Late Gate: Parametric Flight-Delay Stubs
Company technical paper · Version eng-lock.v1 Status: Internal Eng / Product Pricing lock: Takeoff π = $14; Arrival π = $9; B(30/45/60) = $100 / $150 / $200 (both products). Fixed premium; scaled payout. τ internal only.
1. At a glance
Late Gate sells limited-inventory parametric stubs on two outcomes per flight: Takeoff delay and Arrival delay. The traveler picks one delay-buffer tier. The stub pays a fixed cash amount if observed schedule delay meets or exceeds an internal threshold. The house is the sole counterparty in the MVP.
Variables (front-loaded)
| Symbol | Meaning | Locked / default |
|---|---|---|
| π | Premium paid by traveler | Takeoff $14; Arrival $9 |
| B(τ) | Payout if delay ≥ τ | $100 / $150 / $200 at τ = 30 / 45 / 60 |
| τ | Internal threshold (minutes) | Never shown in traveler UI |
| p or p̂ | P(D ≥ τ), true or estimated | Flight-level underwriting input |
| λ | Load = π / E[payout] | Target 1.45 on underwritten book |
| EV_house | π − p · B | Must be > 0 on sold stubs |
Decisions that make the model work
| Decision | Why |
|---|---|
| Fixed π, scaled B | Simple UX; traveler picks Short / Medium / Long buffer |
| Selective underwriting | Locked π is underwater at pool-average p (~18% at ≥30) |
| Target λ = 1.45 | Includes ~20pp adverse-selection / ops buffer vs fair odds |
| CUTOFF + HOT | Shrink purchase-timing adverse selection |
| Inventory caps | FOMO UX and bounded payout exposure |
| Gate-out / gate-in settlement | Matches BTS DepDelay / ArrDelay priors |
Empirical snapshot
BTS Reporting Carrier OTP, winter month + summer month samples, operated non-cancelled non-diverted, n ≈ 1.14M:
| Product proxy | P(≥30) | P(≥45) | P(≥60) |
|---|---|---|---|
Departure / Takeoff (DepDelay) | 18.33% | 13.61% | 10.50% |
Arrival (ArrDelay) | 18.66% | 13.83% | 10.69% |
At locked π and B, expected payout on a random US flight exceeds premium (house EV < 0). Positive house EV is restored on a clean demo / selectively underwritten book (p̂₃₀/₄₅/₆₀ ≈ 8% / 5% / 3.5%).
2. Locked product parameters
| Product code | Traveler name | Settlement metric | BTS proxy | Premium π |
|---|---|---|---|---|
TAKEOFF | Takeoff delay | Gate-out − scheduled dep | DepDelay | $14.00 |
ARRIVAL | Arrival delay | Gate-in − scheduled arr | ArrDelay | $9.00 |
| Internal τ (min) | UI label | Payout B |
|---|---|---|
| 30 | Short buffer | $100 |
| 45 | Medium buffer | $150 |
| 60 | Long buffer | $200 |
Contract rule: Traveler selects one τ per stub. Pays B if observed delay_minutes ≥ τ, else 0 (single-tier, not cumulative).
| Inventory | Default | Demo | Min | Max |
|---|---|---|---|---|
| Stubs per flight × product | 10 | 5 | 3 | 20 |
| Max payout exposure / flight×product | $2,000 | $1,000 | n/a | inventory × $200 |
House is sole counterparty (no user LP) in MVP. Machine-readable twin: rails-pricing-constants.json.
3. Industry baseline rates
3.1 US ≥15 minutes (published)
BTS reporting-carrier late arrivals/departures remain about 20-21% at the ≥15 minute gate standard across recent annual summaries. Marketing-carrier arrival delay is in the same band. Treat ≈20% ±1pp as the industry ≥15 backdrop. Exact ≥30/45/60 come from microdata below (section 3.2), not from the ≥15 reporting standard.
