A signal wins 56% of 200 even-payoff trades. What is the Kelly stake on the point estimate, what is the 95% interval for the win rate, and what stake would you actually run?

Kelly at 56% is a 12% stake, but the 95% interval of 49% to 63% runs from no edge to a 26% stake, so run about half Kelly, 6%. The standard error is 3.5 points. A sceptical prior worth 200 trades at 50% shrinks the estimate to 53%, whose Kelly is 6%. Betting 12% when the truth is 53% earns nothing, while 6% at a true 56% keeps 75% of the maximum growth.

For an even-payoff bet the Kelly stakeThe fraction of capital to bet that maximises the long-run growth rate of wealth, found by maximising the expected logarithm of wealth after each bet. is the win rate minus the loss rate: 0.56 minus 0.44, 12% of capital per trade. At that stake the expected log growth is about 72 basis points a bet, the most any fixed stake can earn if 56% is exact. That if is the whole question. A shopkeeper who stocks up for a festival based on one good season is sizing on a point estimate; the stock that arrives does not care how confident he felt.

Not very. Two standard errors either side gives a 95% interval from 49.1% to 62.9%. Push both ends through the Kelly formula and the stake runs from nothing, because 49.1% is below the 50% at which an even bet has no edge, to 26%. The chance that a coin with no edge at all wins 112 or more of 200 is about 4.4%, so the signal is probably real, but its size is barely pinned down. A Kelly stake inherits all of that uncertainty, multiplied by two.

Because growth is lopsided around the peak. Betting twice the true Kelly stake earns zero growth, while betting half of it keeps three quarters of the maximum. If the true win rate is 53%, its Kelly stake is 6%, and running 12% gives expected log growth of almost exactly zero: all the variance of a strategy, none of the compounding. If instead the truth is 56% and you run 6%, you still collect 54 of the 72 basis points. The drawdowns differ too: in the continuous approximation, full Kelly has a 50% chance of halving capital at some point, and half Kelly about 12.5%.

Shrink the estimate, then apply Kelly to the shrunk number. A backtest win rate is usually the best of several variants tried, so it is biased up. Treat it as if you had also seen 200 trades at 50%: the estimate becomes 53%, its Kelly stake is 6%, which is half Kelly on the raw number. Run that, track the live win rate, and let the stake drift towards the live estimate as trades accumulate. After another 400 live trades the standard error halves, and you will know whether the edge is 53% or 56%.

Say the limits. Kelly assumes independent bets with known odds; real trades cluster, payoffs are not exactly even, and a strategy can stop working altogether, which no amount of shrinkage handles. Half Kelly is a starting point that is robust to the most likely error, not an optimum.