Equation Registry

EQ-CLM-001 Climate CRITICAL Scenario

Transient Climate Response (TCR) warming estimate

\[ T(t) = \mathrm{TCR} \cdot \frac{\ln(C(t)/C_0)}{\ln 2} \]

Warming above pre-industrial is proportional to TCR and the log ratio of current to pre-industrial CO₂ concentration.

Variables

$T(t)$	Global mean surface temperature anomaly (°C)
$TCR$	Transient Climate Response (°C per CO₂ doubling)
$C(t)$	Atmospheric CO₂ concentration at time t (ppm)
$C_0$	Pre-industrial CO₂ baseline (278 ppm)

Assumptions (3)

TCR is treated as a single representative scalar (best estimate 1.8°C, likely 1.2–2.4°C per AR6)
Instantaneous equilibration of forcing to temperature (ignores ocean heat uptake lag)
Log-linear forcing from CO₂ only; non-CO₂ forcings captured in scenario multipliers

Units: °C Source: IPCC AR6 WG1 Chapter 4; Matthews et al. 2009 Nature Uncertainty: TCR uncertainty (±0.6°C at 1σ) dominates temperature pathway uncertainty up to 2050 Reviewed: 2026-01-15 Derived from: EQ-EMI-001

EQ-CLM-002 Climate CRITICAL Scenario

Carbon budget remaining (integrated emissions constraint)

\[ B_{\mathrm{rem}} = B_{\mathrm{total}} - \int_{t_0}^{t} E(\tau)\,d\tau \]

Remaining carbon budget equals total IPCC-assessed budget minus cumulative emissions since the reference year.

Variables

$B_rem$	Remaining carbon budget (GtCO₂)
$B_total$	Total IPCC SR1.5/AR6 budget from reference year (GtCO₂)
$E(τ)$	Annual net CO₂ emissions at time τ (GtCO₂/yr)
$t_0$	Reference year (2020 for AR6 budgets)

Assumptions (3)

Linear interpolation of annual emissions between scenario waypoints
Non-CO₂ forcing converted to CO₂-equivalent using GWP100 from AR6
Permafrost carbon feedbacks partially included per AR6 Table SPM.2 footnotes

Units: GtCO₂ Source: IPCC AR6 WG1 Table SPM.2; Friedlingstein et al. 2022 ESSD Uncertainty: Budget uncertainty ±220 GtCO₂ at 1σ (IPCC AR6); dominated by TCRE distribution Reviewed: 2026-01-15

EQ-CLM-003 Climate HIGH Scenario

Emissions pathway abatement requirement

\[ A(t) = E_{\mathrm{ref}}(t) - E_{\mathrm{target}}(t) \]

Required abatement is the gap between the reference-policy emissions trajectory and the target pathway.

Variables

$A(t)$	Required annual abatement at year t (GtCO₂e/yr)
$E_ref(t)$	Reference policy emissions trajectory (GtCO₂e/yr)
$E_target(t)$	Scenario target pathway (GtCO₂e/yr)

Assumptions (3)

Reference pathway is 'current policies' from UNEP EGR 2024 (57.4 GtCO₂e in 2025)
Linear ramp assumption for abatement deployment between waypoints
LULUCF net emissions treated separately from energy-system abatement

Units: GtCO₂e/yr Source: UNEP Emissions Gap Report 2024; IEA World Energy Outlook 2024 Uncertainty: Reference pathway uncertainty ±3 GtCO₂e/yr; target pathway dependent on budget choice Reviewed: 2026-02-01 Derived from: EQ-CLM-002

EQ-DAM-001 Physical Damage CRITICAL Scenario

DICE-style damage function (quadratic)

\[ D(T) = \frac{\alpha T^2}{1 + \alpha T^2} \]

Fraction of GDP lost to climate damage as a quadratic function of temperature anomaly, calibrated to Nordhaus DICE-2023.

