Risk Mathematics Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Risk Mathematics Ontology within the Kireth Research Semantic Library. Risk mathematics studies the mathematical representation, quantification and aggregation of uncertain losses and adverse outcomes. This ontology supports probability distributions, loss variables, expectation, variance, volatility, quantiles, Value at Risk, Expected Shortfall, tail risk, dependence, copulas, diversification, concentration, stress testing, scenario analysis, credit risk, market risk, operational risk, ruin probabilities, extreme value theory, coherent risk measures and relationships with financial mathematics, probability, statistics, actuarial mathematics and stochastic processes.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Risk Mathematics Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Financial Mathematics / Risk Mathematics"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/financial-mathematics/risk-mathematics-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = RiskMathematics
 SYMBOLIC_FORMS = [
  "E[L]",
  "Var(L)",
  "VaR_alpha(L)",
  "ES_alpha(L)",
  "P(L>x)"
 ]
 ENTITY_CLASS = QuantitativeUncertaintyAndLossFramework
 PRIMARY_CONTEXT = FinancialMathematics
 DEFINITION = "Risk mathematics is the mathematical study of uncertain losses, adverse outcomes, dependence, tail behaviour and risk measures."
 SEMANTIC_ROLE = "Provides machine-readable semantics for loss distributions, expectation, dispersion, quantiles, tail risk, dependence, aggregation, diversification, stress testing, scenario analysis and coherent risk measurement."
}

SCOPE {
 INCLUDE = [
  RiskMathematics,
  Risk,
  Uncertainty,
  LossVariable,
  LossDistribution,
  ProbabilityDistribution,
  ExpectedLoss,
  UnexpectedLoss,
  Variance,
  StandardDeviation,
  Volatility,
  Quantile,
  TailProbability,
  ValueAtRisk,
  VaR,
  ExpectedShortfall,
  ES,
  ConditionalValueAtRisk,
  CVaR,
  TailValueAtRisk,
  TailRisk,
  DownsideRisk,
  Semivariance,
  LowerPartialMoment,
  MaximumLoss,
  RiskMeasure,
  CoherentRiskMeasure,
  ConvexRiskMeasure,
  MonetaryRiskMeasure,
  TranslationInvariance,
  Monotonicity,
  Subadditivity,
  PositiveHomogeneity,
  Convexity,
  Dependence,
  Correlation,
  Covariance,
  Copula,
  TailDependence,
  Diversification,
  ConcentrationRisk,
  AggregationRisk,
  PortfolioLoss,
  MarketRisk,
  CreditRisk,
  OperationalRisk,
  LiquidityRisk,
  ModelRisk,
  DefaultRisk,
  RecoveryRisk,
  ExposureRisk,
  StressTesting,
  ScenarioAnalysis,
  SensitivityAnalysis,
  ExtremeValueTheory,
  GeneralisedExtremeValueDistribution,
  GeneralisedParetoDistribution,
  PeaksOverThreshold,
  RuinProbability,
  SolvencyProbability,
  MonteCarloSimulation,
  HistoricalSimulation,
  ParametricRiskModel
 ]

 EXCLUDE_AS_PRIMARY = [
  GeneralUncertaintyWithoutLossOrAdverseOutcomeContext,
  PurePortfolioReturnAnalysisWithoutRiskFocus,
  DeterministicLossCalculationWithoutUncertainty,
  QualitativeRiskManagementWithoutMathematicalRepresentation
 ]

 DISAMBIGUATION_RULE =
  "When uncertain adverse outcomes, loss distributions, risk aggregation, tail behaviour or quantitative risk measures are central, interpret the context as risk mathematics."
}

CORE_CONCEPTS {

 RiskMathematics {
  definition = "The mathematical framework for uncertain losses and adverse outcomes."
 }

 LossVariable {
  definition = "A random variable representing financial or operational loss."
 }

 ExpectedLoss {
  definition = "The mean of a loss distribution."
 }

 ValueAtRisk {
  definition = "A loss quantile at a specified probability level and horizon."
 }

 ExpectedShortfall {
  definition = "The average loss in the tail beyond a specified quantile under suitable conventions."
 }

 TailRisk {
  definition = "Risk associated with severe outcomes in the extreme tail of a loss distribution."
 }

 CoherentRiskMeasure {
  definition = "A risk measure satisfying monotonicity, translation invariance, subadditivity and positive homogeneity."
 }

 Dependence {
  definition = "Statistical relationship between uncertain variables that affects aggregate risk."
 }

 Diversification {
  definition = "Reduction in aggregate risk through imperfect dependence among exposures."
 }

 StressTesting {
  definition = "Evaluation of losses under severe but defined scenarios."
 }
}

