Hedge Effectiveness & Derivative Projection

M-110 is a Tier 2 hedge-and-derivative model that does two related jobs for a variable-annuity (VA) guarantee / interest-rate hedging program: it projects the derivative book forward through maturity and replacement, and it measures hedge effectiveness against the hedged item under ASC 815 / IFRS 9. Three InsModel components are registered against M-110 (see Components); two are the engines that produce the card's snapshot and one is the registered scenario driver:

• DerivativeProjectionEngine (insmodel.L4.derivative_projection, governance ID M-403) — the forward projection and Greeks/MV/P&L engine. Computes the Output Snapshot.
• HedgeEffectivenessCalculator (utility class in firmmodel/engines/hedge_effectiveness.py, consumed by HedgingEngine, governance ID M-401) — the dollar-offset / regression / variability-reduction effectiveness battery. Computes the effectiveness verdict.
• StochasticEngine (insmodel.L4.stochastic_engine, governance ID M-301, registry category Stochastic) — the correlated Monte-Carlo economic-scenario generator (CIR/Hull-White rates, GBM-with-Merton-jumps equity, O-U inflation, mean-reverting credit spreads, GBM FX) whose calculate() emits a path DataFrame with columns scenario_id, time, interest_rate, equity_return, credit_spread_{aaa,aa,a,bbb,hy}. Registered but not exercised in this card's deterministic snapshot path. It is the intended scenario driver behind the Description's "under stochastic scenarios" language: in a stochastic run its interest_rate / equity_return / credit-spread path columns feed DerivativeProjectionEngine as the scenario_paths input, with one DerivativeProjectionEngine._step per scenario time-step. The deterministic Output Snapshot below instead feeds a single hand-set scenario (interest_rate 4.0%, equity_return +1.0%, vol 20%) so the snapshot is reproducible with no MC draw — so M-301 contributes no numbers to this card's snapshot.

1. Derivative projection (monthly time-stepping). The projection engine carries an internal portfolio state across three instrument classes — IR swaps, equity options, and credit derivatives — initialized from configured notionals and average maturities. Each _step (one month, dt = 1/12) does the following:

1. Roll equity spot by the period equity_return and decay every class's remaining maturity by one month.
2. Mature and replace. A fixed fraction notional / (original_maturity_months) matures each month; replacement_ratio (default 1.0 = full replacement) of that is rolled into a new position, with the class's remaining maturity re-weighted toward the original tenor.
3. Charge hedge cost. New swaps and credit hedges cost hedge_cost_bps of new notional; new equity options cost option_premium_pct of new notional.
4. Mark to market each class: - IR swaps — pay-fixed PV via an annuity factor: PV = N · (rate − fixed_rate) · Σ DF_i·τ_i (positive when float > fixed). - Equity options — a long-put book priced with Black-Scholes (K·e^{−rT}·N(−d2) − S·e^{−qT}·N(−d1)), scaled by notional / K. - Credit derivatives — written-protection spread-duration proxy: MV = −N · spread · min(T, duration) (negative, widens as spreads rise).
5. Recompute portfolio Greeks — analytical Black-Scholes delta/gamma/vega for the equity puts, finite-difference DV01 (1 bp bump-and-reprice) for the IR swap book, and a spread-duration CS01 for credit (folded into portfolio_dv01).
6. Emit derivative_pnl = net_hedge_position − hedge_cost, the per-class MVs, net hedge position, matured/new notional, and the Greeks.

2. Hedge effectiveness (ASC 815 / IFRS 9). HedgeEffectivenessCalculator consumes two aligned series — period fair-value changes of the hedging instrument and of the hedged item — and runs three independent tests:

• Dollar-offset (ASC 815-20-35): ratio = −ΣΔHedge / ΣΔHedgedItem, effective when in [0.80, 1.25].
• OLS regression: fits ΔHedge = α + β·ΔHedgedItem; effective when R² ≥ 0.80, slope β ∈ [−1.25, −0.80], and the F-test p-value < 0.05. The p-value uses scipy.stats.f, with a pure-numpy regularized-incomplete-beta fallback if scipy is unavailable.
• Variability reduction (IFRS 9, prospective): variance_ratio = Var(ΔHedge + ΔHedgedItem) / Var(ΔHedgedItem); effective when the combined (residual) variance is below the unhedged variance.

run_all_tests returns each test plus an overall_effective verdict that is true when at least 2 of 3 tests pass. This is the qualifying gate for hedge accounting and the "residual/unhedged risk" measure the card reports.