Variables

$D(T)$	Fractional GDP damage
$T$	Global mean temperature anomaly above pre-industrial (°C)
$α$	Damage coefficient (0.00267 in DICE-2023 calibration)

Assumptions (4)

Damages are symmetric around global mean temperature (ignores distributional heterogeneity)
Quadratic form implies accelerating but bounded damage — challenged by tipping-point literature
α = 0.00267 from DICE-2023; alternative calibrations range 0.001–0.01
Does not capture catastrophic or non-linear tipping point damage

Units: fraction of GDP Source: Nordhaus (2023) DICE model; Howard & Sterner (2017) meta-analysis Uncertainty: Damage function specification is the single largest uncertainty in long-run economic cost; estimates vary by 10× across literature Reviewed: 2026-01-20 Derived from: EQ-CLM-001

EQ-DAM-002 Physical Damage HIGH Scenario

Annualised average loss (catastrophe actuarial)

\[ \mathrm{AAL} = \int_0^{\infty} L(p)\,dp \approx \sum_i \frac{L_i + L_{i+1}}{2} \cdot (p_i - p_{i+1}) \]

Expected annual loss is the area under the exceedance probability curve — the actuarial integral over all loss-return-period combinations.

Variables

$AAL$	Annualised average loss ($bn/yr)
$L(p)$	Loss at exceedance probability p ($bn)
$p_i$	Exceedance probabilities at each return period level
$L_i$	Ground-up loss at return period i

Assumptions (3)

Poisson process for event occurrence (memoryless, independent events)
Stationarity of hazard distribution (violated under climate change — CE applies non-stationarity adjustment)
Vulnerability functions are sector-constant within broad industry categories

Units: $bn/yr Source: RMS, AIR, catastrophe model industry standard; Guy Carpenter (2023) Uncertainty: Model-to-model AAL variation ±30–50% across major cat model vendors for any given region/peril Reviewed: 2026-01-20

EQ-DAM-003 Physical Damage MEDIUM Scenario

Labour productivity loss from heat stress (WBGT model)

\[ \mathrm{LPL} = \max\!\left(0,\; \beta_1 \cdot (\mathrm{WBGT} - \mathrm{WBGT}_{\mathrm{thresh}}) + \beta_2 \cdot (\mathrm{WBGT} - \mathrm{WBGT}_{\mathrm{thresh}})^2\right) \]

Labour productivity loss increases with wet-bulb globe temperature above a threshold, calibrated to sector-specific outdoor/indoor labour mix.

Variables

$LPL$	Labour productivity loss (fraction of potential output)
$WBGT$	Wet-bulb globe temperature (°C)
$WBGT_thresh$	Sector threshold (27°C outdoor heavy, 32°C indoor light)
$β₁, β₂$	Sector-calibrated coefficients from Kjellstrom et al. 2016

Assumptions (3)

Linear-quadratic form is a first-order approximation; non-linear tipping points not modelled
Adaptation (air conditioning, schedule shifting) is captured in a separate adaptation service
Sector labour shares from ILO 2023; outdoor exposure fractions from WHO

Units: fraction of potential GDP Source: Kjellstrom et al. (2016) Lancet Planetary Health; Burke et al. (2015) Nature Uncertainty: β coefficient uncertainty ±40%; WBGT projection uncertainty from GCM spread dominates Reviewed: 2026-01-20 Derived from: EQ-CLM-001

EQ-ECO-001 Economics CRITICAL Scenario

Marginal abatement cost curve (MAC) — carbon price equilibrium

\[ C^* = \frac{\partial \mathrm{TC}}{\partial A}\bigg|_{A = A^*} \]

The equilibrium carbon price equals the marginal abatement cost at the policy-target abatement level.

