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Search, Evaluate, and Read Evidence","\u002Fen\u002Facademic-writing\u002F02-reading-and-literature-matrix","en\u002Facademic-writing\u002F02-reading-and-literature-matrix",{"title":27,"path":28,"stem":29},"3. From Sources to Synthesis and Argument","\u002Fen\u002Facademic-writing\u002F03-arguments-outlines-and-paragraphs","en\u002Facademic-writing\u002F03-arguments-outlines-and-paragraphs",{"title":31,"path":32,"stem":33},"4. Literature Review and a Defensible Gap","\u002Fen\u002Facademic-writing\u002F04-writing-the-literature-review","en\u002Facademic-writing\u002F04-writing-the-literature-review",{"title":35,"path":36,"stem":37},"5. Research Design, Evidence, and Core Sections","\u002Fen\u002Facademic-writing\u002F05-core-sections-and-evidence","en\u002Facademic-writing\u002F05-core-sections-and-evidence",{"title":39,"path":40,"stem":41},"6. Citation, Paraphrasing, Integrity, and AI","\u002Fen\u002Facademic-writing\u002F06-citation-paraphrasing-and-integrity","en\u002Facademic-writing\u002F06-citation-paraphrasing-and-integrity",{"title":43,"path":44,"stem":45},"7. Revision, Review, and Submission","\u002Fen\u002Facademic-writing\u002F07-revision-style-and-submission","en\u002Facademic-writing\u002F07-revision-style-and-submission",{"title":47,"path":48,"stem":49},"8. Research Workbook","\u002Fen\u002Facademic-writing\u002F08-literature-review-checklist-and-template","en\u002Facademic-writing\u002F08-literature-review-checklist-and-template",{"title":51,"path":52,"stem":53},"9. Research Workflow and Tools","\u002Fen\u002Facademic-writing\u002F09-tools-for-academic-writing-and-literature-management","en\u002Facademic-writing\u002F09-tools-for-academic-writing-and-literature-management",{"title":55,"path":56,"stem":57},"10. Worked Project — From Question to Defensible Conclusion","\u002Fen\u002Facademic-writing\u002F10-worked-example-from-block-structure-to-question-chain","en\u002Facademic-writing\u002F10-worked-example-from-block-structure-to-question-chain",{"title":59,"path":60,"stem":61,"children":62,"page":249},"Accounting","\u002Fen\u002Faccounting","en\u002Faccounting",[63,69,87,93,193],{"title":64,"path":65,"stem":66,"children":67},"Accounting — From Evidence to Decisions","\u002Fen\u002Faccounting\u002F00-index","en\u002Faccounting\u002F00-index",[68],{"title":64,"path":65,"stem":66},{"title":70,"path":71,"stem":72,"children":73},"Accounting Appendix","\u002Fen\u002Faccounting\u002Fappendix","en\u002Faccounting\u002Fappendix\u002Findex",[74,75,79,83],{"title":70,"path":71,"stem":72},{"title":76,"path":77,"stem":78},"Worked Examples and Error Diagnosis","\u002Fen\u002Faccounting\u002Fappendix\u002F24-examples-pitfalls","en\u002Faccounting\u002Fappendix\u002F24-examples-pitfalls",{"title":80,"path":81,"stem":82},"Accounting Glossary","\u002Fen\u002Faccounting\u002Fappendix\u002F26-glossary","en\u002Faccounting\u002Fappendix\u002F26-glossary",{"title":84,"path":85,"stem":86},"Reading and Evidence Map","\u002Fen\u002Faccounting\u002Fappendix\u002F27-reading-map","en\u002Faccounting\u002Fappendix\u002F27-reading-map",{"title":88,"path":89,"stem":90,"children":91},"Northstar Record-to-Decision Capstone","\u002Fen\u002Faccounting\u002Fcapstone","en\u002Faccounting\u002Fcapstone\u002Findex",[92],{"title":88,"path":89,"stem":90},{"title":94,"path":95,"stem":96,"children":97},"Financial Accounting","\u002Fen\u002Faccounting\u002Ffinancial-accounting","en\u002Faccounting\u002Ffinancial-accounting\u002Findex",[98,99,117,131,165,179],{"title":94,"path":95,"stem":96},{"title":100,"path":101,"stem":102,"children":103},"1. Foundations","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations","en\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002Findex",[104,105,109,113],{"title":100,"path":101,"stem":102},{"title":106,"path":107,"stem":108},"Objectives and Qualitative Characteristics","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F01-objectives-qualitative-characteristics","en\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F01-objectives-qualitative-characteristics",{"title":110,"path":111,"stem":112},"Equation and Elements","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F02-equation-elements","en\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F02-equation-elements",{"title":114,"path":115,"stem":116},"Accrual Basis, Estimates and Periods","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F03-accounting-bases-principles","en\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F03-accounting-bases-principles",{"title":118,"path":119,"stem":120,"children":121},"2. Recording Transactions","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions","en\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002Findex",[122,123,127],{"title":118,"path":119,"stem":120},{"title":124,"path":125,"stem":126},"Double-Entry and Debit\u002FCredit","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002F04-double-entry-dr-cr","en\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002F04-double-entry-dr-cr",{"title":128,"path":129,"stem":130},"Accounting Cycle","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002F05-accounting-cycle","en\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002F05-accounting-cycle",{"title":132,"path":133,"stem":134,"children":135},"3. Measurement and Adjustments","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002Findex",[136,137,141,145,149,153,157,161],{"title":132,"path":133,"stem":134},{"title":138,"path":139,"stem":140},"Revenue Recognition","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F07-revenue-recognition","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F07-revenue-recognition",{"title":142,"path":143,"stem":144},"Inventory","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F08-inventory","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F08-inventory",{"title":146,"path":147,"stem":148},"Receivables and Expected Credit Losses","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F09-receivables","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F09-receivables",{"title":150,"path":151,"stem":152},"Property, Plant and Equipment","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F10-ppe","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F10-ppe",{"title":154,"path":155,"stem":156},"Intangible Assets","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F11-intangibles","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F11-intangibles",{"title":158,"path":159,"stem":160},"Leases","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F12-leases","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F12-leases",{"title":162,"path":163,"stem":164},"Current and Deferred Income Tax","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F13-income-tax","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F13-income-tax",{"title":166,"path":167,"stem":168,"children":169},"4. Reporting and Cash","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash","en\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002Findex",[170,171,175],{"title":166,"path":167,"stem":168},{"title":172,"path":173,"stem":174},"Linked Financial Statements","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002F06-financial-statements","en\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002F06-financial-statements",{"title":176,"path":177,"stem":178},"Cash, Reconciliation and Internal Control","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002F14-cash-control","en\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002F14-cash-control",{"title":180,"path":181,"stem":182,"children":183},"5. Analysis and Comparison","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison","en\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002Findex",[184,185,189],{"title":180,"path":181,"stem":182},{"title":186,"path":187,"stem":188},"Ratio Analysis","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002F15-ratio-analysis","en\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002F15-ratio-analysis",{"title":190,"path":191,"stem":192},"IFRS versus US GAAP","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002F25-ifrs-gaap","en\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002F25-ifrs-gaap",{"title":194,"path":195,"stem":196,"children":197},"Management Accounting","\u002Fen\u002Faccounting\u002Fmanagement-accounting","en\u002Faccounting\u002Fmanagement-accounting\u002Findex",[198,199,217,235],{"title":194,"path":195,"stem":196},{"title":200,"path":201,"stem":202,"children":203},"1. Cost Foundations","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations","en\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002Findex",[204,205,209,213],{"title":200,"path":201,"stem":202},{"title":206,"path":207,"stem":208},"Cost Concepts and Behaviour","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F16-cost-concepts","en\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F16-cost-concepts",{"title":210,"path":211,"stem":212},"Costing Systems","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F17-costing-systems","en\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F17-costing-systems",{"title":214,"path":215,"stem":216},"Variable and Absorption Costing","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F19-variable-vs-absorption","en\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F19-variable-vs-absorption",{"title":218,"path":219,"stem":220,"children":221},"2. Planning and Control","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control","en\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002Findex",[222,223,227,231],{"title":218,"path":219,"stem":220},{"title":224,"path":225,"stem":226},"Cost–Volume–Profit Analysis","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F18-cvp-analysis","en\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F18-cvp-analysis",{"title":228,"path":229,"stem":230},"Budgeting and Variance Analysis","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F20-budgeting-variances","en\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F20-budgeting-variances",{"title":232,"path":233,"stem":234},"Performance