What actually happens in a trough

Two moving parts. That is the whole model.

log(fund) = m × log(index) − (m²−m)/2 × variance

The first term is direction and it scales linearly with leverage. The second is cost and it scales with leverage squared. Nothing else enters. Not the order of the days, not where the trough sits, not how scary it felt.

1. The coefficient, and the sign flip

Why the bear fund bleeds twice as fast as the bull fund.

m² is identical for +3 and −3. Subtracting m is what flips: the coefficient goes 3 → 6. The inverse fund carries exactly double the drag of its bull twin. That is the "sign flip" — not a separate phenomenon, just the −m term in the same formula.

2. A real trough, day by day

SOXX (the index), SOXL (3×), SOXS (−3×). Jan–Oct 2025, all indexed to 0% on 15 Jan.

SOXX — the index, 1× SOXL — 3× long SOXS — 3× short

At the trough on 8 Apr the index was down 30.5% and SOXL was down 72.4%. By 31 Oct the index had recovered to +38.1% — and SOXL was at +60.6%, not 3× that. SOXS, the bear fund, ended at −83.0% having been up 137% at the bottom. It gave back everything and more.

3. The order of the days does not matter at all

Six different arrangements of the same twenty daily moves (ten +5%, ten −5%).

Every path lands on exactly the same place: index −2.47%, 3× fund −20.35%. Because log(fund) = Σ log(1 + m·r) is a sum, and addition does not care about order. So a trough is not something that happens to you in a sequence you could have dodged. It is just a stretch with high variance, and variance is the only thing that enters.

4. Same index move, different volatility

This is the part that has been missing.

1× 2× 3× −3× (bear)

Read the −3× bars. The index falls 30% in both cases. In calm 15% volatility the bear fund makes +155%. In wild 40% volatility it makes +12%. Same index move.

So the bear fund is a bet on direction and a bet against volatility — and those two fight each other, because the volatility that creates the crash is the thing that eats the bear fund.

A crash is not a direction event you can harvest with leverage. It is a variance event, and every leveraged product — bull or bear — is short variance.

Table view

Same numbers, for reading rather than looking.

Prices: Alpaca SIP, split and dividend adjusted. Panels 1, 3 and 4 are computed from the identity; panel 2 is measured. The identity is a continuous-time approximation and stops being a usable point estimate once annualised volatility approaches 100%.