Variables

$C*$	Equilibrium carbon price ($/tCO₂e)
$TC$	Total transition cost function ($tn)
$A$	Annual abatement (GtCO₂e/yr)
$A*$	Policy-target abatement level

Assumptions (3)

Smooth, convex MAC curve — in practice, MAC curves have discontinuities at technology step changes
Perfect competition in carbon markets (no market power, no transaction costs)
MAC curve calibrated to IEA NZE scenario technology cost projections

Units: $/tCO₂e Source: IPCC AR6 WG3 Chapter 3; IEA Net Zero 2050 (2021) Uncertainty: Carbon price range for 1.5°C: $50–250/t by 2030 (IPCC AR6 Table 13.SM.2) Reviewed: 2026-02-01 Derived from: EQ-CLM-003

EQ-ECO-002 Economics HIGH Scenario

Structural growth with climate adjustment (augmented Solow)

\[ Y(t) = A(t) \cdot K(t)^\alpha \cdot L(t)^{1-\alpha} \cdot \Omega(T,P) \]

Output is an augmented Cobb-Douglas production function where Ω(T,P) is a climate-policy multiplier reducing productivity under both physical damage and transition costs.

Variables

$Y(t)$	Real GDP
$A(t)$	Total factor productivity
$K(t)$	Capital stock
$L(t)$	Effective labour
$α$	Capital income share (~0.35)
$Ω(T,P)$	Climate-policy damage multiplier (0–1)

Assumptions (3)

Constant returns to scale in K and L
Capital income share α = 0.35 (OECD average; varies significantly across developing economies)
Climate damage enters multiplicatively (not additively) — implies no threshold catastrophe

Units: USD (real) Source: Solow (1956); Dietz & Stern (2015) DICE extension; Burke et al. (2015) Uncertainty: Ω specification drives long-run GDP uncertainty; α uncertainty ±0.05 has minor near-term impact Reviewed: 2026-02-01 Derived from: EQ-DAM-001

EQ-ECO-003 Economics CRITICAL Scenario

Social Cost of Carbon (Ramsey discounting)

\[ \mathrm{SCC} = \int_t^{\infty} \frac{\partial D(\tau)}{\partial E_t} \cdot e^{-\rho(\tau - t)}\,d\tau \]

The SCC is the present value of all future marginal damages from one additional tonne of CO₂ emitted today, discounted at the pure rate of time preference ρ.

Variables

$SCC$	Social cost of carbon ($/tCO₂)
$D(τ)$	Climate damage at time τ
$E_t$	Emissions at time t
$ρ$	Pure rate of time preference (Nordhaus: 1.5%; Stern: 0.1%)

Assumptions (3)

Ramsey utility discounting with CRRA utility function
SCC range driven almost entirely by choice of ρ: Nordhaus ≈$20, Stern ≈$200, EPA 2023 ≈$190
Damage function specification second-largest driver (see EQ-DAM-001)

Units: $/tCO₂ Source: Nordhaus (2017) AER; Stern (2006); US EPA (2023) revised SCC Uncertainty: SCC values range $15–$400 in mainstream literature; discount rate choice is fundamentally an ethical/political decision Reviewed: 2026-02-01 Derived from: EQ-DAM-001, EQ-ECO-002

EQ-FIS-001 Fiscal MEDIUM Scenario

Carbon revenue as fraction of GDP

\[ R_c = \tau_c \cdot E_{\mathrm{covered}} \cdot \frac{1}{\mathrm{GDP}} \]

Carbon revenue (as share of GDP) equals the carbon tax/ETS price times covered emissions divided by GDP.