Measurement","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F21-performance-measurement","en\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F21-performance-measurement",{"title":236,"path":237,"stem":238,"children":239},"3. Decisions and Investment","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment","en\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002Findex",[240,241,245],{"title":236,"path":237,"stem":238},{"title":242,"path":243,"stem":244},"Short-Term Decisions","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002F22-short-term-decisions","en\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002F22-short-term-decisions",{"title":246,"path":247,"stem":248},"Capital Budgeting","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002F23-capital-budgeting","en\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002F23-capital-budgeting",false,{"title":251,"path":252,"stem":253,"children":254,"page":249},"Business Analytics","\u002Fen\u002Fbusiness-analytics","en\u002Fbusiness-analytics",[255,261,279,305,335,365,383,400],{"title":256,"path":257,"stem":258,"children":259},"Business Analytics — From Data to Defensible Action","\u002Fen\u002Fbusiness-analytics\u002F00-index","en\u002Fbusiness-analytics\u002F00-index",[260],{"title":256,"path":257,"stem":258},{"title":262,"path":263,"stem":264,"children":265},"0. Decision and Data Foundations","\u002Fen\u002Fbusiness-analytics\u002F00-intro","en\u002Fbusiness-analytics\u002F00-intro\u002Findex",[266,267,271,275],{"title":262,"path":263,"stem":264},{"title":268,"path":269,"stem":270},"Decision Framing","\u002Fen\u002Fbusiness-analytics\u002F00-intro\u002F01-decision-framing","en\u002Fbusiness-analytics\u002F00-intro\u002F01-decision-framing",{"title":272,"path":273,"stem":274},"Data Contracts","\u002Fen\u002Fbusiness-analytics\u002F00-intro\u002F02-data-contracts","en\u002Fbusiness-analytics\u002F00-intro\u002F02-data-contracts",{"title":276,"path":277,"stem":278},"Reproducible Analytics Workflow","\u002Fen\u002Fbusiness-analytics\u002F00-intro\u002F03-reproducible-workflow","en\u002Fbusiness-analytics\u002F00-intro\u002F03-reproducible-workflow",{"title":280,"path":281,"stem":282,"children":283},"1. Descriptive Analytics","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive","en\u002Fbusiness-analytics\u002F01-descriptive\u002Findex",[284,285,289,293,297,301],{"title":280,"path":281,"stem":282},{"title":286,"path":287,"stem":288},"Distributions and Exploratory Data Analysis","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive\u002F04-distributions-eda","en\u002Fbusiness-analytics\u002F01-descriptive\u002F04-distributions-eda",{"title":290,"path":291,"stem":292},"KPIs and Denominators","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive\u002F05-kpis-denominators","en\u002Fbusiness-analytics\u002F01-descriptive\u002F05-kpis-denominators",{"title":294,"path":295,"stem":296},"Segments, Cohorts and Funnels","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive\u002F06-segmentation-cohorts-funnels","en\u002Fbusiness-analytics\u002F01-descriptive\u002F06-segmentation-cohorts-funnels",{"title":298,"path":299,"stem":300},"Visual Evidence and Data Stories","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive\u002F07-visualisation-story","en\u002Fbusiness-analytics\u002F01-descriptive\u002F07-visualisation-story",{"title":302,"path":303,"stem":304},"Experiments and Causal Boundaries","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive\u002F08-experiments-causal-boundary","en\u002Fbusiness-analytics\u002F01-descriptive\u002F08-experiments-causal-boundary",{"title":306,"path":307,"stem":308,"children":309},"2. Predictive Analytics","\u002Fen\u002Fbusiness-analytics\u002F02-predictive","en\u002Fbusiness-analytics\u002F02-predictive\u002Findex",[310,311,315,319,323,327,331],{"title":306,"path":307,"stem":308},{"title":312,"path":313,"stem":314},"Validation, Baselines and Leakage","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F09-validation-leakage","en\u002Fbusiness-analytics\u002F02-predictive\u002F09-validation-leakage",{"title":316,"path":317,"stem":318},"Regression and Forecasting","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F10-regression-forecasting","en\u002Fbusiness-analytics\u002F02-predictive\u002F10-regression-forecasting",{"title":320,"path":321,"stem":322},"Classification and Calibration","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F11-classification-calibration","en\u002Fbusiness-analytics\u002F02-predictive\u002F11-classification-calibration",{"title":324,"path":325,"stem":326},"Trees and Ensembles","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F12-trees-ensembles","en\u002Fbusiness-analytics\u002F02-predictive\u002F12-trees-ensembles",{"title":328,"path":329,"stem":330},"Decision Metrics and Thresholds","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F13-decision-metrics-thresholds","en\u002Fbusiness-analytics\u002F02-predictive\u002F13-decision-metrics-thresholds",{"title":332,"path":333,"stem":334},"Explainability, Monitoring and Drift","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F14-explainability-drift","en\u002Fbusiness-analytics\u002F02-predictive\u002F14-explainability-drift",{"title":336,"path":337,"stem":338,"children":339},"3. Prescriptive Analytics","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive","en\u002Fbusiness-analytics\u002F03-prescriptive\u002Findex",[340,341,345,349,353,357,361],{"title":336,"path":337,"stem":338},{"title":342,"path":343,"stem":344},"Decision-Making under Uncertainty","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F15-decision-uncertainty","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F15-decision-uncertainty",{"title":346,"path":347,"stem":348},"Linear Optimisation","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F16-linear-optimization","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F16-linear-optimization",{"title":350,"path":351,"stem":352},"Inventory and Allocation","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F17-inventory-allocation","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F17-inventory-allocation",{"title":354,"path":355,"stem":356},"Queueing and Simulation","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F18-queueing-simulation","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F18-queueing-simulation",{"title":358,"path":359,"stem":360},"Pricing and Revenue Management","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F19-pricing-experimentation","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F19-pricing-experimentation",{"title":362,"path":363,"stem":364},"Causal Targeting and Policy Learning","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F20-causal-targeting","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F20-causal-targeting",{"title":366,"path":367,"stem":368,"children":369},"4. Deployment and Governance","\u002Fen\u002Fbusiness-analytics\u002F04-deployment","en\u002Fbusiness-analytics\u002F04-deployment\u002Findex",[370,371,375,379],{"title":366,"path":367,"stem":368},{"title":372,"path":373,"stem":374},"Data Products and Monitoring","\u002Fen\u002Fbusiness-analytics\u002F04-deployment\u002F21-data-products-monitoring","en\u002Fbusiness-analytics\u002F04-deployment\u002F21-data-products-monitoring",{"title":376,"path":377,"stem":378},"Governance, Fairness and Privacy","\u002Fen\u002Fbusiness-analytics\u002F04-deployment\u002F22-governance-fairness-privacy","en\u002Fbusiness-analytics\u002F04-deployment\u002F22-governance-fairness-privacy",{"title":380,"path":381,"stem":382},"Adoption and Business Value","\u002Fen\u002Fbusiness-analytics\u002F04-deployment\u002F23-adoption-value","en\u002Fbusiness-analytics\u002F04-deployment\u002F23-adoption-value",{"title":384,"path":385,"stem":386,"children":387},"Appendix and Revision Tools","\u002Fen\u002Fbusiness-analytics\u002Fappendix","en\u002Fbusiness-analytics\u002Fappendix\u002Findex",[388,389,393,397],{"title":384,"path":385,"stem":386},{"title":390,"path":391,"stem":392},"Formula and Decision Map","\u002Fen\u002Fbusiness-analytics\u002Fappendix\u002F24-formula-map","en\u002Fbusiness-analytics\u002Fappendix\u002F24-formula-map",{"title":394,"path":395,"stem":396},"Worked Examples and Pitfalls","\u002Fen\u002Fbusiness-analytics\u002Fappendix\u002F25-worked-examples-pitfalls","en\u002Fbusiness-analytics\u002Fappendix\u002F25-worked-examples-pitfalls",{"title":84,"path":398,"stem":399},"\u002Fen\u002Fbusiness-analytics\u002Fappendix\u002F26-reading-map","en\u002Fbusiness-analytics\u002Fappendix\u002F26-reading-map",{"title":401,"path":402,"stem":403,"children":404},"Capstone — The Evening Delivery Promise","\u002Fen\u002Fbusiness-analytics\u002Fcapstone","en\u002Fbusiness-analytics\u002Fcapstone\u002Findex",[405],{"title":401,"path":402,"stem":403},{"title":407,"path":408,"stem":409,"children":410,"page":249},"Financial Economic Time Series","\u002Fen\u002Ffinancial-economic-time-series","en\u002Ffinancial-economic-time-series",[411,417,423,429,435,441,447,453,459,465,471],{"title":412,"path":413,"stem":414,"children":415},"Financial and Economic Time Series — Course Guide","\u002Fen\u002Ffinancial-economic-time-series\u002F00-intro","en\u002Ffinancial-economic-time-series\u002F00-intro\u002Findex",[416],{"title":412,"path":413,"stem":414},{"title":418,"path":419,"stem":420,"children":421},"Three Versions of Time Series","\u002Fen\u002Ffinancial-economic-time-series\u002F01-bridge","en\u002Ffinancial-economic-time-series\u002F01-bridge\u002Findex",[422],{"title":418,"path":419,"stem":420},{"title":424,"path":425,"stem":426,"children":427},"Data, Clocks, and Transformations","\u002Fen\u002Ffinancial-economic-time-series\u002F02-data-transformations","en\u002Ffinancial-economic-time-series\u002F02-data-transformations\u002Findex",[428],{"title":424,"path":425,"stem":426},{"title":430,"path":431,"stem":432,"children":433},"Predictive Regressions and Persistent Predictors","\u002Fen\u002Ffinancial-economic-time-series\u002F03-predictive-regressions","en\u002Ffinancial-economic-time-series\u002F03-predictive-regressions\u002Findex",[434],{"title":430,"path":431,"stem":432},{"title":436,"path":437,"stem":438,"children":439},"Volatility, Tails, and Financial