0.545 standard deviations above zero. The normal curve puts 29.3 per cent of seasons below it.
The answer
p = 41.2 per cent, mean +0.03, standard deviation about 1.23; expected +15 units against a spread of 27.5, so roughly 29.3 per cent of such seasons lose.
2,000 bets at evens with a 5 per cent edge
Now 2,000 one-unit bets, all at evens, with a true 5 per cent edge. The same four numbers.
Win probability from the edge
An edge e at decimal odds O means p × O = 1 + e, so p = 1.05 ÷ 2.00 = 52.5 per cent. Break-even is 1 ÷ 2.00 = 50.0 per cent.
Mean per one-unit bet
p × (2.00 − 1) − (1 − p) = +0.05 units: the edge, as it should be.
Spread per bet
A winner pays 1.00, a loser costs 1. The variance is p × 1.00² + (1 − p) − mean² = 0.9975, so the standard deviation is 0.999 units.
Scale to the season
Means add and variances add: expected +100 units, standard deviation 0.999 × √2,000 = 44.7 units.
Chance of still being behind
The expected profit sits 2.239 standard deviations above zero. The normal curve puts 1.3 per cent of seasons below it.
The answer
p = 52.5 per cent, mean +0.05, standard deviation about 1.00; expected +100 units against 44.7, so only about 1.3 per cent of seasons lose.
Your own plan, the season one year in twenty
Run your own planned staking for next season through a simulation and read off the bad-year result, the one you would expect about once in twenty seasons.
Enter what you honestly expect
Number of bets, a typical price, and the edge you can defend from your records, not the one you hope for. If you have no record yet, use an edge of zero first and see what variance alone does.
Choose the staking
Level stakes in units, or a fraction of Kelly as a share of the current bank. Quarter Kelly is the setting chapter 29 recommends.
Read the bad season, not the median
The simulator shows the 5th, 25th, 50th, 75th and 95th percentile finishing banks and the worst drawdown at each. The 5th-percentile row is the one-in-twenty season. Plan your bank so that row is survivable.
Keep the seed
The address bar holds every input, including the random seed, so the same link reproduces the same thousand seasons. Write the figure down before the season starts.
The answer
Your own figure. Report the 5th-percentile finishing bank and drawdown from the simulator and size the bank so that season does not hurt.
These are the workings. The argument for them, and the exercises themselves, are in chapter 28 of the book, Simulate Before You Speculate.
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