Triaxial Regime Hazard (TRH)

1Motivation

Hamilton (1989) formalised the idea that asset returns are generated by a small number of hidden regimes. Diebold, Lee & Weinbach (1994) and Filardo (1994) let the transition probabilities between regimes vary with observable information — the TVTP family. Three practical weaknesses persist in desk use. First, all covariates enter a single logistic, so the reasons for a rising transition probability are not separable: a market that is stretched and a market where selling has started receive the same treatment. Second, the regime is latent and inferred by filtering; the filtered probability is noisy and reverses on small data revisions. Third, TVTP has no policy layer — there is nothing between the filtered probability and the label a human reads, so labels whipsaw.

Systemic-risk work points at the missing structure. Adrian, Boyarchenko & Giannone (2019) separate vulnerabilities (slow-moving conditions that make the left tail fatter) from shocks (the events that realise it). Greenwood, Shleifer & You (2019) show that a price run-up alone does not predict a crash, but a run-up combined with valuation and volatility extension roughly doubles the odds of a subsequent drawdown. Abreu & Brunnermeier (2003) show why: a bubble persists until informed traders synchronise — a coordination event that is observable in positioning, not in price. TRH operationalises exactly these three ideas as three axes.

2The decomposition

Let P₀ be the calibrated probability of a ≥5% SPY drawdown within the next 10 sessions given the technical state, m_frag the rotation-cycle fragility multiplier, m_macro the macro-fragility multiplier (§5c), m_event the scheduled-release event-window multiplier (§5b), and τ ∈ {STABLE, ROTATION, BROAD DRAWDOWN} the observed trigger state. TRH defines the posterior hazard

odds(p)  = p / (1 − p)
P*        = odds⁻¹( m_frag · m_macro · m_event · odds(P₀) )
          ⇔   logit(P*) = logit(P₀) + ln m_frag + ln m_macro + ln m_event
verdict   = f( fragility rank , 𝟙[τ = BROAD DRAWDOWN] )
conviction = g( hazard rank )

The multiplicative form is deliberate. In odds space the fragility and event-window terms act exactly like proportional-hazards covariates (Cox, 1972): they scale the hazard without changing what triggers it, and because the terms are additive in log-odds each one is separately disclosable. Adding the axes linearly would let a saturated tape manufacture a drawdown call with no seller in sight, and would let a trigger inflate the probability on days when the tape has plenty of room to absorb it. Under TRH the roles cannot bleed into one another: fragility never opens an event, the trigger never alters the probability, and the base hazard sets conviction.

3Axis 1 — Calibration (hazard)

The Live Market Thermometer composite T ∈ [0, 100] aggregates seven modules (intraday direction, 5-day trend, volume quality, VIX regime, breadth, smart-money flow, cross-asset confirmation) with asymmetric hysteresis on the composite itself. T is mapped to a drawdown probability with an inverse logistic anchored on SPY 1990–2025 quantile statistics:

P₀(T) = clip( 1 / (1 + exp( k · (T − 42) )) ,  0.03 , 0.70 )        k = 0.075
P₁₀%,₃₀d(T) = clip( 1 / (1 + exp( 0.055 · (T − 38) )) , 0.04 , 0.55 )

P₀ is then held on a six-rung ladder with entry and exit thresholds two points apart. Rising risk commits immediately; easing requires two consecutive sessions below the exit threshold and steps down one rung at a time. Inside the buffer band the label holds. The neutral-tape anchor (~12%) is the base rate against which any rung should be read: MODERATELY ELEVATED at 22% is 1.8× the neutral base rate.

4Axis 2 — Market Defender (observed trigger)

Where TVTP infers the regime, TRH observes it. The Market Defender scores revealed institutional positioning — accumulation in inverse and volatility ETFs, breadth divergence against price, VIX term-structure inversion, rotation tilt into defensives, and stealth hedging in low-beta sleeves — and classifies the event type as STABLE, ROTATION (with a sub-type: inverse accumulation, out of tech/semis, into defensives, generic) or BROAD DRAWDOWN (with a severity band: PULLBACK 1–3%, CORRECTION 5–10%, CRASH 10–20%, PANIC 15–30%). Only BROAD DRAWDOWN counts as a trigger.