Risk","\u002Fen\u002Ffinancial-economic-time-series\u002F04-volatility-risk","en\u002Ffinancial-economic-time-series\u002F04-volatility-risk\u002Findex",[440],{"title":436,"path":437,"stem":438},{"title":442,"path":443,"stem":444,"children":445},"Unit Roots, Cointegration, and Error Correction","\u002Fen\u002Ffinancial-economic-time-series\u002F05-unit-roots-cointegration","en\u002Ffinancial-economic-time-series\u002F05-unit-roots-cointegration\u002Findex",[446],{"title":442,"path":443,"stem":444},{"title":448,"path":449,"stem":450,"children":451},"VAR, Structural Identification, and Local Projections","\u002Fen\u002Ffinancial-economic-time-series\u002F06-var-identification","en\u002Ffinancial-economic-time-series\u002F06-var-identification\u002Findex",[452],{"title":448,"path":449,"stem":450},{"title":454,"path":455,"stem":456,"children":457},"State Space, Mixed Frequency, and Nowcasting","\u002Fen\u002Ffinancial-economic-time-series\u002F07-state-space-nowcasting","en\u002Ffinancial-economic-time-series\u002F07-state-space-nowcasting\u002Findex",[458],{"title":454,"path":455,"stem":456},{"title":460,"path":461,"stem":462,"children":463},"Forecast Evaluation for Decisions","\u002Fen\u002Ffinancial-economic-time-series\u002F08-forecast-evaluation","en\u002Ffinancial-economic-time-series\u002F08-forecast-evaluation\u002Findex",[464],{"title":460,"path":461,"stem":462},{"title":466,"path":467,"stem":468,"children":469},"Integrated R Laboratory","\u002Fen\u002Ffinancial-economic-time-series\u002F09-r-laboratory","en\u002Ffinancial-economic-time-series\u002F09-r-laboratory\u002Findex",[470],{"title":466,"path":467,"stem":468},{"title":472,"path":473,"stem":474,"children":475},"Capstones, Data, and Reading Ladder","\u002Fen\u002Ffinancial-economic-time-series\u002F10-capstone-readings","en\u002Ffinancial-economic-time-series\u002F10-capstone-readings\u002Findex",[476],{"title":472,"path":473,"stem":474},{"title":478,"path":479,"stem":480,"children":481,"page":249},"Intro To Economics","\u002Fen\u002Fintro-to-economics","en\u002Fintro-to-economics",[482,486,490,494,498,502,506,510,514,518,522,526,530],{"title":483,"path":484,"stem":485},"Introduction to Economics — Decisions, Markets, and the Macroeconomy","\u002Fen\u002Fintro-to-economics\u002F00-intro","en\u002Fintro-to-economics\u002F00-intro",{"title":487,"path":488,"stem":489},"Chapter 1 — Choice, Opportunity Cost, and Trade","\u002Fen\u002Fintro-to-economics\u002F01-foundations","en\u002Fintro-to-economics\u002F01-foundations",{"title":491,"path":492,"stem":493},"Chapter 2 — Demand, Supply, Equilibrium, and Welfare","\u002Fen\u002Fintro-to-economics\u002F02-demand-and-supply","en\u002Fintro-to-economics\u002F02-demand-and-supply",{"title":495,"path":496,"stem":497},"Chapter 3 — Elasticity, Revenue, and Tax Incidence","\u002Fen\u002Fintro-to-economics\u002F03-elasticity","en\u002Fintro-to-economics\u002F03-elasticity",{"title":499,"path":500,"stem":501},"Chapter 4 — Firms, Market Power, and Market Failure","\u002Fen\u002Fintro-to-economics\u002F04-market-structures","en\u002Fintro-to-economics\u002F04-market-structures",{"title":503,"path":504,"stem":505},"Chapter 5 — GDP, Income, Wealth, and Welfare","\u002Fen\u002Fintro-to-economics\u002F05-gdp-and-wealth","en\u002Fintro-to-economics\u002F05-gdp-and-wealth",{"title":507,"path":508,"stem":509},"Chapter 6 — Inflation, Purchasing Power, and Labour Markets","\u002Fen\u002Fintro-to-economics\u002F06-inflation-and-unemployment","en\u002Fintro-to-economics\u002F06-inflation-and-unemployment",{"title":511,"path":512,"stem":513},"Chapter 7 — Productivity, Technology, and Economic Growth","\u002Fen\u002Fintro-to-economics\u002F07-economic-growth","en\u002Fintro-to-economics\u002F07-economic-growth",{"title":515,"path":516,"stem":517},"Chapter 8 — Money, Credit, and Banking","\u002Fen\u002Fintro-to-economics\u002F08-money-and-banking","en\u002Fintro-to-economics\u002F08-money-and-banking",{"title":519,"path":520,"stem":521},"Chapter 9 — Business Cycles, AD–AS, and Monetary Policy","\u002Fen\u002Fintro-to-economics\u002F09-monetary-policy-and-ad-as","en\u002Fintro-to-economics\u002F09-monetary-policy-and-ad-as",{"title":523,"path":524,"stem":525},"Chapter 10 — Fiscal Policy, Distribution, and Public Debt","\u002Fen\u002Fintro-to-economics\u002F10-fiscal-policy","en\u002Fintro-to-economics\u002F10-fiscal-policy",{"title":527,"path":528,"stem":529},"Chapter 11 — Trade, Capital Flows, and Exchange Rates","\u002Fen\u002Fintro-to-economics\u002F11-open-economy-and-exchange-rates","en\u002Fintro-to-economics\u002F11-open-economy-and-exchange-rates",{"title":531,"path":532,"stem":533},"Chapter 12 — Integrated Economic Analysis Studio","\u002Fen\u002Fintro-to-economics\u002F12-review-and-case-studies","en\u002Fintro-to-economics\u002F12-review-and-case-studies",{"title":535,"path":536,"stem":537,"children":538,"page":249},"Microeconometrics","\u002Fen\u002Fmicroeconometrics","en\u002Fmicroeconometrics",[539,557,563,581,603,629,651,673,691,709],{"title":540,"path":541,"stem":542,"children":543},"0. Causal Questions and Designs","\u002Fen\u002Fmicroeconometrics\u002F00-foundations","en\u002Fmicroeconometrics\u002F00-foundations\u002Findex",[544,545,549,553],{"title":540,"path":541,"stem":542},{"title":546,"path":547,"stem":548},"Causal Questions and Estimands","\u002Fen\u002Fmicroeconometrics\u002F00-foundations\u002F01-causal-question-estimands","en\u002Fmicroeconometrics\u002F00-foundations\u002F01-causal-question-estimands",{"title":550,"path":551,"stem":552},"Potential Outcomes and Experiments","\u002Fen\u002Fmicroeconometrics\u002F00-foundations\u002F02-potential-outcomes-experiments","en\u002Fmicroeconometrics\u002F00-foundations\u002F02-potential-outcomes-experiments",{"title":554,"path":555,"stem":556},"Causal Diagrams and Controls","\u002Fen\u002Fmicroeconometrics\u002F00-foundations\u002F03-dags-controls","en\u002Fmicroeconometrics\u002F00-foundations\u002F03-dags-controls",{"title":558,"path":559,"stem":560,"children":561},"Microeconometrics — Designing Credible Counterfactuals","\u002Fen\u002Fmicroeconometrics\u002F00-index","en\u002Fmicroeconometrics\u002F00-index",[562],{"title":558,"path":559,"stem":560},{"title":564,"path":565,"stem":566,"children":567},"1. Regression and Inference","\u002Fen\u002Fmicroeconometrics\u002F01-regression","en\u002Fmicroeconometrics\u002F01-regression\u002Findex",[568,569,573,577],{"title":564,"path":565,"stem":566},{"title":570,"path":571,"stem":572},"OLS as a Projection","\u002Fen\u002Fmicroeconometrics\u002F01-regression\u002F04-ols-projection","en\u002Fmicroeconometrics\u002F01-regression\u002F04-ols-projection",{"title":574,"path":575,"stem":576},"FWL, Selection and Controls","\u002Fen\u002Fmicroeconometrics\u002F01-regression\u002F05-fwl-selection-controls","en\u002Fmicroeconometrics\u002F01-regression\u002F05-fwl-selection-controls",{"title":578,"path":579,"stem":580},"Inference and Clustering","\u002Fen\u002Fmicroeconometrics\u002F01-regression\u002F06-inference-clustering","en\u002Fmicroeconometrics\u002F01-regression\u002F06-inference-clustering",{"title":582,"path":583,"stem":584,"children":585},"2. Instruments and Panel Data","\u002Fen\u002Fmicroeconometrics\u002F02-iv-panel","en\u002Fmicroeconometrics\u002F02-iv-panel\u002Findex",[586,587,591,595,599],{"title":582,"path":583,"stem":584},{"title":588,"path":589,"stem":590},"Instrumental Variables","\u002Fen\u002Fmicroeconometrics\u002F02-iv-panel\u002F07-iv-identification","en\u002Fmicroeconometrics\u002F02-iv-panel\u002F07-iv-identification",{"title":592,"path":593,"stem":594},"Weak Instruments and LATE","\u002Fen\u002Fmicroeconometrics\u002F02-iv-panel\u002F08-weak-iv-late","en\u002Fmicroeconometrics\u002F02-iv-panel\u002F08-weak-iv-late",{"title":596,"path":597,"stem":598},"Panel Fixed Effects","\u002Fen\u002Fmicroeconometrics\u002F02-iv-panel\u002F09-panel-fixed-effects","en\u002Fmicroeconometrics\u002F02-iv-panel\u002F09-panel-fixed-effects",{"title":600,"path":601,"stem":602},"Dynamic Panels and Limits","\u002Fen\u002Fmicroeconometrics\u002F02-iv-panel\u002F10-dynamic-panel","en\u002Fmicroeconometrics\u002F02-iv-panel\u002F10-dynamic-panel",{"title":604,"path":605,"stem":606,"children":607},"3. Policy Evaluation Designs","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs","en\u002Fmicroeconometrics\u002F03-policy-designs\u002Findex",[608,609,613,617,621,625],{"title":604,"path":605,"stem":606},{"title":610,"path":611,"stem":612},"Difference-in-Differences","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs\u002F11-did-core","en\u002Fmicroeconometrics\u002F03-policy-designs\u002F11-did-core",{"title":614,"path":615,"stem":616},"Staggered DID and Event Studies","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs\u002F12-staggered-event-studies","en\u002Fmicroeconometrics\u002F03-policy-designs\u002F12-staggered-event-studies",{"title":618,"path":619,"stem":620},"Regression Discontinuity","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs\u002F13-rdd","en\u002Fmicroeconometrics\u002F03-policy-designs\u002F13-rdd",{"title":622,"path":623,"stem":624},"Matching, Weighting and Overlap","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs\u002F14-matching-weighting","en\u002Fmicroeconometrics\u002F03-policy-designs\u002F14-matching-weighting",{"title":626,"path":627,"stem":628},"Synthetic Control","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs\u002F15-synthetic-control","en\u002Fmicroeconometrics\u002F03-policy-designs\u002F15-synthetic-control",{"title":630,"path":631,"stem":632,"children":633},"Module 4 — Choice and Limited Outcomes","\u002Fen\u002Fmicroeconometrics\u002F04-outcomes-choice","en\u002Fmicroeconometrics\u002F04-outcomes-choice\u002Findex",[634,635,639,643,647],{"title":630,"path":631,"stem":632},{"title":636,"path":637,"stem":638},"Binary Choice","\u002Fen\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F16-binary-choice","en\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F16-binary-choice",{"title":640,"path":641,"stem":642},"Multinomial and Ordered Choice","\u002Fen\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F17-multinomial-choice","en\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F17-multinomial-choice",{"title":644,"path":645,"stem":646},"Count