A signal forecasts a 30 basis point move per trade and impact follows 0.5*sigma*sqrt(x) in the participation rate x, with a daily volatility of 2%. At what participation rate does impact eat the whole forecast, and where should the desk stop instead?

A signal forecasts a 30 basis point move per trade and impact follows 0.5*sigma*sqrt(x) in the participation rate x, with a daily volatility of 2%. At what participation rate does impact eat the whole forecast, and where should the desk stop instead?

Approach: First set impact equal to the forecast. Then maximise total profit, which is size times profit per share, and notice the two answers are far apart.

9%. Setting 0.5*0.02*sqrt(x) = 0.003 gives sqrt(x) = 0.3 and x = 9% of daily volume, the breakeven participation rate where the whole forecast is paid away in market impact. Trading there earns nothing while carrying the full risk. Total profit is x*V*(0.003 - 0.01*sqrt(x)), and differentiating gives 0.003 = 0.015*sqrt(x), so sqrt(x) = 0.2 and the profit maximising rate is x = 4%. At that rate impact costs 0.20% and a third of the forecast survives, so the capacity of the strategy in dollars is 4% of the daily volume of the names it fires on, multiplied by how often it fires. A capacity study that reports the breakeven number rather than the optimum overstates the fund's size by a factor of more than two.

Follow-up: How does the optimal participation rate change if impact has a permanent component that is linear in size?

Key concepts: capacity, market impact, participation rate, breakeven.