The severity band is held on its own ladder (STABLE < ROTATION < PULLBACK < CORRECTION < CRASH < PANIC): the read escalates one rung per evaluation with a 60-minute gate between steps, eases one rung after two consecutive confirming sessions, and a raw read back at the held rung resets the easing streak. An inverse-ETF escalator bumps the band one tier when at least two institutional inverse vehicles accumulate simultaneously — the synchronisation event of Abreu & Brunnermeier. The Defender is never an input to P₀ or P*; it is the gate that decides which column of the verdict matrix is active.

5Axis 3 — Market Saturation (fragility)

Fragility is measured as rotation-cycle exhaustion: how much of the market's weight still has room to absorb rotation. The universe is the eleven SPDR sectors, a set of liquid industry ETFs and the eight mega-caps (M8), each weighted by a desk prior ≈ S&P capitalisation share × earnings share × strategic role, with the M8 split by live market capitalisation. Each asset receives a saturation score S ∈ [0, 100] as the mean of four piecewise-linear components:

Index    = Σ wᵢ · Sᵢ  /  Σ wᵢ                       (weighted mean, 0–100)
Runway   = Σ wᵢ · 𝟙[Sᵢ < 50]  /  Σ wᵢ                (share of weight that still has room)
m(level) : EARLY 0.70 · MID CYCLE 1.00 · LATE 1.25 · SATURATED 1.60 · EXHAUSTED 2.20

The multiplier range is calibrated so that the extreme roughly doubles the odds of the drawdown regime — the magnitude the run-up/valuation literature supports (Greenwood, Shleifer & You, 2019) — and so that an early-cycle tape with abundant liquidity sinks discounts the hazard (0.7×). The index runs on the same hysteresis ladder logic as Axis 1 and additionally reports a depth anchor (the extension of the saturated groups above their 200-dma) that bounds the plausible drawdown depth.

5bThe event window — scheduled releases as known-timing shocks (v1.1)

Scheduled macro releases are neither fragility nor a trigger: they are calendar-dated volatility concentrators. Savor & Wilson (2013) show that scheduled announcement days carry a disproportionate share of the equity premium and of return variance; Lucca & Moench (2015) document the pre-FOMC drift. Because P₀ is a 10-session hazard, the density of tier-1 releases inside that window changes how much variance the window can realise. TRH v1.1 therefore adds a second proportional term to the odds:

density  = Σ wₖ  over scheduled releases inside the next 10 NYSE sessions
m_event  = 1 + 0.30 · min( 1 , density / 2.0 )                      → 1.00 … 1.30
weights  : FOMC 1.00 · CPI 0.85 · NFP 0.85 · PCE 0.50 · ISM-Mfg 0.40 · GDP-adv 0.35
           PPI / Retail / ISM-Svc 0.30 · FOMC minutes / quad OPEX 0.15 · weekly claims 0.12 · UMich 0.10

The reference density 2.0 (≈ FOMC + CPI + one tier-2 print) saturates the multiplier, so a maximally loaded fortnight raises drawdown odds by 30% and a quiet one adds nothing. Fixed dates come from the official BLS, BEA, Census and Federal Reserve 2026 schedules; weekly claims (Thursday), ISM (first/third business day), UMich preliminary (second Friday), quarterly expiration (third Friday) and FOMC minutes (decision + three weeks) are rule-generated. The Pro Dashboard's Event Risk Radar shows every release in the window with the average |SPY| close-to-close move on this year's past dates of the same type against the all-session baseline, so the weights can be audited against realised behaviour. After a print, the surprise is not a probability input: if it matters it shows up in positioning within the hour and is read by Axis 2.