Outcomes","\u002Fen\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F18-count-outcomes","en\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F18-count-outcomes",{"title":648,"path":649,"stem":650},"Censoring, Truncation and Selection","\u002Fen\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F19-censoring-selection","en\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F19-censoring-selection",{"title":652,"path":653,"stem":654,"children":655},"Module 5 — Modern Causal Analysis","\u002Fen\u002Fmicroeconometrics\u002F05-modern-causal","en\u002Fmicroeconometrics\u002F05-modern-causal\u002Findex",[656,657,661,665,669],{"title":652,"path":653,"stem":654},{"title":658,"path":659,"stem":660},"Double Machine Learning","\u002Fen\u002Fmicroeconometrics\u002F05-modern-causal\u002F20-dml","en\u002Fmicroeconometrics\u002F05-modern-causal\u002F20-dml",{"title":662,"path":663,"stem":664},"Heterogeneity and Policy Learning","\u002Fen\u002Fmicroeconometrics\u002F05-modern-causal\u002F21-heterogeneity-policy","en\u002Fmicroeconometrics\u002F05-modern-causal\u002F21-heterogeneity-policy",{"title":666,"path":667,"stem":668},"Sensitivity and Partial Identification","\u002Fen\u002Fmicroeconometrics\u002F05-modern-causal\u002F22-sensitivity-partial-id","en\u002Fmicroeconometrics\u002F05-modern-causal\u002F22-sensitivity-partial-id",{"title":670,"path":671,"stem":672},"External Validity and Transport","\u002Fen\u002Fmicroeconometrics\u002F05-modern-causal\u002F23-external-validity","en\u002Fmicroeconometrics\u002F05-modern-causal\u002F23-external-validity",{"title":674,"path":675,"stem":676,"children":677},"Module 6 — From Data to Auditable Evidence","\u002Fen\u002Fmicroeconometrics\u002F06-workflow","en\u002Fmicroeconometrics\u002F06-workflow\u002Findex",[678,679,683,687],{"title":674,"path":675,"stem":676},{"title":680,"path":681,"stem":682},"Data Provenance","\u002Fen\u002Fmicroeconometrics\u002F06-workflow\u002F24-data-provenance","en\u002Fmicroeconometrics\u002F06-workflow\u002F24-data-provenance",{"title":684,"path":685,"stem":686},"Reproducibility and Reporting","\u002Fen\u002Fmicroeconometrics\u002F06-workflow\u002F25-reproducibility-reporting","en\u002Fmicroeconometrics\u002F06-workflow\u002F25-reproducibility-reporting",{"title":688,"path":689,"stem":690},"Paper and Evidence Audit","\u002Fen\u002Fmicroeconometrics\u002F06-workflow\u002F26-paper-audit","en\u002Fmicroeconometrics\u002F06-workflow\u002F26-paper-audit",{"title":692,"path":693,"stem":694,"children":695},"Appendix — Revision and Evidence Map","\u002Fen\u002Fmicroeconometrics\u002Fappendix","en\u002Fmicroeconometrics\u002Fappendix\u002Findex",[696,697,701,705],{"title":692,"path":693,"stem":694},{"title":698,"path":699,"stem":700},"Formula and Design Map","\u002Fen\u002Fmicroeconometrics\u002Fappendix\u002F27-formula-design-map","en\u002Fmicroeconometrics\u002Fappendix\u002F27-formula-design-map",{"title":702,"path":703,"stem":704},"Ten Worked Microeconometric Pitfalls","\u002Fen\u002Fmicroeconometrics\u002Fappendix\u002F28-worked-pitfalls","en\u002Fmicroeconometrics\u002Fappendix\u002F28-worked-pitfalls",{"title":706,"path":707,"stem":708},"Reading and Software Map","\u002Fen\u002Fmicroeconometrics\u002Fappendix\u002F29-reading-software-map","en\u002Fmicroeconometrics\u002Fappendix\u002F29-reading-software-map",{"title":710,"path":711,"stem":712,"children":713},"Capstone — Should Northbridge Expand Pathways?","\u002Fen\u002Fmicroeconometrics\u002Fcapstone","en\u002Fmicroeconometrics\u002Fcapstone\u002Findex",[714],{"title":710,"path":711,"stem":712},{"title":716,"path":717,"stem":718,"children":719,"page":249},"Microeconomics","\u002Fen\u002Fmicroeconomics","en\u002Fmicroeconomics",[720,726,732,738,744,750,756,762,768,774,780,786,792],{"title":721,"path":722,"stem":723,"children":724},"Advanced Microeconomics — Course Guide","\u002Fen\u002Fmicroeconomics\u002F00-intro","en\u002Fmicroeconomics\u002F00-intro\u002Findex",[725],{"title":721,"path":722,"stem":723},{"title":727,"path":728,"stem":729,"children":730},"Module 1 — Choice, Duality, and Revealed Preference","\u002Fen\u002Fmicroeconomics\u002F01-fundations","en\u002Fmicroeconomics\u002F01-fundations\u002Findex",[731],{"title":727,"path":728,"stem":729},{"title":733,"path":734,"stem":735,"children":736},"Module 2 — Comparative Statics and Welfare Measurement","\u002Fen\u002Fmicroeconomics\u002F02-comparative-statics","en\u002Fmicroeconomics\u002F02-comparative-statics\u002Findex",[737],{"title":733,"path":734,"stem":735},{"title":739,"path":740,"stem":741,"children":742},"Module 3 — Choice under Risk and Insurance","\u002Fen\u002Fmicroeconomics\u002F03-uncertainty","en\u002Fmicroeconomics\u002F03-uncertainty\u002Findex",[743],{"title":739,"path":740,"stem":741},{"title":745,"path":746,"stem":747,"children":748},"Module 4 — General Equilibrium and Welfare","\u002Fen\u002Fmicroeconomics\u002F04-general-equilibrium","en\u002Fmicroeconomics\u002F04-general-equilibrium\u002Findex",[749],{"title":745,"path":746,"stem":747},{"title":751,"path":752,"stem":753,"children":754},"Module 5 — Static, Dynamic, and Repeated Games","\u002Fen\u002Fmicroeconomics\u002F05-game-theory","en\u002Fmicroeconomics\u002F05-game-theory\u002Findex",[755],{"title":751,"path":752,"stem":753},{"title":757,"path":758,"stem":759,"children":760},"Module 6 — Oligopoly, Entry, and Algorithmic Pricing","\u002Fen\u002Fmicroeconomics\u002F06-oligopoly","en\u002Fmicroeconomics\u002F06-oligopoly\u002Findex",[761],{"title":757,"path":758,"stem":759},{"title":763,"path":764,"stem":765,"children":766},"Module 7 — Information Economics and Contracts","\u002Fen\u002Fmicroeconomics\u002F07-information-economics","en\u002Fmicroeconomics\u002F07-information-economics\u002Findex",[767],{"title":763,"path":764,"stem":765},{"title":769,"path":770,"stem":771,"children":772},"Module 8 — Mechanism Design and Auctions","\u002Fen\u002Fmicroeconomics\u002F08-mechanism-design","en\u002Fmicroeconomics\u002F08-mechanism-design\u002Findex",[773],{"title":769,"path":770,"stem":771},{"title":775,"path":776,"stem":777,"children":778},"Module 9 — Behavioural and Experimental Microeconomics","\u002Fen\u002Fmicroeconomics\u002F09-behavioural-economics","en\u002Fmicroeconomics\u002F09-behavioural-economics\u002Findex",[779],{"title":775,"path":776,"stem":777},{"title":781,"path":782,"stem":783,"children":784},"Module 10 — Externalities, Public Goods, and Collective Action","\u002Fen\u002Fmicroeconomics\u002F10-externalities-public-goods","en\u002Fmicroeconomics\u002F10-externalities-public-goods\u002Findex",[785],{"title":781,"path":782,"stem":783},{"title":787,"path":788,"stem":789,"children":790},"Module 11 — Matching and Market Design","\u002Fen\u002Fmicroeconomics\u002F11-market-design","en\u002Fmicroeconomics\u002F11-market-design\u002Findex",[791],{"title":787,"path":788,"stem":789},{"title":793,"path":794,"stem":795,"children":796},"Module 12 — Integrated Microeconomic Design Studio","\u002Fen\u002Fmicroeconomics\u002F12-review","en\u002Fmicroeconomics\u002F12-review\u002Findex",[797],{"title":793,"path":794,"stem":795},{"title":799,"path":800,"stem":801,"children":802},"Playground","\u002Fen\u002Fplayground","en\u002Fplayground\u002Findex",[803,804,808,812,816,820],{"title":799,"path":800,"stem":801},{"title":805,"path":806,"stem":807},"Typst Plugin Playground","\u002Fen\u002Fplayground\u002F01-typstex","en\u002Fplayground\u002F01-typstex",{"title":809,"path":810,"stem":811},"Citation Plugin Test","\u002Fen\u002Fplayground\u002F02-citation","en\u002Fplayground\u002F02-citation",{"title":813,"path":814,"stem":815},"Pyodide Playground","\u002Fen\u002Fplayground\u002F03-pyodide","en\u002Fplayground\u002F03-pyodide",{"title":817,"path":818,"stem":819},"WebR Playground","\u002Fen\u002Fplayground\u002F04-webr","en\u002Fplayground\u002F04-webr",{"title":821,"path":822,"stem":823},"Chart.js 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文献综述的构建与写作","\u002Fzh\u002Facademic-writing\u002F04-writing-the-literature-review","zh\u002Facademic-writing\u002F04-writing-the-literature-review",{"title":1056,"path":1057,"stem":1058},"8. 文献综述清单与模板","\u002Fzh\u002Facademic-writing\u002F08-literature-review-checklist-and-template","zh\u002Facademic-writing\u002F08-literature-review-checklist-and-template",{"title":1060,"path":1061,"stem":1062},"10. 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Start with cumulative observations",[1793,1802,1803,1804,1931,1932,1961,1962,1991],{},"Let ",[1805,1806,1809,1848],"span",{"className":1807},[1808],"katex",[1805,1810,1813],{"className":1811},[1812],"katex-mathml",[1814,1815,1817],"math",{"xmlns":1816},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[1818,1819,1820,1843],"semantics",{},[1821,1822,1823],"mrow",{},[1824,1825,1826,1830],"msub",{},[1827,1828,1829],"mi",{},"C",[1821,1831,1832,1835,1840],{},[1827,1833,1834],{},"i",[1836,1837,1839],"mo",{"separator":1838},"true",",",[1827,1841,1842],{},"j",[1844,1845,1847],"annotation",{"encoding":1846},"application\u002Fx-tex","C_{i,j}",[1805,1849,1852],{"className":1850,"ariaHidden":1838},[1851],"katex-html",[1805,1853,1856,1861],{"className":1854},[1855],"base",[1805,1857],{"className":1858,"style":1860},[1859],"strut","height:0.9694em;vertical-align:-0.2861em;",[1805,1862,1865,1870],{"className":1863},[1864],"mord",[1805,1866,1829],{"className":1867,"style":1869},[1864,1868],"mathnormal","margin-right:0.0715em;",[1805,1871,1874],{"className":1872},[1873],"msupsub",[1805,1875,1879,1922],{"className":1876},[1877,1878],"vlist-t","vlist-t2",[1805,1880,1883,1917],{"className":1881},[1882],"vlist-r",[1805,1884,1888],{"className":1885,"style":1887},[1886],"vlist","height:0.3117em;",[1805,1889,1891,1896],{"style":1890},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[1805,1892],{"className":1893,"style":1895},[1894],"pstrut","height:2.7em;",[1805,1897,1903],{"className":1898},[1899,1900,1901,1902],"sizing","reset-size6","size3","mtight",[1805,1904,1906,1909,1913],{"className":1905},[1864,1902],[1805,1907,1834],{"className":1908},[1864,1868,1902],[1805,1910,1839],{"className":1911},[1912,1902],"mpunct",[1805,1914,1842],{"className":1915,"style":1916},[1864,1868,1902],"margin-right:0.0572em;",[1805,1918,1921],{"className":1919},[1920],"vlist-s","​",[1805,1923,1925],{"className":1924},[1882],[1805,1926,1929],{"className":1927,"style":1928},[1886],"height:0.2861em;",[1805,1930],{}," be