Financial Mathematics, Statistics, and Data Foundations

Mathematical Language and Quantitative Reasoning #

Variables, Constants, Expressions, and Equations verified variablesConstantsExpressionsAndEquations(input)
Functions, Domains, Ranges, and Graphs verified functionsDomainsRangesAndGraphs(input)
Ratios, Proportions, Rates, and Percentages verified ratiosProportionsRatesAndPercentages(input)
Percentage Change, Percentage Points, and Basis Points verified percentageChangePercentagePointsAndBasisPoints(input)
Exponents, Roots, and Logarithms verified exponentsRootsAndLogarithms(input)
Summation, Products, and Index Notation verified summationProductsAndIndexNotation(input)
Linear Equations and Systems verified linearEquationsAndSystems(input)
Inequalities, Bounds, and Constraints verified inequalitiesBoundsAndConstraints(input)
Units, Dimensions, Scale, and Normalization verified unitsDimensionsScaleAndNormalization(input)
Rounding, Precision, Tolerance, and Significant Digits verified roundingPrecisionToleranceAndSignificantDigits(input)

Financial Arithmetic, Time Value, and Returns #

Simple Interest verified simpleInterest(input)
Compound Interest verified compoundInterest(input)
Present Value and Future Value verified presentValueAndFutureValue(input)
Discount Factors and Discount Rates verified discountFactorsAndDiscountRates(input)
Cash-Flow Timelines and Net Present Value verified cashFlowTimelinesAndNetPresentValue(input)
Simple Return verified simpleReturn(input)
Log Return verified logReturn(input)
Holding-Period and Cumulative Return verified holdingPeriodAndCumulativeReturn(input)
Arithmetic versus Geometric Average Return verified arithmeticVersusGeometricAverageReturn(input)
Return Annualization and Deannualization verified returnAnnualizationAndDeannualization(input)

Data, Variables, Samples, and Measurement #

Observations, Entities, Variables, and Datasets verified observationsEntitiesVariablesAndDatasets(input)
Numeric, Categorical, Ordinal, and Binary Variables verified numericCategoricalOrdinalAndBinaryVariables(input)
Population, Sample, Census, and Sampling Frame verified populationSampleCensusAndSamplingFrame(input)
Cross-Sectional, Time-Series, Panel, and Event Data verified crossSectionalTimeSeriesPanelAndEventData(input)
Identifiers, Keys, Joins, and Data Grain verified identifiersKeysJoinsAndDataGrain(input)
Timestamps, Time Zones, Calendars, and Observation Time verified timestampsTimeZonesCalendarsAndObservationTime(input)
Missing, Nonfinite, Censored, and Truncated Values verified missingNonfiniteCensoredAndTruncatedValues(input)
Measurement Error, Resolution, Accuracy, and Precision verified measurementErrorResolutionAccuracyAndPrecision(input)
Revisions, Vintages, and Point-in-Time Availability verified revisionsVintagesAndPointInTimeAvailability(input)
Data Provenance, Lineage, Ownership, and Licensing verified dataProvenanceLineageOwnershipAndLicensing(input)

Location, Ranking, and Exploratory Summaries #

Count, Sum, Minimum, Maximum, and Range verified countSumMinimumMaximumAndRange(input)
Arithmetic Mean verified arithmeticMean(input)
Weighted Mean verified weightedMean(input)
Median and Mode verified medianAndMode(input)
Trimmed and Winsorized Means verified trimmedAndWinsorizedMeans(input)
Percentiles, Quantiles, and Quartiles verified percentilesQuantilesAndQuartiles(input)
Ranks, Ties, and Percentile Rank verified ranksTiesAndPercentileRank(input)
Frequency Tables and Relative Frequency verified frequencyTablesAndRelativeFrequency(input)
Histograms and Empirical Distribution Functions verified histogramsAndEmpiricalDistributionFunctions(input)
Five-Number Summary and Box Plot verified fiveNumberSummaryAndBoxPlot(input)

