Kelly Criterion

Kelly Criterion chooses a capital allocation according to the expected logarithm of its multiplier after the next step. With repetition, capital multipliers multiply while their logarithms add. Kelly’s original paper uses this relationship for a long-term growth rate under specified assumptions. Maximizing expected monetary value can produce a different choice, such as an excessive bet with a chance of losing everything. Optimal therefore always refers to a particular objective and model, not to every personal need or every future path. [1]

In the simple model, the fraction f means the share of current capital exposed to complete loss in one step. After each outcome, the monetary amount is recalculated from the new capital. It is therefore neither a constant monetary stake nor automatically a percentage of a trading position’s notional value. Our own checking illustration: with capital of 1000 USD and a fraction of 0.2, the first stake is 200 USD; losing it entirely leaves 800 USD, and the same fraction makes the next stake 160 USD. We assume divisible amounts and no additional costs. [1] [2]

Let p denote the probability of winning and q = 1 - p the probability of losing. A win delivers a net profit of b times the stake, while a loss forfeits the entire stake; b therefore excludes the returned principal. For a nonnegative fraction without borrowing, we maximize g(f) = p ln(1 + b f) + q ln(1 - f), where ln is the natural logarithm. With a nonzero probability of losing, a finite logarithm requires a fraction below one. When both winning and losing have positive probabilities and b is positive, the interior optimum is f = p - q / b. If this is negative, the fraction is zero when taking the opposite bet is prohibited. This formula does not apply unchanged to arbitrary partial losses or multiple different payoffs. [2]

Kelly Criterion does not determine the true probability of winning. If given an incorrect estimate, the calculated size can be unsuitable even when the arithmetic is flawless. Sun and Boyd therefore study a version in which the distribution is not known exactly but belongs to a specified set of possibilities. They optimize the worst logarithmic growth within that set. Even this approach does not protect against possibilities left out of the model. Our example uses two explicitly chosen hypothetical probabilities, not estimates obtained from trading data. [4]

Busseti, Ryu and Boyd separate maximizing long-term growth from constraining the probability of capital falling below a chosen level. Even with a Kelly optimum, capital can fall substantially before growing. Their study adds a separate risk constraint; without it, one cannot claim that the criterion alone guarantees a chosen drawdown limit. A result for long repetition is also no promise of profit after a fixed small number of steps. A practical plan must separately define the threshold beyond which continuing is impossible or unwanted. [3] [5]

Fractional Kelly uses only part of the fraction calculated by the basic criterion. Within the given model, this can change the trade-off between growth and drawdown risk, but it does not universally establish a safe size. Half of an incorrectly estimated fraction still comes from an incorrect input. The risk-constrained study therefore compares growth and risk separately instead of presenting an arbitrary fraction as a guarantee. We recommend no particular fraction here as suitable for a real account. First, it must be clear which outcomes and risks the model includes at all. [3] [4]

A calculation for one isolated binary bet is not automatically a portfolio solution. Simultaneous positions can share adverse scenarios and compete for the same capital. The general Kelly problem models a distribution of joint payoffs and allocates capital among choices. CME also distinguishes leverage, individual losses and total account exposure in a trading plan. Separately calculated fractions therefore cannot simply be added or each applied to the whole account without checking. Joint budget and risk constraints must be preserved. [3] [5]

In a market, a submitted price is not the price of every actual fill. Coinbase describes partial fills, uncertain market-order prices and protection that may not execute during high volatility. Model payoffs therefore need correctly included costs and treatment of the open remainder; the same costs must not be deducted twice. NFA also points to hindsight and the limitations of hypothetical performance. A calculated fraction is therefore not a ready-to-use trading instruction. It is the result of a specified model whose inputs and applicability require separate verification. [6] [7]