3.2 US ≥30 / 45 / 60 (this study)
Method: TranStats Reporting Carrier OTP for one winter month and one summer month; filter Cancelled=0, Diverted=0; share with DepDelay / ArrDelay ≥ τ.
| Sample | Dep ≥30 | Dep ≥45 | Dep ≥60 | Arr ≥30 | Arr ≥45 | Arr ≥60 |
|---|---|---|---|---|---|---|
| Winter month | 15.47% | 11.27% | 8.57% | 15.94% | 11.54% | 8.78% |
| Summer month | 20.77% | 15.61% | 12.15% | 20.99% | 15.80% | 12.32% |
| Weighted | 18.33% | 13.61% | 10.50% | 18.66% | 13.83% | 10.69% |
Artifacts: data/empirical_delay_rates_*.csv, figures/empirical_delay_cdf.png.
Caveats: Two months only; summer worse; cancellations/diversions excluded from the delay CDF (need separate rules); not conditioned on purchase timing (adverse selection).
3.3 Europe (published aggregates)
EUROCONTROL all-causes: arrivals within 15 min of STA around low-to-mid 70s% in recent annual digests (so >15 around mid-to-high 20s%). Monthly briefs publish delay bands (16-30 / 31-60 / >60). Example monthly shares delayed >30 min often sit in the mid-teens. Exact annual EU p≥45 / p≥60 as a single scalar was not extracted from materials reviewed. Treat as unknown pending CODA microdata or Cirium/OAG.
4. Scaling factors
Heterogeneity dominates pool averages. Summer-month departure sample (p≥30) illustrates why selection works.
| Split | Low example | High example |
|---|---|---|
| Airline | HA 6.6% | AA 28.9% |
| Dep time block | 0600-0659 7.6% | 2000-2059 35.6% |
| Origin (top volume) | SEA 14.5% | CLT 35.9% |
| Day of week | Thu 17.5% | Fri 25.7% |
Full tables: data/empirical_delay_splits_jul2024.csv (filename retains sample label; treat as summer-month split file).
Product implication: Prefer morning banks, historically clean carriers/origins, and off-peak DOW for Arrival inventory (Arrival p_max gates are tighter than Takeoff). Origin airport is a practical gate / airport proxy for local congestion risk until gate-level telemetry is wired.
5. Mathematical model and decisions
5.1 Contract
For product k ∈ {TAKEOFF, ARRIVAL} and traveler-selected internal threshold τ ∈ {30, 45, 60}:
Payout = B(τ) · 1{D_k ≥ τ}
B(30)=100, B(45)=150, B(60)=200Premium π_k is fixed per product (not per τ):
π_TAKEOFF = 14
π_ARRIVAL = 9Expected payout: E[payout] = p_k(τ) · B(τ) where p_k(τ) = P(D_k ≥ τ).
5.2 Load and house EV
π = λ · E[payout] = λ · p · B
EV_house = π − p · BTarget λ = 1.45 on the underwritten book (includes ~20pp adverse-selection / ops buffer vs pure fair odds λ=1).
Equivalently, at locked π only sell when:
p ≤ p_max = π / (λ · B)5.3 Why selective underwriting
Locked π and B are product/UX locks. They are not equal to λ · E[payout] at US pool-average p. At pool rates, implied λ is about 0.4-0.8 (house EV deeply negative). The noteworthy mathematical decision:
5.4 Underwriting gates (p_max at λ=1.45)
| Product | τ | p_max (λ=1.45) | Break-even p (λ=1) |
|---|---|---|---|
| Takeoff | 30 | 9.66% | 14.00% |
| Takeoff | 45 | 6.44% | 9.33% |
| Takeoff | 60 | 4.83% | 7.00% |
| Arrival | 30 | 6.21% | 9.00% |
| Arrival | 45 | 4.14% | 6.00% |
| Arrival | 60 | 3.10% | 4.50% |
5.5 Delay definitions → products
| Industry metric | Definition | Late Gate product |
|---|---|---|
| Gate departure delay | Actual gate-out − scheduled dep | Takeoff delay |
| Wheels-off delay | Runway liftoff − schedule | Not primary trigger |
| Gate arrival delay | Actual gate-in − scheduled arr | Arrival delay |
| BTS "late flight" | ≥15 minutes gate late | Reporting standard only |
Recommendation: Settle Takeoff on gate-out vs published STD; Arrival on gate-in vs published STA. Traveler-facing "Takeoff delay" means departure lateness from schedule, not wheels-up. Never display τ minutes in UI.
