Risk Limit Calibration Against Historical Drawdowns

When to Use

Invoke this skill when a strategy needs a maximum-drawdown limit, a daily loss limit and a position-size scalar, and the alternative is picking a round number. A limit set too tight halts the strategy on ordinary noise; a limit set too loose never fires before the capital is gone. This engine derives all three from the strategy's own realized daily returns: observed peak-to-trough drawdown and its duration, the Ulcer Index, historical VaR/Expected Shortfall at a configurable confidence, and — where the history supports it — a peaks-over-threshold generalized Pareto fit of the loss tail.

Every number this skill produces is your own risk policy. Nothing surveyed in references/standards.md sets a drawdown or daily-loss figure for a trading firm. MiFID II RTS 6 Art. 15(4) requires an investment firm to set market and credit risk limits from its own capital base, clearing arrangements, strategy and risk tolerance — it prescribes the obligation to calibrate, not the result. Never present these outputs to an auditor as regulatory minimums.

When NOT to Use

• As the enforcement mechanism. This is an offline calibration step that returns numbers. Runtime enforcement must be an independent control outside strategy logic — kill-switch-and-drawdown-circuit-breakers and portfolio-level-stop-loss-independent-of-strategy-stops.
• On a history shorter than the calibration window. The engine requires 252 observations by default and refuses below 126 (~6 months) whatever you pass. A 99% tail needs at least 100 observations to contain a single loss at all; below that the "99% VaR" is the worst day in a short sample wearing a confidence level it has not earned.
• To learn about a loss the strategy has never taken. HISTORICAL_MAX_DD is bounded above by the worst outcome already in the sample. EXTREME_VALUE_THEORY extrapolates past it, but only in the shape the fitted tail implies — neither invents a gap, a limit-down or a liquidity crunch. Pair with stress-testing-against-historical-crash-scenarios and scenario-based-stress-testing-custom-shocks.
• To get a distribution of drawdowns. Resampling the return series to obtain the 95th-percentile max drawdown across simulated paths is monte-carlo-strategy-robustness-testing. This engine reads one realized path.
• On returns that are not fractional returns on account equity. 0.02 means +2%. Currency P&L compounded as a return produces meaningless drawdowns; a return at or below -1.0 is rejected rather than compounded through zero.
• On serially dependent or volatility-clustered returns, for the horizon-scaled methods. PARAMETRIC_VAR and EXTREME_VALUE_THEORY convert a per-period figure to horizon_days under an IID assumption. HISTORICAL_MAX_DD carries no such assumption and is the safer default when that assumption is doubtful.

Prerequisites

• Daily fractional returns on account equity, chronologically ordered, ending at the last completed session, with no gaps. At least min_observations (default 252); the hard floor is 126.
• Portfolio capital in USD, finite and strictly positive.
• A calibration window that includes at least one adverse regime. Basel's analogue for banks is instructive: the stressed ES observation horizon "must, at a minimum, span back to and include 2007" (MAR33.6). Calibrating on a benign year produces a limit that is only valid in a benign year.
• Policy choices, all of which default but none of which are mandated: stress_buffer_multiplier (>= 1.0), target_confidence_pct, horizon_days, the drawdown-limit floor and cap, the daily-loss VaR multiple, and the position-scalar threshold.

Workflow

1. Validate the return series before measuring anything.

   • Non-finite returns and returns <= -1.0 are rejected, not cleaned. Decision point: a single NaN reaching the daily loss limit makes that limit NaN, and loss > NaN is False for every loss — the control can never fire. Resolve the gap in the data; do not calibrate around it.

2. Compute realized drawdown metrics.

   • Equity curve, peak-to-trough max drawdown, longest run strictly below the running peak, Ulcer Index (Martin & McCann: the root mean square of percentage drawdowns from the running peak), daily mean and volatility.
   • Decision point — if drawdown_unrecovered is true, the series ends below its peak and the duration is right-censored. The true recovery time is not yet observable, so do not report the duration as a recovery time.

3. Compute historical VaR and Expected Shortfall at target_confidence_pct.

   • Order statistics: with n observations and confidence q, k = ceil((1-q)·n); VaR is the k-th smallest return negated, ES the negated mean of the k smallest. ES ≥ VaR by construction.
   • Decision point — if VaR is zero, the sample contains no loss at that confidence. The engine raises rather than issuing a $0 daily loss limit. Extend the window to include a losing regime.

4. Pick a calibration method deliberately — they do not measure the same thing.

   • HISTORICAL_MAX_DD: observed max drawdown × stress buffer. The only method that measures an actual drawdown. Bounded by the sample.
   • PARAMETRIC_VAR: the h-day cumulative loss quantile under IID normal, $\text{loss}_h = -h\mu + z_q\sigma\sqrt{h}$. Drift scales with $h$, volatility with $\sqrt{h}$ — scaling a one-day VaR (which already embeds $-\mu$) by $\sqrt{h}$ mis-scales the drift. This is a fixed-window loss, and therefore a lower bound on the drawdown over a window of the same length: a drawdown maximises over every start point inside the window.
   • EXTREME_VALUE_THEORY: peaks-over-threshold GPD fit of the loss tail, giving a per-day tail VaR/ES, scaled to horizon_days by $\sqrt{h}$. Decision point: the fit raises rather than degrading when the tail has too few exceedances, is degenerate, or when the requested confidence sits below the fitted threshold. A raise is the correct answer; a fallback to another method would put a number in the audit record that the named method did not produce.