cumulative paid loss for origin year ",[1805,1933,1935,1948],{"className":1934},[1808],[1805,1936,1938],{"className":1937},[1812],[1814,1939,1940],{"xmlns":1816},[1818,1941,1942,1946],{},[1821,1943,1944],{},[1827,1945,1834],{},[1844,1947,1834],{"encoding":1846},[1805,1949,1951],{"className":1950,"ariaHidden":1838},[1851],[1805,1952,1954,1958],{"className":1953},[1855],[1805,1955],{"className":1956,"style":1957},[1859],"height:0.6595em;",[1805,1959,1834],{"className":1960},[1864,1868]," at development age ",[1805,1963,1965,1978],{"className":1964},[1808],[1805,1966,1968],{"className":1967},[1812],[1814,1969,1970],{"xmlns":1816},[1818,1971,1972,1976],{},[1821,1973,1974],{},[1827,1975,1842],{},[1844,1977,1842],{"encoding":1846},[1805,1979,1981],{"className":1980,"ariaHidden":1838},[1851],[1805,1982,1984,1988],{"className":1983},[1855],[1805,1985],{"className":1986,"style":1987},[1859],"height:0.854em;vertical-align:-0.1944em;",[1805,1989,1842],{"className":1990,"style":1916},[1864,1868],".",[1993,1994,1995,2018],"table",{},[1996,1997,1998],"thead",{},[1999,2000,2001,2005,2009,2012,2015],"tr",{},[2002,2003,2004],"th",{},"AY",[2002,2006,2008],{"align":2007},"right","Dev 0",[2002,2010,2011],{"align":2007},"Dev 1",[2002,2013,2014],{"align":2007},"Dev 2",[2002,2016,2017],{"align":2007},"Dev 3",[2019,2020,2021,2039,2056,2071],"tbody",{},[1999,2022,2023,2027,2030,2033,2036],{},[2024,2025,2026],"td",{},"2020",[2024,2028,2029],{"align":2007},"100",[2024,2031,2032],{"align":2007},"180",[2024,2034,2035],{"align":2007},"240",[2024,2037,2038],{"align":2007},"280",[1999,2040,2041,2044,2047,2050,2053],{},[2024,2042,2043],{},"2021",[2024,2045,2046],{"align":2007},"120",[2024,2048,2049],{"align":2007},"220",[2024,2051,2052],{"align":2007},"300",[2024,2054,2055],{"align":2007},"—",[1999,2057,2058,2061,2064,2067,2069],{},[2024,2059,2060],{},"2022",[2024,2062,2063],{"align":2007},"140",[2024,2065,2066],{"align":2007},"260",[2024,2068,2055],{"align":2007},[2024,2070,2055],{"align":2007},[1999,2072,2073,2076,2079,2081,2083],{},[2024,2074,2075],{},"2023",[2024,2077,2078],{"align":2007},"160",[2024,2080,2055],{"align":2007},[2024,2082,2055],{"align":2007},[2024,2084,2055],{"align":2007},[1793,2086,2087],{},"Units are £m. The triangle is cumulative, paid, nominal, gross of reinsurance, and valued at year-end 2023.",[1797,2089,2091],{"id":2090},"_2-estimate-age-to-age-factors","2. Estimate age-to-age factors",[1793,2093,2094],{},"The usual volume-weighted factor is a ratio of column sums over rows for which both ages are observed:",[1805,2096,2099],{"className":2097},[2098],"katex-display",[1805,2100,2102,2208],{"className":2101},[1808],[1805,2103,2105],{"className":2104},[1812],[1814,2106,2108],{"xmlns":1816,"display":2107},"block",[1818,2109,2110,2205],{},[1821,2111,2112,2125,2128,2202],{},[1824,2113,2114,2123],{},[2115,2116,2117,2120],"mover",{"accent":1838},[1827,2118,2119],{},"f",[1836,2121,2122],{"stretchy":1838},"^",[1827,2124,1842],{},[1836,2126,2127],{},"=",[2129,2130,2131,2172],"mfrac",{},[1821,2132,2133,2153],{},[2134,2135,2136,2139],"munder",{},[1836,2137,2138],{},"∑",[1821,2140,2141,2143,2146],{},[1827,2142,1834],{},[1836,2144,2145],{},"∈",[1824,2147,2148,2151],{},[1827,2149,2150],{},"I",[1827,2152,1842],{},[1824,2154,2155,2157],{},[1827,2156,1829],{},[1821,2158,2159,2161,2163,2165,2168],{},[1827,2160,1834],{},[1836,2162,1839],{"separator":1838},[1827,2164,1842],{},[1836,2166,2167],{},"+",[2169,2170,2171],"mn",{},"1",[1821,2173,2174,2190],{},[2134,2175,2176,2178],{},[1836,2177,2138],{},[1821,2179,2180,2182,2184],{},[1827,2181,1834],{},[1836,2183,2145],{},[1824,2185,2186,2188],{},[1827,2187,2150],{},[1827,2189,1842],{},[1824,2191,2192,2194],{},[1827,2193,1829],{},[1821,2195,2196,2198,2200],{},[1827,2197,1834],{},[1836,2199,1839],{"separator":1838},[1827,2201,1842],{},[1827,2203,1991],{"mathvariant":2204},"normal",[1844,2206,2207],{"encoding":1846},"\\widehat 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the example:",[1993,2706,2707,2723],{},[1996,2708,2709],{},[1999,2710,2711,2714,2717,2720],{},[2002,2712,2713],{},"Transition",[2002,2715,2716],{"align":2007},"Calculation",[2002,2718,2719],{"align":2007},"Factor",[2002,2721,2722],{"align":2007},"Data supporting it",[2019,2724,2725,2892,3008],{},[1999,2726,2727,2730,2886,2889],{},[2024,2728,2729],{},"0 → 1",[2024,2731,2732],{"align":2007},[1805,2733,2735,2781],{"className":2734},[1808],[1805,2736,2738],{"className":2737},[1812],[1814,2739,2740],{"xmlns":1816},[1818,2741,2742,2778],{},[1821,2743,2744,2748,2750,2752,2754,2756,2758,2761,2764,2766,2768,2770,2772,2774,2776],{},[1836,2745,2747],{"stretchy":2746},"false","(",[2169,2749,2032],{},[1836,2751,2167],{},[2169,2753,2049],{},[1836,2755,2167],{},[2169,2757,2066],{},[1836,2759,2760],{"stretchy":2746},")",[1827,2762,2763],{"mathvariant":2204},"\u002F",[1836,2765,2747],{"stretchy":2746},[2169,2767,2029],{},[1836,2769,2167],{},[2169,2771,2046],{},[1836,2773,2167],{},[2169,2775,2063],{},[1836,2777,2760],{"stretchy":2746},[1844,2779,2780],{"encoding":1846},"(180+220+260)\u002F(100+120+140)",[1805,2782,2784,2807,2826,2856,2874],{"className":2783,"ariaHidden":1838},[1851],[1805,2785,2787,2791,2794,2797,2801,2804],{"className":2786},[1855],[1805,2788],{"className":2789,"style":2790},[1859],"height:1em;vertical-align:-0.25em;",[1805,2792,2747],{"className":2793},[2341],[1805,2795,2032],{"className":2796},[1864],[1805,2798],{"className":2799,"style":2800},[2319],"margin-right:0.2222em;",[1805,2802,2167],{"className":2803},[2668],[1805,2805],{"className":2806,"style":2800},[2319],[1805,2808,2810,2814,2817,2820,2823],{"className":2809},[1855],[1805,2811],{"className":2812,"style":2813},[1859],"height:0.7278em;vertical-align:-0.0833em;",[1805,2815,2049],{"className":2816},[1864],[1805,2818],{"className":2819,"style":2800},[2319],[1805,2821,2167],{"className":2822},[2668],[1805,2824],{"className":2825,"style":2800},[2319],[1805,2827,2829,2832,2835,2838,2841,2844,2847,2850,2853],{"className":2828},[1855],[1805,2830],{"className":2831,"style":2790},[1859],[1805,2833,2066],{"className":2834},[1864],[1805,2836,2760],{"className":2837},[2698],[1805,2839,2763],{"className":2840},[1864],[1805,2842,2747],{"className":2843},[2341],[1805,2845,2029],{"className":2846},[1864],[1805,2848],{"className":2849,"style":2800},[2319],[1805,2851,2167],{"className":2852},[2668],[1805,2854],{"className":2855,"style":2800},[2319],[1805,2857,2859,2862,2865,2868,2871],{"className":2858},[1855],[1805,2860],{"className":2861,"style":2813},[1859],[1805,2863,2046],{"className":2864},[1864],[1805,2866],{"className":2867,"style":2800},[2319],[1805,2869,2167],{"className":2870},[2668],[1805,2872],{"className":2873,"style":2800},[2319],[1805,2875,2877,2880,2883],{"className":2876},[1855],[1805,2878],{"className":2879,"style":2790},[1859],[1805,2881,2063],{"className":2882},[1864],[1805,2884,2760],{"className":2885},[2698],[2024,2887,2888],{"align":2007},"1.8333",[2024,2890,2891],{"align":2007},"3 origin years",[1999,2893,2894,2897,3002,3005],{},[2024,2895,2896],{},"1 → 2",[2024,2898,2899],{"align":2007},[1805,2900,2902,2936],{"className":2901},[1808],[1805,2903,2905],{"className":2904},[1812],[1814,2906,2907],{"xmlns":1816},[1818,2908,2909,2933],{},[1821,2910,2911,2913,2915,2917,2919,2921,2923,2925,2927,2929,2931],{},[1836,2912,2747],{"stretchy":2746},[2169,2914,2035],{},[1836,2916,2167],{},[2169,2918,2052],{},[1836,2920,2760],{"stretchy":2746},[1827,2922,2763],{"mathvariant":2204},[1836,2924,2747],{"stretchy":2746},[2169,2926,2032],{},[1836,2928,2167],{},[2169,2930,2049],{},[1836,2932,2760],{"stretchy":2746},[1844,2934,2935],{"encoding":1846},"(240+300)\u002F(180+220)",[1805,2937,2939,2960,2990],{"className":2938,"ariaHidden":1838},[1851],[1805,2940,2942,2945,2948,2951,2954,2957],{"className":2941},[1855],[1805,2943],{"className":2944,"style":2790},[1859],[1805,2946,2747],{"className":2947},[2341],[1805,2949,2035],{"className":2950},[1864],[1805,2952],{"className":2953,"style":2800},[2319],[1805,2955,2167],{"className":2956},[2668],[1805,2958],{"className":2959,"style":2800},[2319],[1805,2961,2963,2966,2969,2972,2975,2978,2981,2984,2987],{"className":2962},[1855],[1805,2964],{"className":2965,"style":2790},[1859],[1805,2967,2052],{"className":2968},[1864],[1805,2970,2760],{"className":2971},[2698],[1805,2973,2763],{"className":2974},[1864],[1805,2976,2747],{"className":2977},[2341],[1805,2979,2032],{"className":2980},[1864],[1805,2982],{"className":2983,"style":2800},[2319],[1805,2985,2167],{"className":2986},[2668],[1805,2988],{"className":2989,"style":2800},[2319],[1805,2991,2993,2996,2999],{"className":2992},[1855],[1805,2994],{"className":2995,"style":2790},[1859],[1805,2997,2049],{"className":2998},[1864],[1805,3000,2760],{"className":3001},[2698],[2024,3003,3004],{"align":2007},"1.3500",[2024,3006,3007],{"align":2007},"2 origin years",[1999,3009,3010,3013,3048,3051],{},[2024,3011,3012],{},"2 → 3",[2024,3014,3015],{"align":2007},[1805,3016,3018,3036],{"className":3017},[1808],[1805,3019,3021],{"className":3020},[1812],[1814,3022,3023],{"xmlns":1816},[1818,3024,3025,3033],{},[1821,3026,3027,3029,3031],{},[2169,3028,2038],{},[1827,3030,2763],{"mathvariant":2204},[2169,3032,2035],{},[1844,3034,3035],{"encoding":1846},"280\u002F240",[1805,3037,3039],{"className":3038,"ariaHidden":1838},[1851],[1805,3040,3042,3045],{"className":3041},[1855],[1805,3043],{"className":3044,"style":2790},[1859],[1805,3046,3035],{"className":3047},[1864],[2024,3049,3050],{"align":2007},"1.1667",[2024,3052,3053],{"align":2007},"1 origin year",[1793,3055,3056],{},"Volume weighting gives larger cells more influence. It is not automatically superior to an arithmetic or credibility-weighted selection; it is a modelling choice whose sensitivity should be