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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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",[1797,1845,1847,1885],{"className":1846},[1800],[1797,1848,1850],{"className":1849},[1804],[1806,1851,1852],{"xmlns":1808},[1810,1853,1854,1882],{},[1813,1855,1856,1866,1870,1877,1879],{},[1857,1858,1859,1862],"msub",{},[1816,1860,1861],{},"X",[1863,1864,1865],"mn",{},"1",[1867,1868,1869],"mo",{"separator":1827},",",[1857,1871,1872,1874],{},[1816,1873,1861],{},[1863,1875,1876],{},"2",[1867,1878,1869],{"separator":1827},[1867,1880,1881],{},"…",[1820,1883,1884],{"encoding":1822},"X_1,X_2,\\ldots",[1797,1886,1888],{"className":1887,"ariaHidden":1827},[1826],[1797,1889,1891,1895,1952,1956,1961,2001,2004,2007],{"className":1890},[1831],[1797,1892],{"className":1893,"style":1894},[1835],"height:0.8778em;vertical-align:-0.1944em;",[1797,1896,1898,1902],{"className":1897},[1840],[1797,1899,1861],{"className":1900,"style":1901},[1840,1841],"margin-right:0.0785em;",[1797,1903,1906],{"className":1904},[1905],"msupsub",[1797,1907,1911,1943],{"className":1908},[1909,1910],"vlist-t","vlist-t2",[1797,1912,1915,1938],{"className":1913},[1914],"vlist-r",[1797,1916,1920],{"className":1917,"style":1919},[1918],"vlist","height:0.3011em;",[1797,1921,1923,1928],{"style":1922},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[1797,1924],{"className":1925,"style":1927},[1926],"pstrut","height:2.7em;",[1797,1929,1935],{"className":1930},[1931,1932,1933,1934],"sizing","reset-size6","size3","mtight",[1797,1936,1865],{"className":1937},[1840,1934],[1797,1939,1942],{"className":1940},[1941],"vlist-s","​",[1797,1944,1946],{"className":1945},[1914],[1797,1947,1950],{"className":1948,"style":1949},[1918],"height:0.15em;",[1797,1951],{},[1797,1953,1869],{"className":1954},[1955],"mpunct",[1797,1957],{"className":1958,"style":1960},[1959],"mspace","margin-right:0.1667em;",[1797,1962,1964,1967],{"className":1963},[1840],[1797,1965,1861],{"className":1966,"style":1901},[1840,1841],[1797,1968,1970],{"className":1969},[1905],[1797,1971,1973,1993],{"className":1972},[1909,1910],[1797,1974,1976,1990],{"className":1975},[1914],[1797,1977,1979],{"className":1978,"style":1919},[1918],[1797,1980,1981,1984],{"style":1922},[1797,1982],{"className":1983,"style":1927},[1926],[1797,1985,1987],{"className":1986},[1931,1932,1933,1934],[1797,1988,1876],{"className":1989},[1840,1934],[1797,1991,1942],{"className":1992},[1941],[1797,1994,1996],{"className":1995},[1914],[1797,1997,1999],{"className":1998,"style":1949},[1918],[1797,2000],{},[1797,2002,1869],{"className":2003},[1955],[1797,2005],{"className":2006,"style":1960},[1959],[1797,2008,1881],{"className":2009},[2010],"minner",". Aggregate loss is",[1797,2013,2016],{"className":2014},[2015],"katex-display",[1797,2017,2019,2063],{"className":2018},[1800],[1797,2020,2022],{"className":2021},[1804],[1806,2023,2025],{"xmlns":1808,"display":2024},"block",[1810,2026,2027,2060],{},[1813,2028,2029,2032,2035,2052,2058],{},[1816,2030,2031],{},"S",[1867,2033,2034],{},"=",[2036,2037,2038,2041,2050],"munderover",{},[1867,2039,2040],{},"∑",[1813,2042,2043,2046,2048],{},[1816,2044,2045],{},"i",[1867,2047,2034],{},[1863,2049,1865],{},[1816,2051,1818],{},[1857,2053,2054,2056],{},[1816,2055,1861],{},[1816,2057,2045],{},[1867,2059,1869],{"separator":1827},[1820,2061,2062],{"encoding":1822},"S=\\sum_{i=1}^{N}X_i,",[1797,2064,2066,2087],{"className":2065,"ariaHidden":1827},[1826],[1797,2067,2069,2072