You must buy 10% of a stock's daily volume. One day of trading costs 0.5*sigma*sqrt(0.10) in impact with sigma = 2% daily, and five days at 2% participation costs 0.5*sigma*sqrt(0.02) while leaving the unfilled part exposed. Which schedule do you pick and what does it cost?

You must buy 10% of a stock's daily volume. One day of trading costs 0.5*sigma*sqrt(0.10) in impact with sigma = 2% daily, and five days at 2% participation costs 0.5*sigma*sqrt(0.02) while leaving the unfilled part exposed. Which schedule do you pick and what does it cost?

Approach: Price both schedules for impact, then price the timing risk of a linear schedule of length T, whose shortfall has standard deviation sigma*sqrt(T/3). Compare the saving against the extra dispersion.

Trade it in one day unless the alpha survives a week. Five days saves 0.316% - 0.141% = 17.5 basis points of impact and raises the shortfall standard deviation from sigma*sqrt(1/3) = 1.155% to sigma*sqrt(5/3) = 2.582%. The variance rises from 1.33 to 6.67 in units of 10^-4, so indifference needs a risk aversion of 0.00175/0.000533 = 3.3 in units where cost is traded directly against shortfall variance, and most equity desks run well above that. A decaying signal settles the execution horizon on its own: alpha still unexecuted on day five has already been paid away, so the opportunity cost exceeds both the impact saving and the timing risk. The five day schedule only wins for a slow signal in a name where the participation rate would otherwise be extreme.

Follow-up: At what alpha half life does the five day schedule become optimal for this order?

Key concepts: market impact, timing risk, execution horizon, risk aversion.