shown.",[3058,3059,3061,3156,3157,3161],"tip",{"title":3060},"Terminology",[1805,3062,3064,3087],{"className":3063},[1808],[1805,3065,3067],{"className":3066},[1812],[1814,3068,3069],{"xmlns":1816},[1818,3070,3071,3084],{},[1821,3072,3073,3079,3082],{},[1824,3074,3075,3077],{},[1827,3076,2119],{},[1827,3078,1842],{},[1836,3080,3081],{},">",[2169,3083,2171],{},[1844,3085,3086],{"encoding":1846},"f_j>1",[1805,3088,3090,3146],{"className":3089,"ariaHidden":1838},[1851],[1805,3091,3093,3097,3137,3140,3143],{"className":3092},[1855],[1805,3094],{"className":3095,"style":3096},[1859],"height:0.9805em;vertical-align:-0.2861em;",[1805,3098,3100,3103],{"className":3099},[1864],[1805,3101,2119],{"className":3102,"style":2245},[1864,1868],[1805,3104,3106],{"className":3105},[1873],[1805,3107,3109,3129],{"className":3108},[1877,1878],[1805,3110,3112,3126],{"className":3111},[1882],[1805,3113,3115],{"className":3114,"style":1887},[1886],[1805,3116,3117,3120],{"style":2295},[1805,3118],{"className":3119,"style":1895},[1894],[1805,3121,3123],{"className":3122},[1899,1900,1901,1902],[1805,3124,1842],{"className":3125,"style":1916},[1864,1868,1902],[1805,3127,1921],{"className":3128},[1920],[1805,3130,3132],{"className":3131},[1882],[1805,3133,3135],{"className":3134,"style":1928},[1886],[1805,3136],{},[1805,3138],{"className":3139,"style":2320},[2319],[1805,3141,3081],{"className":3142},[2324],[1805,3144],{"className":3145,"style":2320},[2319],[1805,3147,3149,3153],{"className":3148},[1855],[1805,3150],{"className":3151,"style":3152},[1859],"height:0.6444em;",[1805,3154,2171],{"className":3155},[1864]," is a ",[3158,3159,3160],"strong",{},"link ratio"," or age-to-age development factor. Its reciprocal is a completion proportion only under the selected development pattern. Calling both quantities a “grossing-up factor” invites direction errors.",[1797,3163,3165],{"id":3164},"_3-convert-link-ratios-into-cumulative-development-factors","3. Convert link ratios into cumulative development factors",[1793,3167,3168,3169,3197,3198,3229],{},"The cumulative development factor (CDF) from age ",[1805,3170,3172,3185],{"className":3171},[1808],[1805,3173,3175],{"className":3174},[1812],[1814,3176,3177],{"xmlns":1816},[1818,3178,3179,3183],{},[1821,3180,3181],{},[1827,3182,1842],{},[1844,3184,1842],{"encoding":1846},[1805,3186,3188],{"className":3187,"ariaHidden":1838},[1851],[1805,3189,3191,3194],{"className":3190},[1855],[1805,3192],{"className":3193,"style":1987},[1859],[1805,3195,1842],{"className":3196,"style":1916},[1864,1868]," to ultimate age ",[1805,3199,3201,3215],{"className":3200},[1808],[1805,3202,3204],{"className":3203},[1812],[1814,3205,3206],{"xmlns":1816},[1818,3207,3208,3213],{},[1821,3209,3210],{},[1827,3211,3212],{},"J",[1844,3214,3212],{"encoding":1846},[1805,3216,3218],{"className":3217,"ariaHidden":1838},[1851],[1805,3219,3221,3225],{"className":3220},[1855],[1805,3222],{"className":3223,"style":3224},[1859],"height:0.6833em;",[1805,3226,3212],{"className":3227,"style":3228},[1864,1868],"margin-right:0.0962em;"," is",[1805,3231,3233],{"className":3232},[2098],[1805,3234,3236,3328],{"className":3235},[1808],[1805,3237,3239],{"className":3238},[1812],[1814,3240,3241],{"xmlns":1816,"display":2107},[1818,3242,3243,3325],{},[1821,3244,3245,3256,3258,3282,3292,3294,3297,3307,3309,3323],{},[1824,3246,3247,3254],{},[2115,3248,3249,3252],{"accent":1838},[1827,3250,3251],{},"F",[1836,3253,2122],{"stretchy":1838},[1827,3255,1842],{},[1836,3257,2127],{},[3259,3260,3261,3264,3273],"munderover",{},[1836,3262,3263],{},"∏",[1821,3265,3266,3269,3271],{},[1827,3267,3268],{},"k",[1836,3270,2127],{},[1827,3272,1842],{},[1821,3274,3275,3277,3280],{},[1827,3276,3212],{},[1836,3278,3279],{},"−",[2169,3281,2171],{},[1824,3283,3284,3290],{},[2115,3285,3286,3288],{"accent":1838},[1827,3287,2119],{},[1836,3289,2122],{"stretchy":1838},[1827,3291,3268],{},[1836,3293,1839],{"separator":1838},[2319,3295],{"width":3296},"2em",[1824,3298,3299,3305],{},[2115,3300,3301,3303],{"accent":1838},[1827,3302,1793],{},[1836,3304,2122],{"stretchy":1838},[1827,3306,1842],{},[1836,3308,2127],{},[2129,3310,3311,3313],{},[2169,3312,2171],{},[1824,3314,3315,3321],{},[2115,3316,3317,3319],{"accent":1838},[1827,3318,3251],{},[1836,3320,2122],{"stretchy":1838},[1827,3322,1842],{},[1836,3324,1839],{"separator":1838},[1844,3326,3327],{"encoding":1846},"\\widehat F_j=\\prod_{k=j}^{J-1}\\widehat f_k,\n\\qquad\n\\widehat p_j=\\frac{1}{\\widehat F_j},",[1805,3329,3331,3420,3696],{"className":3330,"ariaHidden":1838},[1851],[1805,3332,3334,3338,3411,3414,3417],{"className":3333},[1855],[1805,3335],{"className":3336,"style":3337},[1859],"height:1.2094em;vertical-align:-0.2861em;",[1805,3339,3341,3376],{"className":3340},[1864],[1805,3342,3344],{"className":3343},[1864,2224],[1805,3345,3347],{"className":3346},[1877],[1805,3348,3350],{"className":3349},[1882],[1805,3351,3354,3363],{"className":3352,"style":3353},[1886],"height:0.9233em;",[1805,3355,3356,3359],{"style":2237},[1805,3357],{"className":3358,"style":2241},[1894],[1805,3360,3251],{"className":3361,"style":3362},[1864,1868],"margin-right:0.1389em;",[1805,3364,3367,3370],{"className":3365,"style":3366},[2249],"width:calc(100% - 0.1667em);margin-left:0.1667em;top:-3.6833em;",[1805,3368],{"className":3369,"style":2241},[1894],[1805,3371,3372],{"style":2256},[2258,3373,3374],{"xmlns":2260,"width":2261,"height":2262,"viewBox":2263,"preserveAspectRatio":2264},[2266,3375],{"d":2268},[1805,3377,3379],{"className":3378},[1873],[1805,3380,3382,3403],{"className":3381},[1877,1878],[1805,3383,3385,3400],{"className":3384},[1882],[1805,3386,3388],{"className":3387,"style":1887},[1886],[1805,3389,3391,3394],{"style":3390},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[1805,3392],{"className":3393,"style":1895},[1894],[1805,3395,3397],{"className":3396},[1899,1900,1901,1902],[1805,3398,1842],{"className":3399,"style":1916},[1864,1868,1902],[1805,3401,1921],{"className":3402},[1920],[1805,3404,3406],{"className":3405},[1882],[1805,3407,3409],{"className":3408,"style":1928},[1886],[1805,3410],{},[1805,3412],{"className":3413,"style":2320},[2319],[1805,3415,2127],{"className":3416},[2324],[1805,3418],{"className":3419,"style":2320},[2319],[1805,3421,3423,3427,3509,3512,3594,3597,3601,3604,3687,3690,3693],{"className":3422},[1855],[1805,3424],{"className":3425,"style":3426},[1859],"height:3.2666em;vertical-align:-1.4382em;",[1805,3428,3431],{"className":3429},[2368,3430],"op-limits",[1805,3432,3434,3500],{"className":3433},[1877,1878],[1805,3435,3437,3497],{"className":3436},[1882],[1805,3438,3441,3464,3476],{"className":3439,"style":3440},[1886],"height:1.8283em;",[1805,3442,3444,3448],{"style":3443},"top:-1.8479em;margin-left:0em;",[1805,3445],{"className":3446,"style":3447},[1894],"height:3.05em;",[1805,3449,3451],{"className":3450},[1899,1900,1901,1902],[1805,3452,3454,3458,3461],{"className":3453},[1864,1902],[1805,3455,3268],{"className":3456,"style":3457},[1864,1868,1902],"margin-right:0.0315em;",[1805,3459,2127],{"className":3460},[2324,1902],[1805,3462,1842],{"className":3463,"style":1916},[1864,1868,1902],[1805,3465,3467,3470],{"style":3466},"top:-3.05em;",[1805,3468],{"className":3469,"style":3447},[1894],[1805,3471,3472],{},[1805,3473,3263],{"className":3474},[2368,2372,3475],"large-op",[1805,3477,3479,3482],{"style":3478},"top:-4.3em;margin-left:0em;",[1805,3480],{"className":3481,"style":3447},[1894],[1805,3483,3485],{"className":3484},[1899,1900,1901,1902],[1805,3486,3488,3491,3494],{"className":3487},[1864,1902],[1805,3489,3212],{"className":3490,"style":3228},[1864,1868,1902],[1805,3492,3279],{"className":3493},[2668,1902],[1805,3495,2171],{"className":3496},[1864,1902],[1805,3498,1921],{"className":3499},[1920],[1805,3501,3503],{"className":3502},[1882],[1805,3504,3507],{"className":3505,"style":3506},[1886],"height:1.4382em;",[1805,3508],{},[1805,3510],{"className":3511,"style":2468},[2319],[1805,3513,3515,3558],{"className":3514},[1864],[1805,3516,3518],{"className":3517},[1864,2224],[1805,3519,3521,3550],{"className":3520},[1877,1878],[1805,3522,3524,3547],{"className":3523},[1882],[1805,3525,3527,3535],{"className":3526,"style":2234},[1886],[1805,3528,3529,3532],{"style":2237},[1805,3530],{"className":3531,"style":2241},[1894],[1805,3533,2119],{"className":3534,"style":2245},[1864,1868],[1805,3536,3538,3541],{"className":3537,"style":2250},[2249],[1805,3539],{"className":3540,"style":2241},[1894],[1805,3542,3543],{"style":2256},[2258,3544,3545],{"xmlns":2260,"width":2261,"height":2262,"viewBox":2263,"preserveAspectRatio":2264},[2266,3546],{"d":2268},[1805,3548,1921],{"className":3549},[1920],[1805,3551,3553],{"className":3552},[1882],[1805,3554,3556],{"className":3555,"style":2278},[1886],[1805,3557],{},[1805,3559,3561],{"className":3560},[1873],[1805,3562,3564,3585],{"className":3563},[1877,1878],[1805,3565,3567,3582],{"className":3566},[1882],[1805,3568,3571],{"className":3569,"style":3570},[1886],"height:0.3361em;",[1805,3572,3573,3576],{"style":2295},[1805,3574],{"className":3575,"style":1895},[1894],[1805,3577,3579],{"className":3578},[1899,1900,1901,1902],[1805,3580,3268],{"className":3581,"style":3457},[1864,1868,1902],[1805,3583,1921],{"className":3584},[1920],[1805,3586,3588],{"className":3587},[1882],[1805,3589,3592],{"className":3590,"style":3591},[1886],"height:0.15em;",[1805,3593],{},[1805,3595,1839],{"className":3596},[1912],[1805,3598],{"className":3599,"style":3600},[2319],"margin-right:2em;",[1805,3602],{"className":3603,"style":2468},[2319],[1805,3605,3607,3652],{"className":3606},[1864],[1805,3608,3610],{"className":3609},[1864,2224],[1805,3611,3613,3644],{"className":3612},[1877,1878],[1805,3614,3616,3641],{"className":3615},[1882],[1805,3617,3620,3628],{"className":3618,"style":3619},[1886],"height:0.6706em;",[1805,3621,3622,3625],{"style":2237},[1805,3623],{"className":3624,"style":2241},[1894],[1805,3626,1793],{"className":3627},[1864,1868],[1805,3629,3632,3635],{"className":3630,"style":3631},[2249],"width:calc(100% - 