Variables

$R_c$	Carbon revenue (% GDP)
$τ_c$	Carbon price ($/tCO₂e)
$E_covered$	Covered emissions under the carbon pricing scheme (GtCO₂e/yr)
$GDP$	Nominal GDP ($tn)

Assumptions (3)

100% pass-through of carbon price to covered emitters
Coverage rate: 40% global average (current); 75% in ambitious policy scenario
Carbon revenue recycling mechanism does not affect macro growth in this model (first-order only)

Units: % of GDP Source: IMF Fiscal Monitor 2023; World Bank Carbon Pricing Dashboard Uncertainty: Coverage rate uncertainty ±20 percentage points; pass-through uncertainty ±30% Reviewed: 2026-02-01 Derived from: EQ-ECO-001

EQ-FIN-001 Finance HIGH Scenario

Weighted average cost of capital (WACC) — climate-adjusted

\[ \mathrm{WACC} = \frac{E}{V} R_e + \frac{D}{V} R_d (1-T_c) + \Delta_{\mathrm{climate}} \]

Climate-adjusted WACC adds a climate risk premium to the standard WACC formula, reflecting higher physical and transition risk in the cost of equity and debt.

Variables

$E/V$	Equity share of capital structure
$R_e$	Cost of equity (CAPM-derived)
$D/V$	Debt share of capital structure
$R_d$	Cost of debt (pre-tax)
$T_c$	Corporate tax rate
$Δ_climate$	Climate risk premium addition (0–200bps depending on scenario)

Assumptions (3)

Climate risk premium is additive to baseline WACC
Physical risk premium calibrated to cat bond spreads and TCFD disclosures
Transition risk premium calibrated to fossil fuel stranded asset write-down scenarios

Units: % Source: Modigliani-Miller (1958); IPCC AR6 WG3 Chapter 15; GFANZ (2023) Uncertainty: Climate risk premium estimates vary 0–300bps across methodologies; sector-specific calibration adds ±100bps Reviewed: 2026-02-10 Derived from: EQ-ECO-002, EQ-DAM-001

EQ-UNC-001 Uncertainty MEDIUM Diagnostic

Monte Carlo expected value and confidence interval

\[ \hat{\mu} = \frac{1}{N}\sum_{i=1}^{N} f(\theta_i), \quad \mathrm{CI}_{90} = [P_{5}(f), P_{95}(f)] \]

Monte Carlo mean and 90% confidence interval estimated from N=2000 samples drawn from parameter distributions.

Variables

$μ̂$	Sample mean of output f
$N$	Number of Monte Carlo samples (2000 in CE)
$θ_i$	Parameter vector drawn from joint distribution
$P_5, P_95$	5th and 95th percentiles of output distribution

Assumptions (3)

N=2000 samples sufficient for stable P5/P95 estimates (verified by convergence test)
Parameter distributions are independent (covariance structure partially captured via δ factor)
Normal/triangular/uniform distributions — tails may be heavier in reality

Units: depends on output variable Source: Metropolis & Ulam (1949); IPCC AR6 uncertainty guidance; CE internal calibration Uncertainty: Sampling error in P5/P95 ~±2% at N=2000; parameter distribution assumptions dominate Reviewed: 2026-01-10

EQ-UNC-002 Uncertainty HIGH Scenario

Portfolio abatement de-duplication factor (overlap discount)

\[ A_{\mathrm{net}} = A_{\mathrm{gross}} \cdot (1 - \delta) \]

Net abatement equals gross portfolio abatement discounted by factor δ to account for double-counting of overlapping mitigation pathways (e.g. electrification + grid decarbonisation).

Variables

$A_net$	Net unique abatement (GtCO₂e/yr)
$A_gross$	Sum of all individual technology/policy abatement claims
$δ$	De-duplication factor (default 0.22; range 0.10–0.35)

Assumptions (3)

δ = 0.22 is a central estimate; empirically poorly constrained
Covariance structure: ρ_elec = 0.21 for energy technologies; ρ_CDR = 0.25 for carbon removal
Linear scaling of overlap with portfolio size — likely underestimates overlap at high ambition levels

Units: fraction Source: IEA NZE 2023 scenario accounting; Luderer et al. (2019) Nature Climate Change Uncertainty: δ range 0.10–0.35 translates to ±5–7 GtCO₂e/yr uncertainty in net abatement at 30 Gt gross Reviewed: 2026-02-01

EQ-EMI-001 Emissions MEDIUM Reference

Kaya identity — CO₂ emissions decomposition

\[ E = P \cdot \frac{G}{P} \cdot \frac{E_{\mathrm{prim}}}{G} \cdot \frac{E_{\mathrm{CO_2}}}{E_{\mathrm{prim}}} \]

Total CO₂ emissions equal population × per-capita GDP × energy intensity of GDP × carbon intensity of energy.

