A hedge fund index reports 9.0% annual returns across the 200 funds alive today. Each year 10% of funds close, and closing funds average -12% in their final year while survivors average 9.0%. Estimate the true average return across funds active at the start of a year and the size of the bias.

A hedge fund index reports 9.0% annual returns across the 200 funds alive today. Each year 10% of funds close, and closing funds average -12% in their final year while survivors average 9.0%. Estimate the true average return across funds active at the start of a year and the size of the bias.

Approach: Take the expectation over the full starting population rather than the surviving subset, then compound the annual gap to see the effect on terminal wealth.

6.9%. At the start of a year, 90% of funds will survive and average 9.0% while 10% will close and average -12.0% before leaving the index, so the true cross-sectional mean is 0.9*9.0 + 0.1*(-12.0) = 8.1 - 1.2 = 6.9%. The reported 9.0% overstates it by 2.1 percentage points a year. Compounded over ten years, 1.09^10 = 2.367 against 1.069^10 = 1.949, so the reported figure overstates terminal wealth by about 21%. The mechanism is a selection effect: the index is built by conditioning on survival, and survival is correlated with the return being measured, so the sample average no longer estimates the population mean. Backfill bias adds to it, since a fund that joins a database brings its favourable history with it while a fund that never worked never joins at all.

Follow-up: How would you construct an index from the same data that is free of both effects, and what does that construction require the database to have recorded?

Key concepts: survivorship bias, selection effect, backfill bias, conditioning on survival.