0.1667em);margin-left:0.1667em;top:-3.4306em;",[1805,3633],{"className":3634,"style":2241},[1894],[1805,3636,3637],{"style":2256},[2258,3638,3639],{"xmlns":2260,"width":2261,"height":2262,"viewBox":2263,"preserveAspectRatio":2264},[2266,3640],{"d":2268},[1805,3642,1921],{"className":3643},[1920],[1805,3645,3647],{"className":3646},[1882],[1805,3648,3650],{"className":3649,"style":2278},[1886],[1805,3651],{},[1805,3653,3655],{"className":3654},[1873],[1805,3656,3658,3679],{"className":3657},[1877,1878],[1805,3659,3661,3676],{"className":3660},[1882],[1805,3662,3664],{"className":3663,"style":1887},[1886],[1805,3665,3667,3670],{"style":3666},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[1805,3668],{"className":3669,"style":1895},[1894],[1805,3671,3673],{"className":3672},[1899,1900,1901,1902],[1805,3674,1842],{"className":3675,"style":1916},[1864,1868,1902],[1805,3677,1921],{"className":3678},[1920],[1805,3680,3682],{"className":3681},[1882],[1805,3683,3685],{"className":3684,"style":1928},[1886],[1805,3686],{},[1805,3688],{"className":3689,"style":2320},[2319],[1805,3691,2127],{"className":3692},[2324],[1805,3694],{"className":3695,"style":2320},[2319],[1805,3697,3699,3703,3835],{"className":3698},[1855],[1805,3700],{"className":3701,"style":3702},[1859],"height:2.4209em;vertical-align:-1.0994em;",[1805,3704,3706,3709,3832],{"className":3705},[1864],[1805,3707],{"className":3708},[2341,2342],[1805,3710,3712],{"className":3711},[2129],[1805,3713,3715,3823],{"className":3714},[1877,1878],[1805,3716,3718,3820],{"className":3717},[1882],[1805,3719,3722,3800,3808],{"className":3720,"style":3721},[1886],"height:1.3214em;",[1805,3723,3725,3728],{"style":3724},"top:-2.1867em;",[1805,3726],{"className":3727,"style":2241},[1894],[1805,3729,3731],{"className":3730},[1864],[1805,3732,3734,3766],{"className":3733},[1864],[1805,3735,3737],{"className":3736},[1864,2224],[1805,3738,3740],{"className":3739},[1877],[1805,3741,3743],{"className":3742},[1882],[1805,3744,3746,3754],{"className":3745,"style":3353},[1886],[1805,3747,3748,3751],{"style":2237},[1805,3749],{"className":3750,"style":2241},[1894],[1805,3752,3251],{"className":3753,"style":3362},[1864,1868],[1805,3755,3757,3760],{"className":3756,"style":3366},[2249],[1805,3758],{"className":3759,"style":2241},[1894],[1805,3761,3762],{"style":2256},[2258,3763,3764],{"xmlns":2260,"width":2261,"height":2262,"viewBox":2263,"preserveAspectRatio":2264},[2266,3765],{"d":2268},[1805,3767,3769],{"className":3768},[1873],[1805,3770,3772,3792],{"className":3771},[1877,1878],[1805,3773,3775,3789],{"className":3774},[1882],[1805,3776,3778],{"className":3777,"style":1887},[1886],[1805,3779,3780,3783],{"style":3390},[1805,3781],{"className":3782,"style":1895},[1894],[1805,3784,3786],{"className":3785},[1899,1900,1901,1902],[1805,3787,1842],{"className":3788,"style":1916},[1864,1868,1902],[1805,3790,1921],{"className":3791},[1920],[1805,3793,3795],{"className":3794},[1882],[1805,3796,3798],{"className":3797,"style":1928},[1886],[1805,3799],{},[1805,3801,3802,3805],{"style":2520},[1805,3803],{"className":3804,"style":2241},[1894],[1805,3806],{"className":3807,"style":2528},[2527],[1805,3809,3811,3814],{"style":3810},"top:-3.677em;",[1805,3812],{"className":3813,"style":2241},[1894],[1805,3815,3817],{"className":3816},[1864],[1805,3818,2171],{"className":3819},[1864],[1805,3821,1921],{"className":3822},[1920],[1805,3824,3826],{"className":3825},[1882],[1805,3827,3830],{"className":3828,"style":3829},[1886],"height:1.0994em;",[1805,3831],{},[1805,3833],{"className":3834},[2698,2342],[1805,3836,1839],{"className":3837},[1912],[1793,3839,3840,3841,3912,3913,1991],{},"where ",[1805,3842,3844,3862],{"className":3843},[1808],[1805,3845,3847],{"className":3846},[1812],[1814,3848,3849],{"xmlns":1816},[1818,3850,3851,3859],{},[1821,3852,3853],{},[1824,3854,3855,3857],{},[1827,3856,1793],{},[1827,3858,1842],{},[1844,3860,3861],{"encoding":1846},"p_j",[1805,3863,3865],{"className":3864,"ariaHidden":1838},[1851],[1805,3866,3868,3872],{"className":3867},[1855],[1805,3869],{"className":3870,"style":3871},[1859],"height:0.7167em;vertical-align:-0.2861em;",[1805,3873,3875,3878],{"className":3874},[1864],[1805,3876,1793],{"className":3877},[1864,1868],[1805,3879,3881],{"className":3880},[1873],[1805,3882,3884,3904],{"className":3883},[1877,1878],[1805,3885,3887,3901],{"className":3886},[1882],[1805,3888,3890],{"className":3889,"style":1887},[1886],[1805,3891,3892,3895],{"style":3666},[1805,3893],{"className":3894,"style":1895},[1894],[1805,3896,3898],{"className":3897},[1899,1900,1901,1902],[1805,3899,1842],{"className":3900,"style":1916},[1864,1868,1902],[1805,3902,1921],{"className":3903},[1920],[1805,3905,3907],{"className":3906},[1882],[1805,3908,3910],{"className":3909,"style":1928},[1886],[1805,3911],{}," is the implied percentage paid by age ",[1805,3914,3916,3929],{"className":3915},[1808],[1805,3917,3919],{"className":3918},[1812],[1814,3920,3921],{"xmlns":1816},[1818,3922,3923,3927],{},[1821,3924,3925],{},[1827,3926,1842],{},[1844,3928,1842],{"encoding":1846},[1805,3930,3932],{"className":3931,"ariaHidden":1838},[1851],[1805,3933,3935,3938],{"className":3934},[1855],[1805,3936],{"className":3937,"style":1987},[1859],[1805,3939,1842],{"className":3940,"style":1916},[1864,1868],[1993,3942,3943,3956],{},[1996,3944,3945],{},[1999,3946,3947,3950,3953],{},[2002,3948,3949],{"align":2007},"Latest age",[2002,3951,3952],{"align":2007},"CDF to Dev 3",[2002,3954,3955],{"align":2007},"Implied paid proportion",[2019,3957,3958,4066,4149,4187],{},[1999,3959,3960,3963,4063],{},[2024,3961,3962],{"align":2007},"0",[2024,3964,3965],{"align":2007},[1805,3966,3968,3997],{"className":3967},[1808],[1805,3969,3971],{"className":3970},[1812],[1814,3972,3973],{"xmlns":1816},[1818,3974,3975,3994],{},[1821,3976,3977,3979,3982,3985,3987,3989,3991],{},[2169,3978,2888],{},[1836,3980,3981],{},"×",[2169,3983,3984],{},"1.35",[1836,3986,3981],{},[2169,3988,3050],{},[1836,3990,2127],{},[2169,3992,3993],{},"2.8875",[1844,3995,3996],{"encoding":1846},"1.8333\\times1.35\\times1.1667=2.8875",[1805,3998,4000,4018,4036,4054],{"className":3999,"ariaHidden":1838},[1851],[1805,4001,4003,4006,4009,4012,4015],{"className":4002},[1855],[1805,4004],{"className":4005,"style":2813},[1859],[1805,4007,2888],{"className":4008},[1864],[1805,4010],{"className":4011,"style":2800},[2319],[1805,4013,3981],{"className":4014},[2668],[1805,4016],{"className":4017,"style":2800},[2319],[1805,4019,4021,4024,4027,4030,4033],{"className":4020},[1855],[1805,4022],{"className":4023,"style":2813},[1859],[1805,4025,3984],{"className":4026},[1864],[1805,4028],{"className":4029,"style":2800},[2319],[1805,4031,3981],{"className":4032},[2668],[1805,4034],{"className":4035,"style":2800},[2319],[1805,4037,4039,4042,4045,4048,4051],{"className":4038},[1855],[1805,4040],{"className":4041,"style":3152},[1859],[1805,4043,3050],{"className":4044},[1864],[1805,4046],{"className":4047,"style":2320},[2319],[1805,4049,2127],{"className":4050},[2324],[1805,4052],{"className":4053,"style":2320},[2319],[1805,4055,4057,4060],{"className":4056},[1855],[1805,4058],{"className":4059,"style":3152},[1859],[1805,4061,3993],{"className":4062},[1864],[2024,4064,4065],{"align":2007},"34.63%",[1999,4067,4068,4070,4146],{},[2024,4069,2171],{"align":2007},[2024,4071,4072],{"align":2007},[1805,4073,4075,4098],{"className":4074},[1808],[1805,4076,4078],{"className":4077},[1812],[1814,4079,4080],{"xmlns":1816},[1818,4081,4082,4095],{},[1821,4083,4084,4086,4088,4090,4092],{},[2169,4085,3984],{},[1836,4087,3981],{},[2169,4089,3050],{},[1836,4091,2127],{},[2169,4093,4094],{},"1.5750",[1844,4096,4097],{"encoding":1846},"1.35\\times1.1667=1.5750",[1805,4099,4101,4119,4137],{"className":4100,"ariaHidden":1838},[1851],[1805,4102,4104,4107,4110,4113,4116],{"className":4103},[1855],[1805,4105],{"className":4106,"style":2813},[1859],[1805,4108,3984],{"className":4109},[1864],[1805,4111],{"className":4112,"style":2800},[2319],[1805,4114,3981],{"className":4115},[2668],[1805,4117],{"className":4118,"style":2800},[2319],[1805,4120,4122,4125,4128,4131,4134],{"className":4121},[1855],[1805,4123],{"className":4124,"style":3152},[1859],[1805,4126,3050],{"className":4127},[1864],[1805,4129],{"className":4130,"style":2320},[2319],[1805,4132,2127],{"className":4133},[2324],[1805,4135],{"className":4136,"style":2320},[2319],[1805,4138,4140,4143],{"className":4139},[1855],[1805,4141],{"className":4142,"style":3152},[1859],[1805,4144,4094],{"className":4145},[1864],[2024,4147,4148],{"align":2007},"63.49%",[1999,4150,4151,4154,4184],{},[2024,4152,4153],{"align":2007},"2",[2024,4155,4156],{"align":2007},[1805,4157,4159,4172],{"className":4158},[1808],[1805,4160,4162],{"className":4161},[1812],[1814,4163,4164],{"xmlns":1816},[1818,4165,4166,4170],{},[1821,4167,4168],{},[2169,4169,3050],{},[1844,4171,3050],{"encoding":1846},[1805,4173,4175],{"className":4174,"ariaHidden":1838},[1851],[1805,4176,4178,4181],{"className":4177},[1855],[1805,4179],{"className":4180,"style":3152},[1859],[1805,4182,3050],{"className":4183},[1864],[2024,4185,4186],{"align":2007},"85.71%",[1999,4188,4189,4192,4222],{},[2024,4190,4191],{"align":2007},"3",[2024,4193,4194],{"align":2007},[1805,4195,4197,4210],{"className":4196},[1808],[1805,4198,4200],{"className":4199},[1812],[1814,4201,4202],{"xmlns":1816},[1818,4203,4204,4208],{},[1821,4205,4206],{},[2169,4207,2171],{},[1844,4209,2171],{"encoding":1846},[1805,4211,4213],{"className":4212,"ariaHidden":1838},[1851],[1805,4214,4216,4219],{"className":4215},[1855],[1805,4217],{"className":4218,"style":3152},[1859],[1805,4220,2171],{"className":4221},[1864],[2024,4223,2261],{"align":2007},[1793,4225,4226],{},"A “100%” at Dev 3 is an assumption of this example. Real long-tailed business generally requires a selected tail factor beyond the observed triangle.",[1797,4228,4230],{"id":4229},"_4-project-ultimate-loss-and-reserve","4. Project ultimate loss and reserve",[1793,4232,4233,4234,4262,4263,1839],{},"If the latest observed value for origin year ",[1805,4235,4237,4250],{"className":4236},[1808],[1805,4238,4240],{"className":4239},[1812],[1814,4241,4242],{"xmlns":1816},[1818,4243,4244,4248],{},[1821,4245,4246],{},[1827,4247,1834],{},[1844,4249,1834],{"encoding":1846},[1805,4251,4253],{"className":4252,"ariaHidden":1838},[1851],[1805,4254,4256,4259],{"className":4255},[1855],[1805,4257],{"className":4258,"style":1957},[1859],[1805,4260,1834],{"className":4261},[1864,1868]," is 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the youngest year dominates",[1793,5183,5184],{},"AY 2023 supplies only £160m of observed paid loss yet contributes £302m of reserve. It is both the least mature and the most leveraged by assumptions: a 5% change in its selected CDF changes its ultimate by about £23.1m. This is why reserve review should focus on influence, not merely on the number of