A forecast decays exponentially with a five day half life. You rebalance every ten days and hold positions fixed in between. What fraction of the signal's full strength do you capture on average?

A forecast decays exponentially with a five day half life. You rebalance every ten days and hold positions fixed in between. What fraction of the signal's full strength do you capture on average?

Approach: Average the exponential decay over the holding interval, then compare the alpha captured at different intervals against the trading each one implies.

54.1%. A five day half life gives lambda = ln(2)/5 = 0.1386 per day, and holding for ten days captures the average of exp(-lambda*t) over t in [0, 10], which is (1 - exp(-1.386))/1.386 = 0.75/1.386 = 0.541. Rebalancing every five days captures (1 - 0.5)/0.6931 = 72.1% and daily rebalancing captures 93.4%. Signal decay is exponential, so each halving of the interval roughly doubles turnover, so the optimal frequency sets the marginal alpha gained against the marginal cost paid. With a 10 basis point round trip and 3% of annual alpha at stake, going from ten days to daily adds 39% of the alpha and multiplies the cost by ten, so the optimum for a five day half life sits at a few days.

Follow-up: What rebalancing interval is optimal when cost is linear in turnover and the alpha decays exponentially?

Key concepts: signal decay, half life, rebalancing frequency, turnover.