A desk has two even-money bets, each winning 55% of the time. Size them with Kelly alone, then jointly when they are independent, and then when their outcomes have correlation 0.5.

Alone, each bet takes 10% of the bankroll; together and independent, 9.9% each; with correlation 0.5, about 6.6% each. Kelly for an even-money bet is 2p - 1. Jointly, the stake that maximises expected log growth is (P(both win) - P(both lose)) / (2 x (P(both win) + P(both lose))), and correlation moves probability into the both-lose corner. Staking 10% on each regardless would cut growth from 0.67% to 0.49% a round at correlation 0.5, and to zero if the bets were identical.

A fruit seller who reinvests everything in tomorrow's stock cares about the growth rate of her money over many days, not the profit on one day, and a single day she cannot recover from matters more than any good day. The Kelly stakeThe fraction of bankroll that maximises the expected logarithm of wealth, and so the long-run growth rate, for a repeated bet. for an even-money bet is 2p - 1, which at p = 0.55 is 10% of the bankroll, and it grows the bankroll by 0.55 x ln(1.1) + 0.45 x ln(0.9) = 0.50% a round on average. Bet 20% and growth falls to about zero; bet more and the bankroll shrinks in the long run despite the edge. That asymmetry, in which overbetting is punished harder than underbetting, is the reason the joint problem matters.

With a stake f on each bet there are four outcomes a round: both win, worth 1 + 2f; one of each, worth exactly 1; both lose, worth 1 - 2f. Expected log growth is P(both win) x ln(1 + 2f) + P(both lose) x ln(1 - 2f), the mixed outcomes contribute nothing, and setting the derivative to zero gives f = (P(both win) - P(both lose)) / (2 x (P(both win) + P(both lose))). By symmetry the two stakes are equal, so one number does both. If the bets were independent, P(both win) = 0.55 squared = 0.3025 and P(both lose) = 0.45 squared = 0.2025, so f = 0.10 / (2 x 0.505) = 9.90%: a shade under 10% each, 19.8% in total, and growth of 1.00% a round, almost double the single bet. Two independent edges are nearly additive, which is the whole case for diversification.

Correlation moves probability from the mixed outcomes into the corners. With rho = 0.5, P(both win) = 0.3025 + 0.5 x 0.2475 = 0.4263, P(both lose) = 0.2025 + 0.5 x 0.2475 = 0.3262, and each mixed outcome falls to 0.1237. The stake is then 0.10 / (2 x 0.7525) = 6.64% on each bet, 13.3% in total, with growth of 0.67% a round: the two bets together are worth only about a third more than one bet alone, because half the time they are the same bet. At correlation 1 the formula gives 5% each, 10% in total, exactly the single-bet answer, since two identical bets are one bet in two accounts.

Run the single-bet 10% on both when the true correlation is 0.5 and growth falls to 0.49% a round, less than one bet alone earns, because 20% of the bankroll now rides on outcomes that lose together 33% of the time. If the bets turned out to be identical, 10% on each is 20% on one bet, double Kelly, and the growth rate is about zero: all that edge, and the bankroll goes nowhere. The practical answer for Rudhira is a fraction of the joint figure, half-Kelly or so, about 3% to 3.5% on each bet. The 55% is itself an estimate from a backtest, the correlation is measured on the same data and rises in stress, and the penalty for overbetting is far steeper than for underbetting. Say that limitation plainly: the formula gives the ceiling, and the desk should run well below it.