Prose. The hedging program is modeled as a long-put equity hedge plus pay-fixed rate swaps, the standard posture for VA guarantee and interest-rate hedging. The long-put book carries negative delta, positive gamma, positive vega (it gains as equities fall / vol rises), which is exactly what offsets a VA guarantee's short-equity-tail exposure. The IR swap book is pay-fixed, so it gains when rates rise — offsetting the duration mismatch on a long-duration liability. Credit derivatives are modeled as written protection (negative MV that worsens as spreads widen), reflecting a yield-enhancement sleeve rather than a hedge. All Greeks are instantaneous sensitivities; discrete monthly rebalancing error is not captured.

Deterministic run — DerivativeProjectionEngine v1.0.0 (one monthly step) + HedgeEffectivenessCalculator (six-period offset series). Reproducible, requires no live firm data (python scripts/model_snapshots.py M-110 in InsModel; the projection path is asserted by tests/test_derivative_projection.py).

Input: calibration-default book (IR swaps $232.7B @ fixed 3.5% / 7yr · equity puts $104.4B / 2yr ATM · credit $37.1B / 5yr) · period-1 step at interest_rate 4.0%, equity_return +1.0%, vol 20%, replacement_ratio 1.0, hedge_cost 5 bps, option premium 3%. Effectiveness series: hedged-item ΔFV [100, −150, 200, −80, 120, −60], hedge ΔFV [−92, 138, −185, 75, −110, 56].

Prose. The projection produces an internally consistent hedge posture: a put book that is short-delta, long-gamma, long-vega (the VA-guarantee hedge profile), pay-fixed swaps marking positive because the 4.0% scenario rate sits above the 3.5% fixed rate, and written credit protection marking negative. The hedge effectiveness battery passes all three tests — a 0.91 dollar-offset ratio, a near-perfect regression with slope −0.92, and a 99.4% variance reduction — so the relationship qualifies for hedge accounting and the residual (unhedged) risk is the 91.67 combined-position variance against a 15,680.56 unhedged variance. Note the absolute projection magnitudes are large because the canonical book uses full-scale calibration-default notionals ($374B aggregate); they are illustrative of the engine mechanics, not a claim about any firm's actual book — see Limitation 6.

Captured 2026-06-04 · deterministic, no live data.

1. Period P&L is a simplified identity, not a full attribution. The projection emits derivative_pnl = net_hedge_position − hedge_cost, i.e. current MV minus roll cost — it does not subtract the prior-period MV. So the first-period "P&L" is essentially the standing mark plus cost, not a true period change. The richer delta/gamma/vega/theta/rho/basis/unexplained attribution lives in HedgingEngine._calculate_pnl_attribution (M-401) and is not wired into the projection path. A card consumer wanting period-over-period P&L must difference net_hedge_position across rows or invoke M-401 separately.
2. Flat volatility, no skew or term structure. Equity options use a single default_equity_vol per class; there is no smile, skew, or vol-term-structure. For a put-heavy VA hedge this understates the cost and vega of deep-OTM tail protection.
3. No CVA/DVA/FVA or counterparty risk. Neither engine prices counterparty credit, funding, or collateral valuation adjustments. Net hedge MV is a clean mid-market mark.
4. Credit leg is a spread-duration proxy only. Credit derivatives are valued as −N·spread·min(T, duration) with no default-leg / premium-leg decomposition, no recovery assumption, and no CDS curve. CS01 is similarly a duration proxy folded into portfolio_dv01 rather than reported as a separate output.
5. Effectiveness inputs — projection-integrated, production-inputs still exogenous. HedgeEffectivenessCalculator consumes ΔHedge / ΔHedgedItem series. As of the RAT-110-v1.0.0 remediation the two halves of M-110 are integration-tested: tests/test_hedge_projection_integration.py drives the DerivativeProjectionEngine forward and feeds its produced equity-option ΔMV and delta-matched hedged-item series into the effectiveness battery (recorded result: dollar-offset 0.7868 ineffective, regression R² 0.9998 / slope −0.9544 effective, variability 99.77% effective → overall effective, 2-of-3). The Output Snapshot below still uses a deterministic illustration series; and a production run must feed real per-period hedge and hedged-item fair-value changes (e.g. from the VA-guarantee reserve engine) rather than the calibration-input derivation. The remaining gap is the production data feed, not the engine wiring.
6. Calibration-default notionals carry a 10-K provenance string that is NOT an independently verified claim (BV-032). The $232.7B / $104.4B / $37.1B defaults and the "PRU FY2024 Note 5" string come from the engine's legacy_metadata. This card makes no 10-K accuracy claim; the figures are engine calibration defaults used to make the snapshot deterministic. Subject to the BV-032 firm-data divergence caveat, they should not be read as a firm's reported derivative book.
7. (Closed by RAT-110-v1.0.0 remediation.) HedgeEffectivenessCalculator now has a dedicated test module (tests/test_hedge_effectiveness.py, 22 tests) covering the dollar-offset / OLS-regression / IFRS 9 variability paths and the 2-of-3 verdict including the effective/ineffective band boundaries. The derivative projection retains its thorough suite (tests/test_derivative_projection.py, 50 tests). See Validation Packet.
8. Greeks are instantaneous; monthly rebalancing only. Discrete intra-month rebalancing error and path-dependent gamma P&L are not captured; rebalancing is a once-a-month roll.