,2076,2080,2084],{"className":2068},[1831],[1797,2070],{"className":2071,"style":1836},[1835],[1797,2073,2031],{"className":2074,"style":2075},[1840,1841],"margin-right:0.0576em;",[1797,2077],{"className":2078,"style":2079},[1959],"margin-right:0.2778em;",[1797,2081,2034],{"className":2082},[2083],"mrel",[1797,2085],{"className":2086,"style":2079},[1959],[1797,2088,2090,2094,2171,2174,2215],{"className":2089},[1831],[1797,2091],{"className":2092,"style":2093},[1835],"height:3.106em;vertical-align:-1.2777em;",[1797,2095,2099],{"className":2096},[2097,2098],"mop","op-limits",[1797,2100,2102,2162],{"className":2101},[1909,1910],[1797,2103,2105,2159],{"className":2104},[1914],[1797,2106,2109,2131,2144],{"className":2107,"style":2108},[1918],"height:1.8283em;",[1797,2110,2112,2116],{"style":2111},"top:-1.8723em;margin-left:0em;",[1797,2113],{"className":2114,"style":2115},[1926],"height:3.05em;",[1797,2117,2119],{"className":2118},[1931,1932,1933,1934],[1797,2120,2122,2125,2128],{"className":2121},[1840,1934],[1797,2123,2045],{"className":2124},[1840,1841,1934],[1797,2126,2034],{"className":2127},[2083,1934],[1797,2129,1865],{"className":2130},[1840,1934],[1797,2132,2134,2137],{"style":2133},"top:-3.05em;",[1797,2135],{"className":2136,"style":2115},[1926],[1797,2138,2139],{},[1797,2140,2040],{"className":2141},[2097,2142,2143],"op-symbol","large-op",[1797,2145,2147,2150],{"style":2146},"top:-4.3em;margin-left:0em;",[1797,2148],{"className":2149,"style":2115},[1926],[1797,2151,2153],{"className":2152},[1931,1932,1933,1934],[1797,2154,2156],{"className":2155},[1840,1934],[1797,2157,1818],{"className":2158,"style":1842},[1840,1841,1934],[1797,2160,1942],{"className":2161},[1941],[1797,2163,2165],{"className":2164},[1914],[1797,2166,2169],{"className":2167,"style":2168},[1918],"height:1.2777em;",[1797,2170],{},[1797,2172],{"className":2173,"style":1960},[1959],[1797,2175,2177,2180],{"className":2176},[1840],[1797,2178,1861],{"className":2179,"style":1901},[1840,1841],[1797,2181,2183],{"className":2182},[1905],[1797,2184,2186,2207],{"className":2185},[1909,1910],[1797,2187,2189,2204],{"className":2188},[1914],[1797,2190,2193],{"className":2191,"style":2192},[1918],"height:0.3117em;",[1797,2194,2195,2198],{"style":1922},[1797,2196],{"className":2197,"style":1927},[1926],[1797,2199,2201],{"className":2200},[1931,1932,1933,1934],[1797,2202,2045],{"className":2203},[1840,1841,1934],[1797,2205,1942],{"className":2206},[1941],[1797,2208,2210],{"className":2209},[1914],[1797,2211,2213],{"className":2212,"style":1949},[1918],[1797,2214],{},[1797,2216,1869],{"className":2217},[1955],[1793,2219,2220,2221,2274,2275,2326],{},"with 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",[1797,2276,2278,2296],{"className":2277},[1800],[1797,2279,2281],{"className":2280},[1804],[1806,2282,2283],{"xmlns":1808},[1810,2284,2285,2293],{},[1813,2286,2287,2289,2291],{},[1816,2288,1818],{},[1867,2290,2034],{},[1863,2292,2239],{},[1820,2294,2295],{"encoding":1822},"N=0",[1797,2297,2299,2317],{"className":2298,"ariaHidden":1827},[1826],[1797,2300,2302,2305,2308,2311,2314],{"className":2301},[1831],[1797,2303],{"className":2304,"style":1836},[1835],[1797,2306,1818],{"className":2307,"style":1842},[1840,1841],[1797,2309],{"className":2310,"style":2079},[1959],[1797,2312,2034],{"className":2313},[2083],[1797,2315],{"className":2316,"style":2079},[1959],[1797,2318,2320,2323],{"className":2319},[1831],[1797,2321],{"className":2322,"style":2270},[1835],[1797,2324,2239],{"className":2325},[1840],".",[2328,2329,2331],"h2",{"id":2330},"_1-baseline-assumptions","1. Baseline