5.6 Alternatives if Eng reopens pricing (not locked)
For an unselective US pool book at λ=1.45 and current B: π ≈ $27-$31 both products. Or keep π and reduce B to about $30-$55 (poor UX). Recommendation: keep locked π/B + selection for demo; revisit risk-based π later.
6. House EV under pool vs clean prior
6.1 Pool-average book (underwater)
Weighted winter + summer microdata:
| Product | τ | p | E[payout] | π | House EV | Implied λ |
|---|---|---|---|---|---|---|
| Takeoff | 30 | 18.33% | $18.33 | $14 | −$4.33 | 0.76 |
| Takeoff | 45 | 13.61% | $20.41 | $14 | −$6.41 | 0.69 |
| Takeoff | 60 | 10.50% | $21.00 | $14 | −$7.00 | 0.67 |
| Arrival | 30 | 18.66% | $18.66 | $9 | −$9.66 | 0.48 |
| Arrival | 45 | 13.83% | $20.75 | $9 | −$11.75 | 0.43 |
| Arrival | 60 | 10.69% | $21.38 | $9 | −$12.38 | 0.42 |
6.2 Clean demo prior (recommended)
Assume selectively listed flight with p̂ = {8%, 5%, 3.5%} at {30, 45, 60}:
| Product | τ | E[payout] | House EV | λ |
|---|---|---|---|---|
| Takeoff | 30 | $8.00 | +$6.00 | 1.75 |
| Takeoff | 45 | $7.50 | +$6.50 | 1.87 |
| Takeoff | 60 | $7.00 | +$7.00 | 2.00 |
| Arrival | 30 | $8.00 | +$1.00 | 1.12 |
| Arrival | 45 | $7.50 | +$1.50 | 1.20 |
| Arrival | 60 | $7.00 | +$2.00 | 1.29 |
Arrival at $9 is tight (λ ≈ 1.1-1.3 on this prior). Prefer morning / historically clean flights for Arrival inventory.
6.3 Scenario ladder (research)
| Scenario | p̂30/45/60 | Takeoff EV @τ30 | Arrival EV @τ30 |
|---|---|---|---|
| ultra_clean | 5 / 3 / 2% | +$9.00 | +$4.00 |
| clean_demo | 8 / 5 / 3.5% | +$6.00 | +$1.00 |
| HA-like (summer dep) | 6.6 / 3.6 / 2.4% | +$7.40 | +$2.40 |
| AS-like | 13.3 / 7.9 / 5.2% | +$0.70 | −$4.30 |
| pool_avg | ~18.5 / 13.7 / 10.6% | loss | loss |
See data/pricing_selective_scenarios.csv and figures/house_ev_sensitivity.png.
7. Underwriting gates and risk controls
7.1 Refusal codes
| Code | Rule |
|---|---|
| HOT | Freeze new sales if delay already developing, weather watch, or soft p̂ warn |
| CUTOFF | No sales inside 4h of STD (demo 6h) |
| FULL | Inventory exhausted |
| DUPLICATE | Max 1 open stub per user per flight×product |
| UNDERWRITE_REJECT | p̂ > p_max for product×τ |
| EXPOSURE_CAP | Portfolio / cluster limit hit |
| CANCELLED | MVP: refund premium |
| DIVERTED | Arrival: pay max B or review |
7.2 Portfolio / weather caps
| Limit | USD |
|---|---|
| Max net payout exposure / airport-day | 25,000 |
| Max net payout exposure / airline-day | 40,000 |
| Max net payout exposure / book-day | 150,000 |
| Correlation cluster cap | 15,000 |
| Weather-watch airport remaining-capacity multiplier | 0.35 |
Prefer diversifying away from single-hub evening banks.
7.3 Traveler-safe wording
| Avoid (UI) | Prefer |
|---|---|
| insurance, policy, claim, premium | stub, locked price, payout |
| gamble, bet, odds | delay cushion, buffer tier |
| show "30/45/60 minutes" as τ | Short / Medium / Long buffer |
| house edge, λ, p̂ | Limited stubs on this flight |
Paper/actuarial language is fine internally. Traveler surfaces should stay assistance/ancillary-protection tone, without claiming regulated insurance unless licensed.