RELATIONS {
 RiskMathematics --is_child_of--> FinancialMathematics
 LossVariable --has--> LossDistribution
 ExpectedLoss --is_expectation_of--> LossVariable
 Variance --measures--> Dispersion
 StandardDeviation --is_square_root_of--> Variance
 ValueAtRisk --is_quantile_of--> LossDistribution
 ExpectedShortfall --depends_on--> TailOfLossDistribution
 TailRisk --concerns--> ExtremeLosses
 CoherentRiskMeasure --requires--> Monotonicity
 CoherentRiskMeasure --requires--> TranslationInvariance
 CoherentRiskMeasure --requires--> Subadditivity
 CoherentRiskMeasure --requires--> PositiveHomogeneity
 Diversification --depends_on--> Dependence
 Correlation --summarises--> LinearDependence
 Copula --models--> DependenceStructure
 TailDependence --affects--> JointExtremeLoss
 PortfolioLoss --aggregates--> ComponentLosses
 StressTesting --uses--> ScenarioAnalysis
 ExtremeValueTheory --models--> TailBehaviour
 MonteCarloSimulation --estimates--> RiskMeasure
 HistoricalSimulation --estimates--> EmpiricalLossDistribution
}

OPERATORS {
 EXPECTED_LOSS(L) :=
  EL=E[L]

 LOSS_VARIANCE(L) :=
  Var(L)=E[(L-E[L])^2]

 LOSS_STANDARD_DEVIATION(L) :=
  sigma_L=sqrt(Var(L))

 VALUE_AT_RISK(L,alpha) :=
  VaR_alpha=inf{x : P(L<=x)>=alpha}

 EXPECTED_SHORTFALL(L,alpha) :=
  ES_alpha=(1/(1-alpha))*integral_alpha^1 VaR_u(L) du
  under continuous_or_generalised quantile convention

 TAIL_PROBABILITY(L,x) :=
  P_tail=P(L>x)

 PORTFOLIO_LOSS(weights,losses) :=
  L_p=sum_i weights_i*L_i

 PORTFOLIO_VARIANCE(w,Sigma) :=
  Var(L_p)=w^T*Sigma*w

 CORRELATION(X,Y) :=
  rho=Cov(X,Y)/(sigma_X*sigma_Y)

 LOWER_PARTIAL_MOMENT(R,target,p) :=
  LPM_p=E[max(target-R,0)^p]

 STRESS_LOSS(model,scenario) :=
  L_stress=model(scenario)

 RUIN_PROBABILITY(U_t) :=
  psi(u)=P(inf_t U_t<0 | U_0=u)

 MONTE_CARLO_RISK(samples,alpha) :=
  estimate distribution_quantiles_and_tail means_from_samples
}

DEFINITIONS {
 D1 = "Risk mathematics represents adverse uncertainty through random losses or equivalent adverse-state variables."
 D2 = "Expected loss captures average loss but not tail severity."
 D3 = "Variance and volatility measure dispersion, not necessarily downside-only risk."
 D4 = "Value at Risk is a quantile and does not describe losses beyond that quantile."
 D5 = "Expected Shortfall measures average tail loss beyond a confidence threshold."
 D6 = "Dependence determines how individual risks aggregate."
 D7 = "Correlation does not fully characterise nonlinear or tail dependence."
 D8 = "Copulas separate marginal distributions from dependence structure."
 D9 = "Stress testing evaluates specified adverse scenarios rather than probabilistic quantiles alone."
 D10 = "Extreme value theory focuses explicitly on tail behaviour and rare events."
}

AXIOMS {
 A1_PROBABILISTIC_REPRESENTATION:
  Quantitative risk requires a defined loss variable distribution or scenario structure

 A2_HORIZON:
  Risk measures depend on an explicit time horizon

 A3_SIGN_CONVENTION:
  Loss-versus-return sign convention must be explicit

 A4_DEPENDENCE:
  Aggregate risk cannot generally be derived from marginal distributions alone

 A5_TAIL_SENSITIVITY:
  Tail-risk measures depend strongly on distributional assumptions and data availability

 A6_COHERENCE:
  Coherent risk measures satisfy monotonicity translation invariance subadditivity and positive homogeneity

 A7_MODEL_RISK:
  Risk estimates depend on modelling assumptions calibration and implementation

 A8_VALIDATION:
  Risk models require backtesting stress testing sensitivity analysis and distributional diagnostics where applicable
}

REASONING {

 R1_RISK_CONTEXT {
  INPUT = uncertain_financial_or_operational problem
  IDENTIFY = loss_variable
  IDENTIFY = horizon
  IDENTIFY = probability_or_scenario model
  IDENTIFY = confidence level_if relevant
  RETURN = risk_context
 }