Variables

$E$	CO₂ emissions (GtCO₂/yr)
$P$	Population (billions)
$G/P$	Per-capita GDP ($/person)
$E_prim/G$	Primary energy intensity (EJ/$tn GDP)
$E_CO₂/E_prim$	Carbon intensity of energy (GtCO₂/EJ)

Assumptions (3)

Additively separable drivers — in reality, population, income, and technology co-evolve
Non-CO₂ GHGs expressed as CO₂-equivalent using GWP100 (AR6 values)
LULUCF emissions accounted separately

Units: GtCO₂/yr Source: Kaya (1990); IPCC AR6 WG3 Figure SPM.4 Uncertainty: Identity is exact; uncertainty enters through projection of each driver Reviewed: 2026-01-15

EQ-EMI-002 Emissions MEDIUM Diagnostic

Transition pressure composite index

\[ P_{\mathrm{trans}} = w_1 \cdot s_{\mathrm{path}} + w_2 \cdot s_{\mathrm{policy}} + w_3 \cdot s_{\mathrm{emission}} + w_4 \cdot s_{\mathrm{shock}} \]

CE's internal transition pressure score is a weighted composite of pathway intensity, policy regime stringency, sector emissions profile, and shock overlay.

Variables

$P_trans$	Transition pressure score (0–1 normalised)
$s_path$	Scenario pathway score (SSP1.9=1.0 … SSP5.8.5=0.1)
$s_policy$	Policy regime score (net-zero=1.0 … delayed=0.1)
$s_emission$	Sector emissions intensity score
$s_shock$	Active shock overlay score
$w_1–w_4$	Calibrated weights (0.40, 0.30, 0.20, 0.10)

Assumptions (3)

Weights w_1–w_4 are expert-calibrated, not empirically estimated
Linear combination — assumes no interaction effects between pathway and policy
Score normalisation maps to qualitative risk tiers: Low <0.4, Medium 0.4–0.7, High >0.7

Units: dimensionless (0–1) Source: CE internal calibration against NGFS Phase IV scenario outputs Uncertainty: Weight uncertainty ±0.1 per component translates to ±0.15 composite score uncertainty Reviewed: 2026-02-10

\(T(t)\)	Global mean surface temperature anomaly (°C)
\(TCR\)	Transient Climate Response (°C per CO₂ doubling)
\(C(t)\)	Atmospheric CO₂ concentration at time t (ppm)
\(C_0\)	Pre-industrial CO₂ baseline (278 ppm)

\(B_rem\)	Remaining carbon budget (GtCO₂)
\(B_total\)	Total IPCC SR1.5/AR6 budget from reference year (GtCO₂)
\(E(τ)\)	Annual net CO₂ emissions at time τ (GtCO₂/yr)
\(t_0\)	Reference year (2020 for AR6 budgets)

\(A(t)\)	Required annual abatement at year t (GtCO₂e/yr)
\(E_ref(t)\)	Reference policy emissions trajectory (GtCO₂e/yr)
\(E_target(t)\)	Scenario target pathway (GtCO₂e/yr)

\(D(T)\)	Fractional GDP damage
\(T\)	Global mean temperature anomaly above pre-industrial (°C)
\(α\)	Damage coefficient (0.00267 in DICE-2023 calibration)

\(AAL\)	Annualised average loss ($bn/yr)
\(L(p)\)	Loss at exceedance probability p ($bn)
\(p_i\)	Exceedance probabilities at each return period level
\(L_i\)	Ground-up loss at return period i