Tracked for ratification (not applied in this documentation pass). The following are output-changing / modeling-code items left for ratification, noted here for transparency: integrate the projection and effectiveness engines so dollar-offset / regression / variability run on real projected hedge + hedged-item MV series instead of the deterministic illustration, and wire greek-attribution P&L from HedgingEngine (M-401) into the projection path (INV-019); add a dedicated test module for HedgeEffectivenessCalculator plus a projection→effectiveness integration test and record a real validation result for last_validated_on; replace the simplified derivative_pnl = net_hedge_position − hedge_cost identity with a true period change carrying the greek/theta attribution that currently exists only in HedgingEngine._calculate_pnl_attribution. No model outputs were fabricated or changed in this documentation pass.

Overall status: scoped Tier-2 evidence assembled; 2L re-review pending. Registry lifecycle.last_validated_on is 2026-06-06 — a recorded validation result now exists (the projection→effectiveness integration verdict, below). The 2L technical ratification is RAT-110-v1.0.0 (decision: conditionally_approved). Following the RAT-110-v1.0.0 remediation: (1) a dedicated test module for HedgeEffectivenessCalculator exists (tests/test_hedge_effectiveness.py, 22 tests — COND-110-01); (2) a projection→effectiveness integration test exists (tests/test_hedge_projection_integration.py — COND-110-02); (3) the recorded validation result populates last_validated_on; (4) the scoped evidence pack is assembled at modelling/validation_evidence/M-110/v1.0.0/README.md. What remains for unconditional sign-off: 2L re-review of COND-110-01 / COND-110-02 (and the external countersign, pending per Decision 028). The table below records the validation that exists.

Validation evidence note (BV-032 / honesty): the projection path, the effectiveness path, and the projection↔effectiveness integration are now all tested (79 tests total across the three modules). M-110 should be read as "two validated component engines wired through a tested integration on calibration-input series" — the remaining honest gap is a production data feed (real per-period hedge / hedged-item fair-value changes), not engine wiring. Full scoped evidence: modelling/validation_evidence/M-110/v1.0.0/README.md.

Accounting / regulatory: - FASB ASC 815 Derivatives and Hedging — hedge accounting; dollar-offset (ASC 815-20-35) and regression effectiveness methods; effectiveness window [0.80, 1.25]. - IFRS 9 Financial Instruments §6.4 — prospective effectiveness assessment; the variability-reduction basis for the IFRS 9 test. - Federal Reserve SR 26-2 (April 2026, supersedes SR 11-7) — model risk management; the governance metadata (TestGovernance) encodes the SR 11-7 narrative.