 R2_DISTRIBUTION_REASONING {
  INPUT = loss_data_or_model
  SELECT = empirical_parametric_mixture_or_simulation representation
  ESTIMATE = distribution parameters
  RETURN = loss_distribution
 }

 R3_MOMENT_REASONING {
  INPUT = loss_distribution
  COMPUTE = expectation_variance_standard deviation_and_higher moments_if useful
  RETURN = moment_profile
 }

 R4_QUANTILE_RISK_REASONING {
  INPUT = loss_distribution_and_alpha
  COMPUTE = VaR_and_ES
  RETURN = quantile_tail_profile
 }

 R5_DEPENDENCE_REASONING {
  INPUT = multiple_risk factors
  ESTIMATE = covariance_correlation_or_copula
  ANALYSE = tail dependence
  RETURN = dependence_profile
 }

 R6_AGGREGATION_REASONING {
  INPUT = component_losses_and_dependence
  COMBINE = joint_loss distribution
  COMPUTE = portfolio_or_enterprise risk
  RETURN = aggregate_risk
 }

 R7_STRESS_REASONING {
  INPUT = defined_adverse scenarios
  REVALUE = positions_or_processes
  COMPUTE = scenario losses
  RETURN = stress_profile
 }

 R8_EXTREME_VALUE_REASONING {
  INPUT = tail_data
  SELECT = block_maxima_or_peaksOverThreshold
  FIT = GEV_or_GPD model
  RETURN = extreme_tail_profile
 }

 R9_MODEL_VALIDATION_REASONING {
  INPUT = predicted_risk_and_realised outcomes
  PERFORM = backtesting_calibration_diagnostics_and_sensitivity
  RETURN = validation_profile
 }

 R10_REPORTING_REASONING {
  INPUT = risk_results
  ATTACH = horizon_confidence level_method_assumptions_and_uncertainty
  RETURN = interpretable_risk_output
 }
}

CONSTRAINTS {
 C1 = "VaR is not generally subadditive for all distributions."
 C2 = "Expected Shortfall estimates can be unstable with limited tail data."
 C3 = "Normal-distribution assumptions can materially understate heavy-tail risk."
 C4 = "Correlation does not capture all dependence structures."
 C5 = "Historical simulation assumes historical observations are informative for future risk."
 C6 = "Monte Carlo accuracy depends on sample size and model correctness."
 C7 = "Stress scenarios should not be assigned probabilities unless explicitly modelled."
 C8 = "Risk measures from different horizons are not directly comparable without a scaling model."
 C9 = "Square-root-of-time scaling is not universally valid."
 C10 = "Backtesting success does not eliminate structural model risk."
}

SEMANTIC_EQUIVALENCE {
 "RiskMathematics" <=> "MathematicalRiskAnalysis"
 "ValueAtRisk" <=> "VaR"
 "ExpectedShortfall" <=> "ES"
 "ConditionalValueAtRisk" <=> "CVaR"
 "TailValueAtRisk" <=> "TVaR"
 "ExpectedShortfall" <=> "ConditionalTailExpectation"
}

VALIDATION {
 V1 = "Loss definition and sign convention are explicit."
 V2 = "Risk horizon is explicit."
 V3 = "Confidence level is explicit for quantile measures."
 V4 = "Probability or scenario model is documented."
 V5 = "Dependence assumptions are explicit."
 V6 = "VaR and ES estimation methods are documented."
 V7 = "Tail-model thresholds and calibration are documented."
 V8 = "Stress scenarios are clearly specified."
 V9 = "Model backtesting and sensitivity are assessed."
 V10 = "Model uncertainty and data limitations are disclosed."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 RISK_MATHEMATICS_DEFINITION = 0.99
 DISTRIBUTION_REASONING = 0.98
 MOMENT_REASONING = 0.99
 QUANTILE_RISK_REASONING = 0.99
 DEPENDENCE_REASONING = 0.98
 AGGREGATION_REASONING = 0.98
 STRESS_REASONING = 0.98
 EXTREME_VALUE_REASONING = 0.97
 VALIDATION_REASONING = 0.98