\(LPL\)	Labour productivity loss (fraction of potential output)
\(WBGT\)	Wet-bulb globe temperature (°C)
\(WBGT_thresh\)	Sector threshold (27°C outdoor heavy, 32°C indoor light)
\(β₁, β₂\)	Sector-calibrated coefficients from Kjellstrom et al. 2016

\(C*\)	Equilibrium carbon price ($/tCO₂e)
\(TC\)	Total transition cost function ($tn)
\(A\)	Annual abatement (GtCO₂e/yr)
\(A*\)	Policy-target abatement level

\(Y(t)\)	Real GDP
\(A(t)\)	Total factor productivity
\(K(t)\)	Capital stock
\(L(t)\)	Effective labour
\(α\)	Capital income share (~0.35)
\(Ω(T,P)\)	Climate-policy damage multiplier (0–1)

\(SCC\)	Social cost of carbon ($/tCO₂)
\(D(τ)\)	Climate damage at time τ
\(E_t\)	Emissions at time t
\(ρ\)	Pure rate of time preference (Nordhaus: 1.5%; Stern: 0.1%)

\(R_c\)	Carbon revenue (% GDP)
\(τ_c\)	Carbon price ($/tCO₂e)
\(E_covered\)	Covered emissions under the carbon pricing scheme (GtCO₂e/yr)
\(GDP\)	Nominal GDP ($tn)

\(E/V\)	Equity share of capital structure
\(R_e\)	Cost of equity (CAPM-derived)
\(D/V\)	Debt share of capital structure
\(R_d\)	Cost of debt (pre-tax)
\(T_c\)	Corporate tax rate
\(Δ_climate\)	Climate risk premium addition (0–200bps depending on scenario)

\(μ̂\)	Sample mean of output f
\(N\)	Number of Monte Carlo samples (2000 in CE)
\(θ_i\)	Parameter vector drawn from joint distribution
\(P_5, P_95\)	5th and 95th percentiles of output distribution

\(A_net\)	Net unique abatement (GtCO₂e/yr)
\(A_gross\)	Sum of all individual technology/policy abatement claims
\(δ\)	De-duplication factor (default 0.22; range 0.10–0.35)

\(E\)	CO₂ emissions (GtCO₂/yr)
\(P\)	Population (billions)
\(G/P\)	Per-capita GDP ($/person)
\(E_prim/G\)	Primary energy intensity (EJ/$tn GDP)
\(E_CO₂/E_prim\)	Carbon intensity of energy (GtCO₂/EJ)

\(P_trans\)	Transition pressure score (0–1 normalised)
\(s_path\)	Scenario pathway score (SSP1.9=1.0 … SSP5.8.5=0.1)
\(s_policy\)	Policy regime score (net-zero=1.0 … delayed=0.1)
\(s_emission\)	Sector emissions intensity score
\(s_shock\)	Active shock overlay score
\(w_1–w_4\)	Calibrated weights (0.40, 0.30, 0.20, 0.10)

∑ Equation Registry

Transient Climate Response (TCR) warming estimate

Variables

Carbon budget remaining (integrated emissions constraint)

Variables

Emissions pathway abatement requirement

Variables

DICE-style damage function (quadratic)

Variables

Annualised average loss (catastrophe actuarial)

Variables

Labour productivity loss from heat stress (WBGT model)

Variables

Marginal abatement cost curve (MAC) — carbon price equilibrium

Variables

Structural growth with climate adjustment (augmented Solow)

Variables

Social Cost of Carbon (Ramsey discounting)

Variables

Carbon revenue as fraction of GDP

Variables

Weighted average cost of capital (WACC) — climate-adjusted

Variables

Monte Carlo expected value and confidence interval

Variables

Portfolio abatement de-duplication factor (overlap discount)

Variables

Kaya identity — CO₂ emissions decomposition

Variables

Transition pressure composite index

Variables