5cMacro fragility — the second vulnerability term (v1.1)

Rotation-cycle saturation is a market vulnerability. The macro backdrop is a second, slower one: when labour momentum fades, credit spreads widen or the curve un-inverts, the same trigger travels further. Following Adrian, Boyarchenko & Giannone (2019), macro vulnerability scales the left tail rather than causing the shock, so it enters TRH exactly as saturation does — a proportional odds term with a deliberately narrow range:

index    = Σ wᵢ · sᵢ   sᵢ ∈ [0,100] fragility score of a realised FRED series
series   : initial claims 4w/26w (0.20) · Sahm rule real-time (0.20) · payroll 3m momentum (0.10) · CPI 3m annualised (0.10)
           retail sales 3m annualised (0.10) · industrial production 6m (0.05) · 10y–2y curve (0.10) · HY OAS level + 60d Δ (0.10) · UMich (0.05)
regime   : SUPPORTIVE <30 → 0.90 · NEUTRAL ≥30 → 1.00 · SOFTENING ≥45 → 1.10 · DETERIORATING ≥60 → 1.20 · STRESSED ≥75 → 1.30
           (2-point hysteresis buffer on the way down)

Only realised data is used — no consensus feed — so the index measures the trend of the macro backdrop rather than day-of surprises. Post-release surprises are handled where they belong: if a print matters, positioning reacts within the hour and Axis 2 reads it. The consensus-surprise extension (a Citi-style surprise index) is deferred until a licensed consensus source is available.

5dNews: hypothesis, not evidence

Headlines are never an input to P₀, the multipliers or the trigger. A tariff headline that does not move VIX term structure, credit, gold, the dollar or Treasuries is not an event. TRH therefore uses news only for attribution: given the flow-revealed positioning state and the last ~18 hours of market-level headlines (Fed, geopolitics, energy, tariffs, credit, macro data), an attribution model names the most plausible driver and states whether flows confirm it — or states that the read is flow-driven with no dominant headline. The output is a label on the AB-AI read; removing it changes nothing in the probability.

6Composition and the verdict matrix

The fragility rank (Axis 3) and the trigger state (Axis 2) cross into four cells. Conviction is read from the hazard rank (Axis 1): a risk-off call is HIGH conviction only when the calibrated hazard is at least MODERATELY ELEVATED; a benign call is HIGH conviction only when the hazard is at most MODERATE.

P* runs its own ladder (same rungs as Axis 1) and is the single source of truth for every public surface: the home-page drawdown pill reads the P* rung, the Pro Dashboard shows P₀ → P* with the multiplier, and the AB-AI Market Read folds the verdict into its top-line synthesis so no surface can contradict another.

6bCompanion layer — the Level Map (v1.1)

TRH answers how likely and what kind; it says nothing about where. The Level Map is a companion layer, deliberately outside the probability: multi-timeframe support and resistance for SPY and QQQ — 200/50/20-day averages, 52-week and prior-year/quarter extremes, monthly and weekly pivots, the YTD anchored VWAP, round numbers, prior-week and prior-day extremes, fractal swing points, unfilled gaps and the 5-day VWAP — clustered into zones when within 0.35% and scored by confluence (×2), long-term membership (×1.5) and tests over 250 sessions (×0.5, capped). The Defender severity band supplies a depth range; the Level Map returns the strongest confluent support inside it, which becomes the verdict's price and its invalidation: "a CORRECTION-band move lands at 693–732, which contains the 729 swing-low cluster; the temporary-dips read holds while 729 holds." A break of a MAJOR zone on volume is an observable event and is read by Axis 2 — it never changes P₀ or the multipliers.