 POLICY =
  "Reduce confidence when loss definition, horizon, confidence level, tail model, dependence structure, stress scenarios, historical sample size, simulation size or validation evidence are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  RiskMathematicsReasoning,
  LossDistributionReasoning,
  ExpectedLossReasoning,
  VarianceReasoning,
  VolatilityReasoning,
  VaRReasoning,
  ExpectedShortfallReasoning,
  TailRiskReasoning,
  CoherentRiskMeasureReasoning,
  DependenceReasoning,
  CopulaReasoning,
  DiversificationReasoning,
  ConcentrationRiskReasoning,
  StressTestingReasoning,
  ScenarioAnalysisReasoning,
  ExtremeValueReasoning,
  RuinProbabilityReasoning,
  MonteCarloRiskReasoning,
  ModelRiskReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "risk mathematics" -> RiskMathematics
  "mathematical risk analysis" -> RiskMathematics
  "loss distribution" -> LossDistribution
  "value at risk" -> ValueAtRisk
  "VaR" -> ValueAtRisk
  "expected shortfall" -> ExpectedShortfall
  "ES" -> ExpectedShortfall
  "conditional value at risk" -> ConditionalValueAtRisk
  "CVaR" -> ConditionalValueAtRisk
  "tail risk" -> TailRisk
 }

 INTERPRETATION_ORDER = [
  detect_risk_mathematics_context,
  identify loss_variable_horizon_probability model_confidence level_and_dependence structure,
  estimate or construct loss_distribution,
  compute expectation_dispersion_and_quantile risk measures,
  analyse dependence_and_aggregate risk,
  if_tailRisk apply extreme_value_or_tail model,
  if_stressTesting apply declared scenarios,
  if_simulation estimate uncertainty_and_sampling error,
  validate through backtesting_sensitivity_and_model diagnostics,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("risk mathematics")
  OR uncertain_loss_quantification_is_primary
  THEN activate(RiskMathematicsOntology)

 E2:
  IF quantile_loss_measure_requested
  THEN compute ValueAtRisk

 E3:
  IF tail_average_requested
  THEN compute ExpectedShortfall

 E4:
  IF multiple_risks_are_aggregated
  THEN require dependence model

 E5:
  IF correlation_is_insufficient_for_joint extremes
  THEN consider Copula_or_tail dependence reasoning

 E6:
  IF severe_scenario_analysis
  THEN apply StressTesting

 E7:
  IF rare_extreme_losses
  THEN consider ExtremeValueTheory

 E8:
  IF simulation_based
  THEN report sampling uncertainty

 E9:
  IF coherent_measure_requested
  THEN verify coherence properties
}

RECURSION {
 ENTRY = RiskMathematics

 EXPAND(RiskMathematics) -> [
  Risk,
  Uncertainty,
  LossVariable,
  LossDistribution,
  ProbabilityDistribution,
  ExpectedLoss,
  UnexpectedLoss,
  Variance,
  StandardDeviation,
  Volatility,
  Quantile,
  TailProbability,
  ValueAtRisk,
  ExpectedShortfall,
  ConditionalValueAtRisk,
  TailValueAtRisk,
  TailRisk,
  DownsideRisk,
  Semivariance,
  LowerPartialMoment,
  MaximumLoss,
  RiskMeasure,
  CoherentRiskMeasure,
  ConvexRiskMeasure,
  MonetaryRiskMeasure,
  TranslationInvariance,
  Monotonicity,
  Subadditivity,
  PositiveHomogeneity,
  Convexity,
  Dependence,
  Correlation,
  Covariance,
  Copula,
  TailDependence,
  Diversification,
  ConcentrationRisk,
  AggregationRisk,
  PortfolioLoss,
  MarketRisk,
  CreditRisk,
  OperationalRisk,
  LiquidityRisk,
  ModelRisk,
  DefaultRisk,
  RecoveryRisk,
  ExposureRisk,
  StressTesting,
  ScenarioAnalysis,
  SensitivityAnalysis,
  ExtremeValueTheory,
  GeneralisedExtremeValueDistribution,
  GeneralisedParetoDistribution,
  PeaksOverThreshold,
  RuinProbability,
  SolvencyProbability,
  MonteCarloSimulation,
  HistoricalSimulation,
  ParametricRiskModel
 ]

 EXPAND(RiskMeasures) -> [
  ExpectedLoss,
  Variance,
  StandardDeviation,
  ValueAtRisk,
  ExpectedShortfall,
  Semivariance,
  LowerPartialMoment,
  MaximumLoss
 ]

 EXPAND(TailFramework) -> [
  TailProbability,
  TailRisk,
  ExpectedShortfall,
  ExtremeValueTheory,
  GeneralisedExtremeValueDistribution,
  GeneralisedParetoDistribution,
  PeaksOverThreshold
 ]

 STOP_CONDITION =
  "Stop when loss variable, horizon, distribution, dependence, risk measure, tail assumptions, scenario structure and validation evidence are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Financial Mathematics"

