Path dependency: why the order of returns matters

Multiplication is commutative: 0.6 times 1.3 times 1.3 gives the same answer in any order. So for an unleveraged holder, the sequence of returns genuinely does not matter — only the set does. Add borrowing and that stops being true, sharply.

The reason is that a maintenance threshold is a level test applied continuously. If a path crosses the level, something irreversible happens: collateral is sold. The position is permanently smaller from then on, so it participates in any recovery at reduced size. Multiplication is commutative; forced sales are not.

Work the standard setup with three annual returns of minus 40, plus 30 and plus 30 percent. Unleveraged, both orderings end at $608,400 — identical, as the algebra requires. With a $300,000 balance and a 70 percent threshold, the two orderings end $110,400 apart.

Here is why. If the decline comes first, collateral falls to $360,000 against a $300,000 balance — an 83 percent loan-to-value, well past the threshold. The cure requires selling $160,000 of collateral, from the forced-sale formula in Track 1. What remains is $200,000 of collateral against $140,000 of balance, and only that smaller base compounds through the two good years. If the decline comes last, no threshold is ever crossed, nothing is sold, and the full base compounds.

This is the single most important interaction between the two halves of this curriculum. Leverage converts a question about the distribution of outcomes into a question about the distribution of paths, and paths are a much larger and less tractable object.