Circular Block Bootstrap: Full-Horizon Return Invariance

Circular block bootstrap: A full-horizon circular block rotates each return once, preserving terminal compounding while changing path risk.

A full-horizon circular bootstrap block can preserve terminal return while changing the observed drawdown. That property follows from the arithmetic of multiplying the same return factors, not evidence that a strategy is stable. This original Tradewink numerical study compares the implemented diagnostic with an independent scalar oracle. The path is synthetic and has no market or account provenance.

Circular block bootstrap: construct a small transparent history

Start at 100 equity units. Supply twenty returns: -10%, -10%, +25%, followed by seventeen zeros. The terminal growth factor is 0.9 × 0.9 × 1.25 = 1.0125. Terminal equity is 101.25 and the simple return is 1.25%. The zero returns are measured fictional cash/flat periods, not placeholders for missing data.

A circular block of length twenty starts at one of those twenty positions, wraps from the sample's end to its beginning, and uses every return exactly once. The multiplication order changes, but the product does not. Our scalar oracle enumerates every possible starting position and verifies the same 1.25% terminal return. The production research helper's 1,000 seeded full-block paths yield matching terminal percentiles.

Drawdown depends on the path

With the two losses adjacent after the initial peak, equity falls from 100 to 90 to 81. Observed drawdown reaches 19%. Start between those losses and the gain separates them: the path can fall 10%, rise to a new peak, then fall 10% again. Maximum drawdown is then 10%. The oracle finds full-block depths ranging from 10% to 19%, despite identical terminal return.

This makes a constant terminal-return interval easy to misread. It says this resampling construction reused the same multiset of returns. It does not say future returns are known or that the supplied return factors recur in the next market period. A nearly empty sensitivity interval can be a property of the sampling rule rather than strong evidence.

Compare block rules without selecting the comforting row

The companion experiment also computes block lengths one and twenty with seed 42 and 1,000 paths. Length one permits different frequencies and ordering of individual returns. Length twenty has only one approximate history block. Those are different declared dependence assumptions, not rival forecasts whose smallest drawdown should win.

The helper exposes thin-history warnings and labels block length in equity observations. Twenty intraday marks and twenty daily marks cover different periods. Neither twenty observations nor the heuristic observation-count divided by block length establishes twenty independent events. The fixture intentionally avoids claiming an effective independent sample size.

What the independent check establishes

The scalar oracle uses a loop over the original return factors, updates a running peak and calculates the largest decline. It does not call the bootstrap implementation to produce its reference depths. Assertions compare terminal values and the 10%/19% extrema against expected arithmetic. This is a contract check for one construction, not a statistical calibration study.

Download the standalone numerical checks and run python3 equity-research-labs.py. It enumerates every full-horizon rotation independently; it does not reproduce the separate NumPy random stream. Preserve its inputs, seed and output before changing an assumption. If you rerun many seeds, display their variability rather than quietly replacing an inconvenient result. A seed makes a computation reproducible; it does not make it representative.

Read the method and the limits together

The arch circular-bootstrap reference describes fixed-length wrapping blocks. Our code implements its own declared circular sampling rule; this article does not claim the arch package generated the reported numbers.

Open Drawdown & recovery in Advisory Research and compare the block rows, history support and unrecovered outcomes. All frequencies remain conditional simulations. This exercise does not correct strategy-selection bias, model unseen crises or recommend a production stop. It explains an observable arithmetic property that researchers should recognize before interpreting a narrow interval.

Continue the research

Intraday Equity Sampling: Audit Short-Hold Risk and Trading Research Costs: Build a Vendor Budget develop adjacent checks. Use a small synthetic fixture first; these educational diagnostics do not validate a trading edge or change account settings.