 IMMEDIATE_CONTEXTS = [
  "Financial Mathematics Ontology",
  "Probability & Statistics Ontologies",
  "Stochastic Process Ontologies",
  "Actuarial Mathematics Ontologies"
 ]

 RELATED_ONTOLOGIES = [
  "Financial Mathematics",
  "Financial Mathematics Ontology",
  "Probability & Statistics Ontologies",
  "Stochastic Process Ontologies",
  "Actuarial Mathematics Ontologies",
  "Portfolio Mathematics Ontologies",
  "Credit Risk Ontologies",
  "Operational Risk Ontologies",
  "Extreme Value Ontologies",
  "Copula Ontologies",
  "Monte Carlo Ontologies",
  "Optimisation Ontologies",
  "Computational Finance Ontologies"
 ]

 ROUTE_RULES = [
  "Risk Mathematics -> Financial Mathematics for parent semantics",
  "Risk Mathematics -> Probability & Statistics for distributions and estimation",
  "Risk Mathematics -> Stochastic Processes for evolving risk factors",
  "Risk Mathematics -> Actuarial Mathematics for ruin and solvency",
  "Risk Mathematics -> Portfolio Mathematics for aggregated investment risk",
  "Risk Mathematics -> Credit Risk for default and recovery loss",
  "Risk Mathematics -> Extreme Value Theory for severe tail behaviour",
  "Risk Mathematics -> Copula Ontologies for nonlinear dependence",
  "Risk Mathematics -> Monte Carlo for simulation-based estimation",
  "Risk Mathematics -> Optimisation for risk-constrained decisions"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible risk measures, dependence models or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve loss-variable semantics.",
  "Preserve horizon-and-confidence semantics.",
  "Preserve VaR-versus-Expected-Shortfall distinction.",
  "Preserve marginal-versus-dependence distinction.",
  "Preserve stress-test-versus-probabilistic-risk distinction.",
  "Preserve tail-risk semantics.",
  "Preserve model-risk and validation semantics.",
  "Preserve machine-readable risk-mathematics semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Risk mathematics quantifies uncertain adverse outcomes through loss distributions, dependence models and mathematical risk measures."

 EXPECTATION_SUMMARY =
  "Expected loss measures average adverse outcome while variance and volatility describe dispersion."

 VAR_SUMMARY =
  "Value at Risk is a specified loss quantile over a declared horizon and confidence level."

 ES_SUMMARY =
  "Expected Shortfall measures average tail loss beyond a specified confidence threshold."

 DEPENDENCE_SUMMARY =
  "Aggregate risk depends on dependence structure; correlation alone may not capture nonlinear or tail dependence."

 STRESS_SUMMARY =
  "Stress testing evaluates losses under explicitly defined severe scenarios."

 TAIL_SUMMARY =
  "Extreme value methods model rare severe losses using specialised tail distributions."

 VALIDATION_SUMMARY =
  "Risk estimates should be checked through backtesting, sensitivity analysis, scenario analysis and explicit disclosure of model assumptions."
}

FLOW {
 INPUT
 -> detect_risk_mathematics_context
 -> identify loss_variable_sign convention_horizon_confidence level_data_probability model_and_dependence
 -> construct empirical_parametric_simulated_or_scenario loss representation
 -> compute expected_loss_variance_standard deviation_and_other dispersion measures
 -> compute VaR_ES_or_other declared risk measure
 -> if_multiple exposures model covariance_correlation_copula_or_tail dependence
 -> aggregate component losses
 -> if_extreme tail apply EVT_GEV_GPD_or_peaksOverThreshold reasoning
 -> if_stress requested apply explicit adverse scenarios
 -> if_simulation estimate sampling error
 -> validate with backtesting_sensitivity_calibration_and_model diagnostics
 -> route to financial_mathematics probability_statistics stochastic_processes actuarial_mathematics portfolio credit_risk operational_risk extreme_value copula MonteCarlo optimisation_or_computational_finance ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Risk Mathematics Ontology defines quantitative analysis of uncertain losses. It supports loss variables and distributions, expected and unexpected loss, variance, standard deviation and volatility, quantiles and tail probabilities, Value at Risk, Expected Shortfall, CVaR and tail risk, downside measures and partial moments, coherent, convex and monetary risk measures, dependence, covariance, correlation, copulas and tail dependence, diversification and concentration, portfolio, market, credit, operational, liquidity and model risk, stress testing and scenarios, sensitivity analysis, extreme value theory, GEV and GPD models, peaks-over-threshold methods, ruin and solvency probabilities, Monte Carlo, historical simulation and parametric risk models."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-RISK-MATHEMATICS-ONTOLOGY-V1"
}

}

END_FLOW