Chapter 29 · Worked solutions
Chapter 29 workings: How Much to Stake: Kelly Is Responsible Gambling in Disguise
Chapter twenty-nine worked: full and quarter Kelly stakes for four prices from 9/4 to evens, and which of them should not be bet at all.
Kelly stakes a share of the current bank equal to (bp − q) ÷ b, where b is the fractional odds as a number (9/4 is 2.25), p is your probability and q = 1 − p. A negative answer means no bet. Quarter Kelly is the setting I use.
| Price | Your p | Edge | Full Kelly | Quarter | Verdict |
|---|---|---|---|---|---|
| 9/4 | 35 per cent | +13.75% | 6.11% | 1.53% | Bet |
| 5/2 | 30 per cent | +5.00% | 2.00% | 0.50% | Bet |
| 11/10 | 50 per cent | +5.00% | 4.55% | 1.14% | Bet |
| evens | 48 per cent | −4.00% | −4.00% | −1.00% | No bet |
The exercises
9/4 at 35 per cent
Full and quarter Kelly at 9/4 with your probability at 35 per cent, and whether to bet.
Edge
p × decimal odds − 1 = 0.35 × 3.25 − 1 = +13.75 per cent.
Full Kelly
(bp − q) ÷ b, with b = 2.25 the fractional odds: (2.25 × 0.35 − 0.65) ÷ 2.25 = 6.11 per cent of the bank.
Quarter Kelly
6.11 ÷ 4 = 1.53 per cent.
Decision
Positive, so a bet at the quarter stake.
The answer
Edge +13.75 per cent; full Kelly 6.11 per cent, quarter 1.53 per cent. Bet.
5/2 at 30 per cent
Full and quarter Kelly at 5/2 with your probability at 30 per cent, and whether to bet.
Edge
p × decimal odds − 1 = 0.30 × 3.50 − 1 = +5.00 per cent.
Full Kelly
(bp − q) ÷ b, with b = 2.5 the fractional odds: (2.5 × 0.30 − 0.70) ÷ 2.5 = 2.00 per cent of the bank.
Quarter Kelly
2.00 ÷ 4 = 0.50 per cent.
Decision
Positive, so a bet at the quarter stake.
The answer
Edge +5.00 per cent; full Kelly 2.00 per cent, quarter 0.50 per cent. Bet.
11/10 at 50 per cent
Full and quarter Kelly at 11/10 with your probability at 50 per cent, and whether to bet.
Edge
p × decimal odds − 1 = 0.50 × 2.10 − 1 = +5.00 per cent.
Full Kelly
(bp − q) ÷ b, with b = 1.1 the fractional odds: (1.1 × 0.50 − 0.50) ÷ 1.1 = 4.55 per cent of the bank.
Quarter Kelly
4.55 ÷ 4 = 1.14 per cent.
Decision
Positive, so a bet at the quarter stake.
The answer
Edge +5.00 per cent; full Kelly 4.55 per cent, quarter 1.14 per cent. Bet.
evens at 48 per cent
Full and quarter Kelly at evens with your probability at 48 per cent, and whether to bet.
Edge
p × decimal odds − 1 = 0.48 × 2.00 − 1 = −4.00 per cent.
Full Kelly
(bp − q) ÷ b, with b = 1 the fractional odds: (1 × 0.48 − 0.52) ÷ 1 = −4.00 per cent of the bank.
Quarter Kelly
−4.00 ÷ 4 = −1.00 per cent.
Decision
Negative: the formula is telling you the bet is on the wrong side of fair. Do not bet.
The answer
Edge −4.00 per cent; full Kelly −4.00 per cent, quarter −1.00 per cent. No bet.
Explaining 0.8 per cent to my nan
Take the chapter's headline example, 13/8 at 40 per cent, and justify the quarter-Kelly stake in one plain sentence.
The numbers
Full Kelly = (1.625 × 0.40 − 0.60) ÷ 1.625 = 3.08 per cent; a quarter of that is 0.77 per cent.
Why so small
Kelly assumes your 40 per cent is exactly right. It is an estimate, and staking the full amount on an overestimate shrinks the bank. A quarter keeps most of the growth and takes away most of the pain.
The sentence
My version: “Nan, I'm only risking pennies in the pound because even when I'm right I lose more than half of these, and I want to still be here after the bad month.”
The answer
Full Kelly 3.08 per cent, quarter 0.77 per cent: small because the probability is an estimate and the losing runs are long.
These are the workings. The argument for them, and the exercises themselves, are in chapter 29 of the book, How Much to Stake: Kelly Is Responsible Gambling in Disguise.
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