5. Apply the policy floor and cap, and record which one bound.

   • The limit is clipped into [drawdown_limit_floor_pct, drawdown_limit_cap_pct] for every method. Decision point — if floor_binding or cap_binding is true, the limit was set by policy, not by the return sample. Do not report it as a measurement of the strategy's risk.

6. Derive the daily loss limit and position scalar, then hand the record to review.

   • Daily loss limit = capital × historical VaR × daily_loss_var_multiple, computed from the unrounded VaR.
   • Position scalar = position_scalar_threshold_pct / observed max drawdown when the observed drawdown exceeds the threshold, otherwise 1.0.
   • CalibratedRiskLimits carries metrics, tail_fit, limit_basis, the binding flags and audit_notes, so the record states what the number is and how it was produced.

Full procedure: see references/workflows.md. Standards reference: see references/standards.md. Printable pre-flight checklist: see assets/checklist.md.

Common Pitfalls

• Treating the three methods as interchangeable estimates of one quantity. A daily VaR is not a drawdown. Multiplying a one-day tail loss by an unexplained constant does not turn it into a drawdown limit; the horizon and the scaling rule must be stated, and limit_basis states them.
• Scaling a VaR that contains a drift term by $\sqrt{h}$. Over $h$ days the drift contributes $h\mu$, not $\sqrt{h}\mu$. On a strategy with meaningful positive drift the two differ by several percentage points of drawdown limit — in the direction of an over-generous limit.
• Calling a magic multiplier "EVT". Extreme Value Theory means fitting a tail distribution and extrapolating from the fit, with a threshold, an exceedance count and shape/scale parameters you can inspect. If those are not in the output, whatever produced the number was not EVT.
• Reading a floored or capped limit as a calibration result. A benign sample floored to 5% and a catastrophic sample capped at 50% both look like calibrated numbers and are not. Check floor_binding / cap_binding.
• Reporting a censored drawdown duration as a recovery time. A series that ends underwater has not recovered; its longest underwater run is a lower bound.
• Setting a stress buffer below 1.0. That sets a limit tighter than a loss the strategy has already survived and lived through, guaranteeing a halt the next time it repeats. The engine rejects it.
• Calibrating a 99% tail on a short window. With $n$ observations the 99% tail is the worst $\lceil 0.01n \rceil$ of them; at $n = 10$ that is one observation and the "confidence level" is decoration. The engine refuses below 126 observations and below the count needed to contain a single tail loss.
• Trusting the GPD shape on a very heavy tail. The method-of-moments estimator is structurally bounded above by $\xi = 0.5$, so it cannot represent an infinite-variance tail and will understate one. The module logs a warning at $\xi \ge 0.25$.
• Feeding non-finite or ruinous returns. A NaN produces a limit that no comparison ever breaches; a return below -1.0 drives equity through zero and every subsequent drawdown figure is meaningless.

Verification

• Instantiate DrawdownLimitCalibratorEngine() (defaults: 1.5× buffer, 99% confidence, 252-observation window, 20-day horizon, 5%/50% floor/cap).
• Feed a 252-observation series that peaks at 1.0, falls exactly 25%, recovers to a new high, then takes three −0.1% losses: verify metrics.max_drawdown_pct == 25.0, calibrated_max_drawdown_pct == 37.5 (1.5 × 25), position_size_scalar == 0.8 (20 / 25), and both binding flags false.
• Feed [-0.10] + [0.0] × 251: verify ulcer_index == 10.0 exactly and drawdown_unrecovered is true.
• Feed [0.002] × 249 + [-0.03, -0.04, -0.05]: with k = ceil(0.01 × 252) = 3, verify var_pct == 3.0 and cvar_pct == 4.0.
• Feed [0.001 + 0.01, 0.001 - 0.01] × 126 with PARAMETRIC_VAR: $\mu = 0.001$, $\sigma = 0.01\sqrt{252/251}$, so the 20-day limit is $1.5 \times (-20\mu + z_{99}\sigma\sqrt{20}) = 12.6367%$ — not the $14.9635%$ produced by $\sqrt{h}$-scaling a one-day VaR.
• Verify the GPD fit against a hand-computed case: 246 gains, one loss at the threshold, five exceedances with excesses $0.001\ldots0.005$ gives mean $0.003$ and sample variance $2.5\times10^{-6}$, hence $\xi = -1.3$ and $\beta = 0.0069$ exactly, with POT VaR $= 1.312965%$ and ES $= 1.436072%$ at 99%.
• Negative checks: a NaN return, a return <= -1.0, a 10-observation series, non-positive or non-finite capital, a stress buffer below 1.0, a confidence level the window cannot support, a degenerate tail, and an EVT request with too few exceedances must each raise a CalibrationError subclass.
• Run python -m unittest discover -s skills/risk-limit-calibration-against-historical-drawdowns/scripts and confirm a 100% pass rate.

Related Skills

• kill-switch-and-drawdown-circuit-breakers
• portfolio-level-stop-loss-independent-of-strategy-stops
• monte-carlo-strategy-robustness-testing
• risk-limit-breach-escalation-matrix
• risk-metric-recalculation-frequency-tuning
• stress-testing-against-historical-crash-scenarios
• value-at-risk-var-live-monitoring