7Worked example — today's live read

8Why the ladders matter

Each axis is held on an asymmetric hysteresis ladder — in control terms a Schmitt trigger. Entry thresholds are above exit thresholds, escalation is immediate (risk-off is fast), and easing requires confirmation (risk-on is slow). The consequence is that a ±2-point wobble in the composite, or a single quiet session in the Defender, cannot flip a label. This is the policy layer TVTP lacks: the filtered probability may be as noisy as it likes, the read a human sees changes only when the evidence has persisted.

9Relationship to the literature

TRH is best described as a decomposed TVTP. It keeps the central object of Diebold–Lee–Weinbach and Filardo — a transition probability into a drawdown regime that varies with observable information — but factors it into a base hazard and a proportional fragility term (the Cox structure), replaces the latent regime with an observed trigger state (Abreu–Brunnermeier synchronisation, made measurable), imports the vulnerability-versus-shock separation of Adrian–Boyarchenko–Giannone, and anchors the fragility magnitude on Greenwood–Shleifer–You. The composite's cross-asset and breadth modules echo Kritzman et al. (2011): fragility rises when the tape's variance concentrates into few groups. Campbell & Shiller (1988) supply the valuation component's rationale.

10Limitations and validation programme

TRH is presented as a framework, not yet a validated model. The calibration anchors are priors fitted to historical quantiles, not maximum-likelihood estimates; the fragility multipliers (0.7×–2.2×) are judgment calibrated to the literature; the universe weights are a desk prior. The validation programme, already scheduled, is: (i) a walk-forward backtest of P₀ and P* against realised 10-session SPY drawdowns with Brier score and reliability diagrams; (ii) hit-rate and depth statistics by verdict cell; (iii) comparison against an unconditional base rate and a VIX-only baseline; (iv) re-fitting the sigmoid and the multiplier grid on the accumulated snapshot history. Until then, every read is a statistical observation published by a financial publisher — not a forecast, not advice.

Companion study. TRH prices the odds that a drawdown regime begins; it does not date the drawdown. Suppressed Hazard Convergence (SHC) is the ex-post companion: price pressure against suppression capacity, the tunnel between them, and the touch that dates the moment an elevated hazard realizes — with results on every SPY drawdown ≥ 5 % since 2007.

Live scorecard — item (i) of the validation programme

Loading the scorecard…

11References

1. Abreu, D. & Brunnermeier, M. K. (2003). Bubbles and Crashes. Econometrica, 71(1), 173–204.
2. Adrian, T., Boyarchenko, N. & Giannone, D. (2019). Vulnerable Growth. American Economic Review, 109(4), 1263–1289.
3. Campbell, J. Y. & Shiller, R. J. (1988). Stock Prices, Earnings, and Expected Dividends. Journal of Finance, 43(3), 661–676.
4. Cox, D. R. (1972). Regression Models and Life-Tables. Journal of the Royal Statistical Society B, 34(2), 187–220.
5. Diebold, F. X., Lee, J.-H. & Weinbach, G. C. (1994). Regime Switching with Time-Varying Transition Probabilities. In Nonstationary Time Series Analysis and Cointegration, Oxford University Press.
6. Filardo, A. J. (1994). Business-Cycle Phases and Their Transitional Dynamics. Journal of Business & Economic Statistics, 12(3), 299–308.
7. Greenwood, R., Shleifer, A. & You, Y. (2019). Bubbles for Fama. Journal of Financial Economics, 131(1), 20–43.
8. Hamilton, J. D. (1989). A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle. Econometrica, 57(2), 357–384.
9. Kritzman, M., Li, Y., Page, S. & Rigobon, R. (2011). Principal Components as a Measure of Systemic Risk. Journal of Portfolio Management, 37(4), 112–126.
10. Lucca, D. O. & Moench, E. (2015). The Pre-FOMC Announcement Drift. Journal of Finance, 70(1), 329–371.
11. Savor, P. & Wilson, M. (2013). How Much Do Investors Care About Macroeconomic Risk? Evidence from Scheduled Economic Announcements. Journal of Financial and Quantitative Analysis, 48(2), 343–375.