7.4 Future depositor pool (optional)
MVP: house sole counterparty. Future sketch: depositors post capital; after traveler payouts and ops, residual edge (λ−1)·E[payout] can be shared (example split 40% house ops / 50% depositor yield / 10% reserve). Depositors inherit house edge only after underwriting and reserves. They do not remove the need for p_max gates. No user LP in current product.
8. Settlement definitions
- Oracle: Production: Cirium or OAG (or airline direct). MVP: partner status API; BTS for offline backtest only (lag).
- Metric: Takeoff = gate-out delay; Arrival = gate-in delay vs published times at purchase lock.
- Observe: After actual event or scheduled time + τ + 30 min buffer.
- Auto-pay if high-confidence source and D ≥ τ; else hold.
- Dispute window: 72 hours.
- Cancelled: refund π (MVP). Diverted: Arrival special case.
9. Data sources
Primary free US prior: BTS Airline On-Time Performance (Reporting Carrier). Europe: EUROCONTROL CODA digests and monthly briefs. Commercial settlement candidates: Cirium Sky API, OAG Flight Status. US airport/carrier metrics: FAA ASPM / ASQP. ADS-B tracking (OpenSky) is not CRS schedule delay.
Labeled links live in References. Local study artifacts: data/, figures/.
Blockers: Cirium/OAG paywalled (needed for live settlement). BTS Excel summary tables often blocked to datacenter IPs (prezip CSVs work). OpenSky unsuitable as sole settlement oracle for schedule delay. Full-year microdata not fully ingested here (two-month sample).
10. Sensitivity and assumptions
10.1 House EV vs p at locked π
See figures/house_ev_sensitivity.png. Zero-crossing at break-even p = π/B. Pool p₃₀ (≈18-19%) lies deep in the loss region for both products.
10.2 Inventory × premium income
| Inventory | Takeoff premium income | Max exposure (τ=60) | Arrival premium income |
|---|---|---|---|
| 5 | $70 | $1,000 | $45 |
| 10 | $140 | $2,000 | $90 |
| 20 | $280 | $4,000 | $180 |
House EV scales with Σᵢ (π − pᵢ Bᵢ) over sold stubs. Negative if pᵢ uncontrolled.
10.3 Traveler fair-odds comparison (clean demo prior)
For Takeoff τ=30: fair premium ≈ $8.00; traveler pays $14 → load 1.75. For Arrival τ=30: fair ≈ $8.00; pays $9 → load 1.12 (thin). Communicate value as certainty of payout rules + limited inventory, not "cheap odds."
10.4 Assumptions (labeled)
- A1: Settlement equals BTS-style gate delay (assumption until Cirium/OAG wired).
- A2: Sold-stub p equals historical flight-level p̂ if CUTOFF/HOT enforced (optimistic without live telemetry).
- A3: Single-tier payout (not stacked 30+45+60).
- A4: Cancellations refund π (not a delay win).
- A5: Clean demo prior 8/5/3.5% is a scenario, not a national statistic.
- A6: Full-year microdata not fully ingested. Expand before production rate cards.
10.5 Rails constants
See rails-pricing-constants.md and rails-pricing-constants.json for drop-in thresholds, premiums, payouts, inventory, λ, refusal codes, and exposure limits.
11. References
- BTS Airline On-Time Performance Database
- BTS Download (Reporting Carrier)
- BTS TranStats annual charts
- BTS Marketing Carrier Annual
- BTS on-time gallery
- EUROCONTROL CODA Digest (annual all-causes delays)
- EUROCONTROL monthly delay brief (example PDF)
- EUROCONTROL network overview review PDF
- FAA ASQP definitions
- FAA ASPM overview.html)
- Cirium Sky subscriptions
- OAG Flight Status
- OpenSky terms of use
- Wang et al. delay CCDF (arXiv HTML)
- AXA HK Express parametric protection
- Cover Genius Delay Valet
- Blink Flight Disruption
- Hopper / OAG case study
- Late Gate empirical outputs: local
data/andfigures/under this docs pack