Dispersion, Shape, and Robust Statistics #

Deviation, Absolute Deviation, and Squared Deviation verified deviationAbsoluteDeviationAndSquaredDeviation(input)
Range and Interquartile Range verified rangeAndInterquartileRange(input)
Mean Absolute Deviation verified meanAbsoluteDeviation(input)
Median Absolute Deviation verified medianAbsoluteDeviation(input)
Population and Sample Variance verified populationAndSampleVariance(input)
Population and Sample Standard Deviation verified populationAndSampleStandardDeviation(input)
Coefficient of Variation and Scale Comparability verified coefficientOfVariationAndScaleComparability(input)
Z-Score, Robust Z-Score, and Standardization verified zScoreRobustZScoreAndStandardization(input)
Skewness and Tail Asymmetry verified skewnessAndTailAsymmetry(input)
Kurtosis, Excess Kurtosis, and Tail Weight verified kurtosisExcessKurtosisAndTailWeight(input)

Probability and Random Variables #

Experiments, Outcomes, Sample Spaces, and Events verified experimentsOutcomesSampleSpacesAndEvents(input)
Probability Rules, Complements, Unions, and Intersections verified probabilityRulesComplementsUnionsAndIntersections(input)
Conditional Probability verified conditionalProbability(input)
Independence and Dependence verified independenceAndDependence(input)
Bayes' Theorem and Base Rates verified bayesTheoremAndBaseRates(input)
Discrete and Continuous Random Variables verified discreteAndContinuousRandomVariables(input)
Expected Value verified expectedValue(input)
Variance, Moments, and Moment-Generating Intuition verified varianceMomentsAndMomentGeneratingIntuition(input)
Joint, Marginal, and Conditional Distributions verified jointMarginalAndConditionalDistributions(input)
Covariance and Correlation of Random Variables verified covarianceAndCorrelationOfRandomVariables(input)

Probability Distributions and Simulation Basics #

PMF, PDF, CDF, Survival, and Quantile Functions verified pmfPdfCdfSurvivalAndQuantileFunctions(input)
Bernoulli and Binomial Distributions verified bernoulliAndBinomialDistributions(input)
Poisson Distribution and Event Counts verified poissonDistributionAndEventCounts(input)
Uniform Distribution and Random Sampling verified uniformDistributionAndRandomSampling(input)
Normal Distribution and Standard Normal verified normalDistributionAndStandardNormal(input)
Lognormal Distribution and Positive Quantities verified lognormalDistributionAndPositiveQuantities(input)
Student-t Distribution and Heavy Tails verified studentTDistributionAndHeavyTails(input)
Exponential, Gamma, and Weibull Waiting-Time Models verified exponentialGammaAndWeibullWaitingTimeModels(input)
Mixture Distributions, Multimodality, and Fat Tails verified mixtureDistributionsMultimodalityAndFatTails(input)
Random Sampling and Monte Carlo Intuition verified randomSamplingAndMonteCarloIntuition(input)

Sampling, Estimation, and Statistical Inference #

Parameters, Statistics, Estimands, and Estimators verified parametersStatisticsEstimandsAndEstimators(input)
Sampling Distributions verified samplingDistributions(input)
Estimator Bias, Consistency, Efficiency, and Robustness verified estimatorBiasConsistencyEfficiencyAndRobustness(input)
Law of Large Numbers verified lawOfLargeNumbers(input)
Central Limit Theorem verified centralLimitTheorem(input)
Standard Error verified standardError(input)
Confidence Intervals and Coverage verified confidenceIntervalsAndCoverage(input)
Null and Alternative Hypotheses verified nullAndAlternativeHypotheses(input)
P-Values, Significance, Type I/II Errors, and Power verified pValuesSignificanceTypeIIiErrorsAndPower(input)
Effect Size, Practical Significance, and Multiple Comparisons verified effectSizePracticalSignificanceAndMultipleComparisons(input)

