# Kelly Criterion ## The Idea in Brief The Kelly criterion is a mathematical formula that tells you how much of your capital to risk on a favourable bet or investment. It aims to maximise the long-term growth of wealth by finding the optimal balance between risk and reward. Unlike strategies that focus on short-term gains, the Kelly approach guards against ruin while ensuring that money grows at the fastest possible rate over time. --- ## Key Concepts ### 1. The Formula The basic Kelly formula is: $ f^* = \frac{bp - q}{b} $ - **f*** = fraction of current capital to bet - **b** = net odds received on the wager (e.g. betting £1 at 2-to-1 odds means $b = 2$) - **p** = probability of winning - **q** = probability of losing ($q = 1 - p$) If the result is negative, you should not bet. --- ### 2. Maximising Growth - The Kelly criterion does not maximise expected returns in a single trial. - Instead, it maximises the _expected logarithmic growth rate_ of wealth over many repetitions. - This makes it suited to repeated opportunities rather than one-off bets. --- ### 3. Overbetting vs. Underbetting - **Overbetting** (staking more than the Kelly fraction) can lead to rapid losses and even ruin. - **Underbetting** (staking less) reduces the chance of ruin further but grows wealth more slowly. - Practitioners often use _fractional Kelly_ (e.g. half Kelly) to strike a balance between safety and growth. --- ### 5. Limitations - Requires accurate estimates of probabilities and returns, which are uncertain in real markets. - Ignores behavioural factors such as risk aversion and fear of losses. - Assumes repeated, independent opportunities — which may not always exist in finance. --- ## Example Suppose you are offered a bet with: - 60% chance of winning ($p = 0.6$), - Even odds (1-to-1, so $b = 1$). The Kelly fraction is: $ f∗=(1×0.6)−0.41=0.2f^* = \frac{(1 \times 0.6) - 0.4}{1} = 0.2 $ You should bet **20% of your bankroll** each time.