rows.",[1797,5186,5188],{"id":5187},"_5-reproduce-the-calculation","5. Reproduce the calculation",[5190,5191],"pyodide",{"code64":5192,"layout":5193,"locale":7,"packages":5194,"title":5195},"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","vertical","numpy,pandas","Chain-ladder calculation and factor sensitivity",[1797,5197,5199],{"id":5198},"_6-state-the-assumptions-as-testable-claims","6. State the assumptions as testable claims",[1793,5201,5202],{},"The chain ladder does not merely assume “the past repeats.” It requires more precise claims:",[1993,5204,5205,5218],{},[1996,5206,5207],{},[1999,5208,5209,5212,5215],{},[2002,5210,5211],{},"Assumption",[2002,5213,5214],{},"Observable warning sign",[2002,5216,5217],{},"Response",[2019,5219,5220,5231,5242,5253,5264,5275],{},[1999,5221,5222,5225,5228],{},[2024,5223,5224],{},"comparable origin years",[2024,5226,5227],{},"exposure or mix changes sharply",[2024,5229,5230],{},"on-level, segment, or use an exposure-based method",[1999,5232,5233,5236,5239],{},[2024,5234,5235],{},"stable development by age",[2024,5237,5238],{},"link ratios trend by origin year",[2024,5240,5241],{},"investigate operations, inflation, or model calendar effects",[1999,5243,5244,5247,5250],{},[2024,5245,5246],{},"no unmodelled calendar shock",[2024,5248,5249],{},"residuals align on diagonals",[2024,5251,5252],{},"isolate shock or use a model with calendar terms",[1999,5254,5255,5258,5261],{},[2024,5256,5257],{},"adequate volume",[2024,5259,5260],{},"one cell controls a factor",[2024,5262,5263],{},"use credibility, benchmarks, or a range",[1999,5265,5266,5269,5272],{},[2024,5267,5268],{},"latest cumulative value is meaningful",[2024,5270,5271],{},"large recoveries\u002Freopenings",[2024,5273,5274],{},"reconcile and model the mechanism",[1999,5276,5277,5280,5283],{},[2024,5278,5279],{},"selected tail is adequate",[2024,5281,5282],{},"oldest years still develop",[2024,5284,5285],{},"fit or benchmark a tail factor",[1797,5287,5289],{"id":5288},"_7-three-quick-diagnostics","7. Three quick diagnostics",[5178,5291,5293],{"id":5292},"leave-one-origin-year-out","Leave-one-origin-year-out",[1793,5295,5296],{},"Re-estimate each factor after removing one eligible origin year. Large movement identifies influential observations; it does not automatically justify deleting them.",[5178,5298,5300],{"id":5299},"backtest-a-historical-diagonal","Backtest a historical diagonal",[1793,5302,5303],{},"Pretend the latest diagonal was unavailable, project it using only older data, and compare projection with what actually emerged. Repeat across diagonals where possible.",[5178,5305,5307],{"id":5306},"compare-paid-and-incurred","Compare paid and incurred",[1793,5309,5310],{},"Agreement is not proof, but divergence can reveal settlement-speed changes or case-reserving shifts. Define incurred consistently before comparing.",[1797,5312,5314],{"id":5313},"practice","Practice",[5316,5317,5318,5497,5500],"ol",{},[5319,5320,5321,5322,5410,5411,5496],"li",{},"If ",[1805,5323,5325,5350],{"className":5324},[1808],[1805,5326,5328],{"className":5327},[1812],[1814,5329,5330],{"xmlns":1816},[1818,5331,5332,5347],{},[1821,5333,5334],{},[1824,5335,5336,5338],{},[1827,5337,2119],{},[1821,5339,5340,5342,5345],{},[2169,5341,3962],{},[1836,5343,5344],{},":",[2169,5346,2171],{},[1844,5348,5349],{"encoding":1846},"f_{0:1}",[1805,5351,5353],{"className":5352,"ariaHidden":1838},[1851],[1805,5354,5356,5360],{"className":5355},[1855],[1805,5357],{"className":5358,"style":5359},[1859],"height:0.8889em;vertical-align:-0.1944em;",[1805,5361,5363,5366],{"className":5362},[1864],[1805,5364,2119],{"className":5365,"style":2245},[1864,1868],[1805,5367,5369],{"className":5368},[1873],[1805,5370,5372,5402],{"className":5371},[1877,1878],[1805,5373,5375,5399],{"className":5374},[1882],[1805,5376,5379],{"className":5377,"style":5378},[1886],"height:0.3011em;",[1805,5380,5381,5384],{"style":2295},[1805,5382],{"className":5383,"style":1895},[1894],[1805,5385,5387],{"className":5386},[1899,1900,1901,1902],[1805,5388,5390,5393,5396],{"className":5389},[1864,1902],[1805,5391,3962],{"className":5392},[1864,1902],[1805,5394,5344],{"className":5395},[2324,1902],[1805,5397,2171],{"className":5398},[1864,1902],[1805,5400,1921],{"className":5401},[1920],[1805,5403,5405],{"className":5404},[1882],[1805,5406,5408],{"className":5407,"style":3591},[1886],[1805,5409],{}," is unchanged but ",[1805,5412,5414,5438],{"className":5413},[1808],[1805,5415,5417],{"className":5416},[1812],[1814,5418,5419],{"xmlns":1816},[1818,5420,5421,5435],{},[1821,5422,5423],{},[1824,5424,5425,5427],{},[1827,5426,2119],{},[1821,5428,5429,5431,5433],{},[2169,5430,2171],{},[1836,5432,5344],{},[2169,5434,4153],{},[1844,5436,5437],{"encoding":1846},"f_{1:2}",[1805,5439,5441],{"className":5440,"ariaHidden":1838},[1851],[1805,5442,5444,5447],{"className":5443},[1855],[1805,5445],{"className":5446,"style":5359},[1859],[1805,5448,5450,5453],{"className":5449},[1864],[1805,5451,2119],{"className":5452,"style":2245},[1864,1868],[1805,5454,5456],{"className":5455},[1873],[1805,5457,5459,5488],{"className":5458},[1877,1878],[1805,5460,5462,5485],{"className":5461},[1882],[1805,5463,5465],{"className":5464,"style":5378},[1886],[1805,5466,5467,5470],{"style":2295},[1805,5468],{"className":5469,"style":1895},[1894],[1805,5471,5473],{"className":5472},[1899,1900,1901,1902],[1805,5474,5476,5479,5482],{"className":5475},[1864,1902],[1805,5477,2171],{"className":5478},[1864,1902],[1805,5480,5344],{"className":5481},[2324,1902],[1805,5483,4153],{"className":5484},[1864,1902],[1805,5486,1921],{"className":5487},[1920],[1805,5489,5491],{"className":5490},[1882],[1805,5492,5494],{"className":5493,"style":3591},[1886],[1805,5495],{}," rises from 1.35 to 1.42, calculate AY 2022 ultimate and reserve.",[5319,5498,5499],{},"Why is the Dev 2→3 factor especially uncertain here?",[5319,5501,5502],{},"A £20m payment is moved from Dev 1 to Dev 0 without changing ultimate loss. Predict the direction of the early link ratios.",[5504,5505,5507],"legacy-details",{"title":5506},"Answers",[5316,5508,5509,5671,5674],{},[5319,5510,5511,5512,5625,5626,1991],{},"Ultimate ",[1805,5513,5515,5546],{"className":5514},[1808],[1805,5516,5518],{"className":5517},[1812],[1814,5519,5520],{"xmlns":1816},[1818,5521,5522,5543],{},[1821,5523,5524,5526,5528,5530,5533,5535,5537,5540],{},[1836,5525,2127],{},[2169,5527,2066],{},[1836,5529,3981],{},[2169,5531,5532],{},"1.42",[1836,5534,3981],{},[2169,5536,3050],{},[1836,5538,5539],{},"≈",[2169,5541,5542],{},"430.7",[1844,5544,5545],{"encoding":1846},"=260\\times1.42\\times1.1667\\approx430.7",[1805,5547,5549,5562,5580,5598,5616],{"className":5548,"ariaHidden":1838},[1851],[1805,5550,5552,5556,5559],{"className":5551},[1855],[1805,5553],{"className":5554,"style":5555},[1859],"height:0.3669em;",[1805,5557,2127],{"className":5558},[2324],[1805,5560],{"className":5561,"style":2320},[2319],[1805,5563,5565,5568,5571,5574,5577],{"className":5564},[1855],[1805,5566],{"className":5567,"style":2813},[1859],[1805,5569,2066],{"className":5570},[1864],[1805,5572],{"className":5573,"style":2800},[2319],[1805,5575,3981],{"className":5576},[2668],[1805,5578],{"className":5579,"style":2800},[2319],[1805,5581,5583,5586,5589,5592,5595],{"className":5582},[1855],[1805,5584],{"className":5585,"style":2813},[1859],[1805,5587,5532],{"className":5588},[1864],[1805,5590],{"className":5591,"style":2800},[2319],[1805,5593,3981],{"className":5594},[2668],[1805,5596],{"className":5597,"style":2800},[2319],[1805,5599,5601,5604,5607,5610,5613],{"className":5600},[1855],[1805,5602],{"className":5603,"style":3152},[1859],[1805,5605,3050],{"className":5606},[1864],[1805,5608],{"className":5609,"style":2320},[2319],[1805,5611,5539],{"className":5612},[2324],[1805,5614],{"className":5615,"style":2320},[2319],[1805,5617,5619,5622],{"className":5618},[1855],[1805,5620],{"className":5621,"style":3152},[1859],[1805,5623,5542],{"className":5624},[1864],"; reserve ",[1805,5627,5629,5646],{"className":5628},[1808],[1805,5630,5632],{"className":5631},[1812],[1814,5633,5634],{"xmlns":1816},[1818,5635,5636,5643],{},[1821,5637,5638,5640],{},[1836,5639,5539],{},[2169,5641,5642],{},"170.7",[1844,5644,5645],{"encoding":1846},"\\approx170.7",[1805,5647,5649,5662],{"className":5648,"ariaHidden":1838},[1851],[1805,5650,5652,5656,5659],{"className":5651},[1855],[1805,5653],{"className":5654,"style":5655},[1859],"height:0.4831em;",[1805,5657,5539],{"className":5658},[2324],[1805,5660],{"className":5661,"style":2320},[2319],[1805,5663,5665,5668],{"className":5664},[1855],[1805,5666],{"className":5667,"style":3152},[1859],[1805,5669,5642],{"className":5670},[1864],[5319,5672,5673],{},"Only AY 2020 supplies a fully observed pair, so process variation and that year's peculiarities are inseparable from the selected factor.",[5319,5675,5676],{},"Dev 0 rises and Dev 1 cumulative is unchanged, so the 0→1 ratio falls. The change is payment timing, not necessarily ultimate severity.",[1793,5678,5679,5680,1991],{},"Next, ",[5681,5682,5684],"a",{"href":5683},".\u002F02-average-per-claim","separate claim emergence from average claim cost",{"title":10,"searchDepth":5686,"depth":5686,"links":5687},2,[5688,5689,5690,5691,5695,5696,5697,5702],{"id":1799,"depth":5686,"text":1800},{"id":2090,"depth":5686,"text":2091},{"id":3164,"depth":5686,"text":3165},{"id":4229,"depth":5686,"text":4230,"children":5692},[5693],{"id":5180,"depth":5694,"text":5181},3,{"id":5187,"depth":5686,"text":5188},{"id":5198,"depth":5686,"text":5199},{"id":5288,"depth":5686,"text":5289,"children":5698},[5699,5700,5701],{"id":5292,"depth":5694,"text":5293},{"id":5299,"depth":5694,"text":5300},{"id":5306,"depth":5694,"text":5307},{"id":5313,"depth":5686,"text":5314},"Calculate a deterministic chain-ladder reserve and expose the assumptions behind every step.","md",{"sidebar":5706},{"order":5707},1,true,{"title":842,"description":5703},"7GoKiEOr47o9myi5JVTogtAQjSYRir3_rd9xGfQjhw4",[5712,5714],{"title":836,"path":837,"stem":838,"description":5713,"children":-1},"Build a defensible claims-development data contract before projecting a reserve.",{"title":846,"path":847,"stem":848,"description":5715,"children":-1},"Separate claim emergence from average cost to explain why a triangle develops.",1785754734047]