A fund is down 30% on the year. What percentage gain on the reduced capital returns it to the high water mark, and what is the general rule for a fall of size d?

A fund is down 30% on the year. What percentage gain on the reduced capital returns it to the high water mark, and what is the general rule for a fall of size d?

Approach: Invert the surviving capital multiplier rather than reflecting the percentage, then generalise the result to an arbitrary loss and describe how it behaves as the loss grows.

42.857%. Capital falls to 0.70 of the start, so the multiplier back is the reciprocal 1/0.7 = 10/7 = 1.428571. That is a required gain of 3/7, which is 42.857%. The general rule is that a fall of d needs a rise of d/(1 - d), which exceeds d for every positive d and grows without bound as d approaches 1. A 10% fall needs 11.1%, a 20% fall needs 25%, a 50% fall needs 100% and an 80% fall needs 400%. This asymmetry of returns is why drawdown control dominates return chasing, since the required recovery is convex in the loss and two 30% drawdowns in a row need a 104% gain to undo.

Follow-up: A fund is down 30% and charges 20% performance fees above the high water mark. What gross return does an investor need to be whole?

Key concepts: drawdown, asymmetry of returns, reciprocal.