Dependence, Regression, and Model Foundations #

Scatter Plots, Association, and Nonlinear Patterns verified scatterPlotsAssociationAndNonlinearPatterns(input)
Sample Covariance Calculation and Interpretation verified sampleCovarianceCalculationAndInterpretation(input)
Pearson Correlation Calculation and Interpretation verified pearsonCorrelationCalculationAndInterpretation(input)
Spearman Rank Correlation and Kendall Tau verified spearmanRankCorrelationAndKendallTau(input)
Correlation, Causation, Confounding, and Spurious Relationships verified correlationCausationConfoundingAndSpuriousRelationships(input)
Simple Ordinary Least Squares Regression verified simpleOrdinaryLeastSquaresRegression(input)
Intercepts, Slopes, Coefficients, and Predictions verified interceptsSlopesCoefficientsAndPredictions(input)
Residuals, MAE, MSE, and RMSE verified residualsMaeMseAndRmse(input)
R-Squared and Adjusted R-Squared verified rSquaredAndAdjustedRSquared(input)
Regression Assumptions, Heteroskedasticity, and Multicollinearity verified regressionAssumptionsHeteroskedasticityAndMulticollinearity(input)

Financial Time-Series Foundations #

Time Order, Frequency, Regularity, and Financial Calendars verified timeOrderFrequencyRegularityAndFinancialCalendars(input)
Levels, Changes, Differences, and Returns verified levelsChangesDifferencesAndReturns(input)
Lags, Leads, and Temporal Dependence verified lagsLeadsAndTemporalDependence(input)
Rolling and Expanding Windows verified rollingAndExpandingWindows(input)
Resampling, Aggregation, and Time Alignment verified resamplingAggregationAndTimeAlignment(input)
Trend, Seasonality, Cycles, and Remainder verified trendSeasonalityCyclesAndRemainder(input)
Autocovariance and Autocorrelation verified autocovarianceAndAutocorrelation(input)
Stationarity and Differencing Intuition verified stationarityAndDifferencingIntuition(input)
Smoothing, Baselines, and Naive Forecasts verified smoothingBaselinesAndNaiveForecasts(input)
Look-Ahead Leakage and Time-Aware Data Splits verified lookAheadLeakageAndTimeAwareDataSplits(input)

Financial Risk and Performance Statistics #

Volatility and Annualized Volatility verified volatilityAndAnnualizedVolatility(input)
Downside Deviation and Target Shortfall verified downsideDeviationAndTargetShortfall(input)
Drawdown and Maximum Drawdown verified drawdownAndMaximumDrawdown(input)
Loss Distributions and Loss Quantiles verified lossDistributionsAndLossQuantiles(input)
Value at Risk Intuition verified valueAtRiskIntuition(input)
Expected Shortfall Intuition verified expectedShortfallIntuition(input)
Beta and Market-Relative Risk verified betaAndMarketRelativeRisk(input)
Sharpe, Sortino, and Information Ratio Intuition verified sharpeSortinoAndInformationRatioIntuition(input)
Active Return and Tracking Error verified activeReturnAndTrackingError(input)
Covariance Matrices, Portfolio Variance, and Diversification verified covarianceMatricesPortfolioVarianceAndDiversification(input)

Statistical Computing and Reproducibility #

Floating-Point Representation, Overflow, and Underflow verified floatingPointRepresentationOverflowAndUnderflow(input)
Stable Summation and Mean Calculation verified stableSummationAndMeanCalculation(input)
Stable Online Variance and Welford's Algorithm verified stableOnlineVarianceAndWelfordsAlgorithm(input)
Batch, Rolling, and Streaming Statistic Equivalence verified batchRollingAndStreamingStatisticEquivalence(input)
Missing, Infinite, Invalid, and Unsupported-State Policies verified missingInfiniteInvalidAndUnsupportedStatePolicies(input)
Pseudorandom Numbers, Seeds, and Reproducibility verified pseudorandomNumbersSeedsAndReproducibility(input)
Vectorization, Index Alignment, and Shape Safety verified vectorizationIndexAlignmentAndShapeSafety(input)
Leakage-Free Fitting, Scaling, and Preprocessing verified leakageFreeFittingScalingAndPreprocessing(input)
Fixtures, Numerical Tolerances, and Property Tests verified fixturesNumericalTolerancesAndPropertyTests(input)
Reproducible Analysis, Metadata, and Audit Trails verified reproducibleAnalysisMetadataAndAuditTrails(input)