You measure a coin at 55% heads and stake full Kelly on heads at even money. In truth the coin is 55% heads or 45% heads with equal probability and you cannot tell which. What is your expected log growth per flip?

You measure a coin at 55% heads and stake full Kelly on heads at even money. In truth the coin is 55% heads or 45% heads with equal probability and you cannot tell which. What is your expected log growth per flip?

Approach: Compute the unconditional probability of heads first, then evaluate the expected log growth of the stake you actually placed. Compare it with the stake that maximises the expectation over the distribution of p.

-0.50%. Full Kelly on p = 0.55 at even money stakes f = 2p - 1 = 0.10. Unconditionally the coin lands heads with probability 0.5*0.55 + 0.5*0.45 = 0.5, so the expected log growth is 0.5*ln(1.1) + 0.5*ln(0.9) = 0.5*0.09531 - 0.5*0.10536 = -0.00502, a loss of 0.50% per flip. The estimate of p was unbiased and the sizing rule was optimal given it, and the growth is still negative, because the log is concave and the correct stake maximises the expectation of the log over the whole distribution of p rather than at its mean. Here that optimum is zero. Any sizing rule fed a point estimate inherits this concavity penalty, which is a second reason desks size below full Kelly.

Follow-up: What stake maximises expected log growth if the coin is 55% heads with probability 0.7 and 45% otherwise?

Key concepts: kelly criterion, estimation error, expected log growth, concavity.