Quantitative methods: - Black, F. & Scholes, M. (1973), The Pricing of Options and Corporate Liabilities, JPE — closed-form European option pricing for the equity put book. - Black, F. (1976) — Black's model (used by the sibling HedgingEngine for swaptions). - Hull, J. (2018), Options, Futures, and Other Derivatives, 10th ed. — derivative projection, roll mechanics, Greek evolution. - Fabozzi, F. (2012), Bond Markets, Analysis, and Strategies — DV01 and key-rate duration for the IR swap leg.

Engine source: - firmmodel/engines/derivative_projection.py — DerivativeProjectionEngine (governance ID M-403), the projection / MV / Greeks engine. - firmmodel/engines/hedge_effectiveness.py — HedgeEffectivenessCalculator, the ASC 815 / IFRS 9 effectiveness battery. - firmmodel/engines/hedging_engine.py — HedgingEngine (governance ID M-401), the sibling pricing / Greeks / P&L-attribution engine that consumes the effectiveness calculator.

Tests: - tests/test_derivative_projection.py — 50 tests / 13 classes covering construction, initialization, projection, valuation, Greeks, hedge cost, governance, contract, and edge cases for M-403. - tests/test_hedge_effectiveness.py — 22 tests covering the dollar-offset / OLS-regression / IFRS 9 variability paths, the 2-of-3 verdict, and the effective/ineffective band boundaries of HedgeEffectivenessCalculator (COND-110-01). - tests/test_hedge_projection_integration.py — projection→effectiveness integration: drives DerivativeProjectionEngine and feeds its produced series into HedgeEffectivenessCalculator, recording the validation result (COND-110-02).

Internal: - Engine legacy_metadata: firmmodel/governance/legacy_metadata/DerivativeProjectionEngine.yaml and HedgingEngine.yaml — SR 11-7 narrative, references, assumptions, limitations, and the calibration-default provenance string (BV-032 caveat applies).

Card change history. Code-side change history lives in git log of the component files.

• 2026-05-08 — stub created from registry data per Decision 023 Phase 5 / B-07.
• 2026-06-04 — hand-authored to M-001 depth: Methodology (derivative projection + ASC 815 / IFRS 9 effectiveness battery), Key Assumptions and Their Justification, Output Snapshot (deterministic, no live data), Limitations and Known Gaps, Validation Packet, and References, grounded in DerivativeProjectionEngine (M-403) / HedgeEffectivenessCalculator / HedgingEngine (M-401) source and legacy_metadata. Stub marker advanced to ``.
• 2026-06-06 — code-grounded documentation completeness pass. Documented the third registered component StochasticEngine (M-301, category Stochastic) and its interest_rate / equity_return / credit-spread path output, marking it registered-but-not-exercised in the deterministic snapshot path, and reconciled the "backed by two components" wording with the three engines in Components. Added a Standards Coverage section (ASC 815 / ASC 815-20-35, IFRS 9 §6.4, ASOP 56) to complete the 13-section structure. Added an explicit Validation-Packet honesty status line (validation pending — last_validated_on null, 2L peer-review pending — ratification_ref null, effectiveness-calculator test module absent). Added a D049 documentation-currency note (unstamped / pending baseline) to the header. Noted gated/ratification items (INV-019, effectiveness-test module + integration, derivative_pnl attribution) under Limitations. No model outputs, back-test numbers, or validation results were fabricated or changed.
• 2026-06-06 — RAT-110-v1.0.0 1L remediation (Decision 053 §2.2). Closed COND-110-01 by adding tests/test_hedge_effectiveness.py (22 tests: all three ASC 815 / IFRS 9 paths + 2-of-3 verdict + band boundaries). Closed COND-110-02 by adding tests/test_hedge_projection_integration.py (effectiveness battery on projection-produced series; recorded result: overall effective 2-of-3) and setting lifecycle.last_validated_on to 2026-06-06. Reconciled COND-110-03 (registry documentation_pack.model_card/validation_evidence → present). Assembled the scoped Tier-2 evidence pack at modelling/validation_evidence/M-110/v1.0.0/README.md; re-stamped doc-currency (fingerprint 6c0c3389f350a5bd, 2026-06-06). Output snapshot md5 unchanged (6deda4779f731f26907e3a12ece3f6a0) — no model output changed. No validation results, back-test numbers, or model outputs were fabricated.