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How much should we stake when we have the edge?

March 18, 2017

How much should we bet when we believe the odds are in our favour. The answer to this question was first formalised in 1956, by daredevil pilot, recreational gunslinger and physicist John L. Kelly, Jr. at Bell Labs. The so-called Kelly Criterion is a formula employed to determine the optimal size of a series of bets when we have the advantage, in other words when the odds favour us. It takes account of the size of our edge over the market as well as the adverse impact of volatility. In other words, even when we have the edge, we can still go bankrupt along the way if we stake too much on any individual wager or series of wagers. 

Essentially, the Kelly strategy is to wager a proportion of our capital which is equivalent to our advantage at the available odds. So if we are being offered even money, and we back heads, and we are certain that the coin will come down heads, we have a 100% advantage. So the recommended wager is the total of our capital. If there is a 60% chance of heads, and a 40% chance of tails, our advantage is now 20%, and we are advised to stake accordingly. This is a simplified representation of the literature on Kelly, Half-Kelly, and other derivatives of same, but the bottom line is clear. It is just as important to know how much to stake as it is to gauge when we have the advantage. But it’s not easy unless we can accurately identify that advantage.

Put more technically, the Kelly criterion is the fraction of capital to wager to maximise compounded growth of capital. The problem it seeks to address is that even when there is an edge, beyond some threshold larger bets will result in lower compounded return because of the adverse impact of volatility. The Kelly criterion defines the threshold, and indicates the fraction that should be wagered to maximise compounded return over the long run (F), which is given by:  

F = Pw – (Pl/W)

where

F = Kelly criterion fraction of capital to bet

W = Pounds won per pound wagered (i.e. win size divided by lose size)

Pw = Probability of winning

Pl = Probability of losing

When win size and loss size are equal, W = 1, and the formula reduces to:

F = Pw – Pl

For example, if a trader loses £1,000 on losing trades and gains £1,000 on winning trades, and 60 per cent of all trades are winning trades, the Kelly criterion indicates an optimal  trade size equal to 20 per cent (0.60-0.40 = 0.20). As another example, if a trader wins £2,000 on winning trades and loses £1,000 on losing trades, and the probability of winning and losing are both equal to 50 per cent, the Kelly criterion indicates an optimal trade size equal to 25 per cent of capital: 0.50- (0.50/2) = 0.25.

Proportional over-betting is more harmful than under-betting. For example, betting half the Kelly criterion will reduce compounded return by 25 per cent, while betting double the Kelly criterion will eliminate 100 per cent of the gain. Betting more than double the Kelly criterion will result in an expected negative compounded return, regardless of the edge on any individual bet. The Kelly criterion implicitly assumes that there is no minimum bet size. This assumption prevents the possibility of total loss. If there is a minimum trade size, as is the case in most practical investment and trading situations, then ruin is possible if the amount falls below the minimum possible bet size.

So should we bet the full amount recommended by the Kelly Criterion? Not so according to sports betting legend, Bill Benter. Betting the full amount recommended by the Kelly formula, he says, is unwise for a number of reasons. Notably, he warns that accurate estimation of the advantage of the bets is critical; if we overestimate the advantage by more than a factor of two, Kelly betting will cause a negative rate of capital growth, and he says this is easily done. So, as he puts it “… full Kelly betting is a rough ride.” According to Benter, and I for one will  defer to his advice in these matters, a fractional Kelly betting strategy is advisable, that is, a strategy wherein one bets some fraction of the recommended Kelly bet (e.g. one half or one third). Ironically, John Kelly himself died in 1965, never having used his own criterion to make money.

So that’s the Kelly criterion. In a nutshell, the advice is only to bet when you believe you have the edge, and to do so using a stake size related to the size of the edge. Mathematically, it means betting a fraction of your capital equal to the size of your advantage. So, if you have a 20% edge at the odds, bet 20% of your capital. In the real world, however, we need to allow for errors that can creep in, like uncertainty as to the true edge, if any, that we have at the odds. So, unless we’re happy to risk a very bumpy ride, and we have total confidence in our judgment, a preferred strategy will to be stake a defined fraction of that amount, known as a fractional Kelly strategy. Purists will hate us for it, but it’s not their capital at risk. So if we are going to bet, the advice is to use Kelly, but with due caution, not least in the assessment of our advantage. And when the fun of betting stops, the best advice of all may of course be to just stop. Good luck!

 

Further Reading and Links

https://selectnetworks.net

 

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