assumptions",[1793,2333,2334],{},"The classical compound model assumes:",[2336,2337,2338,2413,2514],"ol",{},[2339,2340,2341,2412],"li",{},[1797,2342,2344,2362],{"className":2343},[1800],[1797,2345,2347],{"className":2346},[1804],[1806,2348,2349],{"xmlns":1808},[1810,2350,2351,2359],{},[1813,2352,2353],{},[1857,2354,2355,2357],{},[1816,2356,1861],{},[1816,2358,2045],{},[1820,2360,2361],{"encoding":1822},"X_i",[1797,2363,2365],{"className":2364,"ariaHidden":1827},[1826],[1797,2366,2368,2372],{"className":2367},[1831],[1797,2369],{"className":2370,"style":2371},[1835],"height:0.8333em;vertical-align:-0.15em;",[1797,2373,2375,2378],{"className":2374},[1840],[1797,2376,1861],{"className":2377,"style":1901},[1840,1841],[1797,2379,2381],{"className":2380},[1905],[1797,2382,2384,2404],{"className":2383},[1909,1910],[1797,2385,2387,2401],{"className":2386},[1914],[1797,2388,2390],{"className":2389,"style":2192},[1918],[1797,2391,2392,2395],{"style":1922},[1797,2393],{"className":2394,"style":1927},[1926],[1797,2396,2398],{"className":2397},[1931,1932,1933,1934],[1797,2399,2045],{"className":2400},[1840,1841,1934],[1797,2402,1942],{"className":2403},[1941],[1797,2405,2407],{"className":2406},[1914],[1797,2408,2410],{"className":2409,"style":1949},[1918],[1797,2411],{}," are independent and identically distributed;",[2339,2414,2415,2443,2444,2513],{},[1797,2416,2418,2431],{"className":2417},[1800],[1797,2419,2421],{"className":2420},[1804],[1806,2422,2423],{"xmlns":1808},[1810,2424,2425,2429],{},[1813,2426,2427],{},[1816,2428,1818],{},[1820,2430,1818],{"encoding":1822},[1797,2432,2434],{"className":2433,"ariaHidden":1827},[1826],[1797,2435,2437,2440],{"className":2436},[1831],[1797,2438],{"className":2439,"style":1836},[1835],[1797,2441,1818],{"className":2442,"style":1842},[1840,1841]," is independent of all ",[1797,2445,2447,2464],{"className":2446},[1800],[1797,2448,2450],{"className":2449},[1804],[1806,2451,2452],{"xmlns":1808},[1810,2453,2454,2462],{},[1813,2455,2456],{},[1857,2457,2458,2460],{},[1816,2459,1861],{},[1816,2461,2045],{},[1820,2463,2361],{"encoding":1822},[1797,2465,2467],{"className":2466,"ariaHidden":1827},[1826],[1797,2468,2470,2473],{"className":2469},[1831],[1797,2471],{"className":2472,"style":2371},[1835],[1797,2474,2476,2479],{"className":2475},[1840],[1797,2477,1861],{"className":2478,"style":1901},[1840,1841],[1797,2480,2482],{"className":2481},[1905],[1797,2483,2485,2505],{"className":2484},[1909,1910],[1797,2486,2488,2502],{"className":2487},[1914],[1797,2489,2491],{"className":2490,"style":2192},[1918],[1797,2492,2493,2496],{"style":1922},[1797,2494],{"className":2495,"style":1927},[1926],[1797,2497,2499],{"className":2498},[1931,1932,1933,1934],[1797,2500,2045],{"className":2501},[1840,1841,1934],[1797,2503,1942],{"className":2504},[1941],[1797,2506,2508],{"className":2507},[1914],[1797,2509,2511],{"className":2510,"style":1949},[1918],[1797,2512],{},";",[2339,2515,2516],{},"losses use one period, currency, and contract basis.",[1793,2518,2519],{},"These assumptions make the derivation possible. They are not universal insurance facts.",[2328,2521,2523],{"id":2522},"_2-mean-by-conditioning","2. Mean by conditioning",[1793,2525,2526,2527,1869],{},"Given ",[1797,2528,2530,2549],{"className":2529},[1800],[1797,2531,2533],{"className":2532},[1804],[1806,2534,2535],{"xmlns":1808},[1810,2536,2537,2546],{},[1813,2538,2539,2541,2543],{},[1816,2540,1818],{},[1867,2542,2034],{},[1816,2544,2545],{},"n",[1820,2547,2548],{"encoding":1822},"N=n",[1797,2550,2552,2570],{"className":2551,"ariaHidden":1827},[1826],[1797,2553,2555,2558,2561,2564,2567],{"className":2554},[1831],[1797,2556],{"className":2557,"style":1836},[1835],[1797,2559,1818],{"className":2560,"style":1842},[1840,1841],[1797,2562],{"className":2563,"style":2079},[1959],[1797,2565,2034],{"className":2566},[2083],[1797,2568],{"className":2569,"style":2079},[1959],[1797,2571,2573,2577],{"className":2572},[1831],[1797,2574],{"className":2575,"style":2576},[1835],"height:0.4306em;",[1797,2578,2545],{"className":2579},[1840,1841],[1797,2581,2583],{"className":2582},[2015],[1797,2584,2586,2634],{"className":2585},[1800],[1797,2587,2589],{"className":2588},[1804],[1806,2590,2591],{"xmlns":1808,"display":2024},[1810,2592,2593,2631],{},[1813,2594,2595,2598,2602,2604,2607,2609,2611,2613,2616,2618,2620,2622,2624,2626,2628],{},[1816,2596,2597],{},"E",[1867,2599,2601],{"stretchy":2600},"false","[",[1816,2603,2031],{},[1867,2605,2606],{},"∣",[1816,2608,1818],{},[1867,2610,2034],{},[1816,2612,2545],{},[1867,2614,2615],{"stretchy":2600},"]",[1867,2617,2034],{},[1816,2619,2545],{},[1816,2621,2597],{},[1867,2623,2601],{"stretchy":2600},[1816,2625,1861],{},[1867,2627,2615],{"stretchy":2600},[1816,2629,2326],{"mathvariant":2630},"normal",[1820,2632,2633],{"encoding":1822},"E[S\\mid N=n]=nE[X].",[1797,2635,2637,2663,2681,2703],{"className":2636,"ariaHidden":1827},[1826],[1797,2638,2640,2644,2647,2651,2654,2657,2660],{"className":2639},[1831],[1797,2641],{"className":2642,"style":2643},[1835],"height:1em;vertical-align:-0.25em;",[1797,2645,2597],{"className":2646,"style":2075},[1840,1841],[1797,2648,2601],{"className":2649},[2650],"mopen",[1797,2652,2031],{"className":2653,"style":2075},[1840,1841],[1797,2655],{"className":2656,"style":2079},[1959],[1797,2658,2606],{"className":2659},[2083],[1797,2661],{"className":2662,"style":2079},[1959],[1797,2664,2666,2669,2672,2675,2678],{"className":2665},[1831],[1797,2667],{"className":2668,"style":1836},[1835],[1797,2670,1818],{"className":2671,"style":1842},[1840,1841],[1797,2673],{"className":2674,"style":2079},[1959],[1797,2676,2034],{"className":2677},[2083],[1797,2679],{"className":2680,"style":2079},[1959],[1797,2682,2684,2687,2690,2694,2697,2700],{"className":2683},[1831],[1797,2685],{"className":2686,"style":2643},[1835],[1797,2688,2545],{"className":2689},[1840,1841],[1797,2691,2615],{"className":2692},[2693],"mclose",[1797,2695],{"className":2696,"style":2079},[1959],[1797,2698,2034],{"className":2699},[2083],[1797,2701],{"className":2702,"style":2079},[1959],[1797,2704,2706,2709,2712,2715,2718,2721,2724],{"className":2705},[1831],[1797,2707],{"className":2708,"style":2643},[1835],[1797,2710,2545],{"className":2711},[1840,1841],[1797,2713,2597],{"className":2714,"style":2075},[1840,1841],[1797,2716,2601],{"className":2717},[2650],[1797,2719,1861],{"className":2720,"style":1901},[1840,1841],[1797,2722,2615],{"className":2723},[2693],[1797,2725,2326],{"className":2726},[1840],[1793,2728,2729],{},"Using the law of total expectation,",[1797,2731,2733],{"className":2732},[2015],[1797,2734,2736,2796],{"className":2735},[1800],[1797,2737,2739],{"className":2738},[1804],[1806,2740,2741],{"xmlns":1808,"display":2024},[1810,2742,2743,2793],{},[1813,2744,2745,2747,2749,2751,2753,2755,2757,2759,2761,2763,2765,2767,2769,2771,2773,2775,2777,2779,2781,2783,2785,2787,2789,2791],{},[1816,2746,2597],{},[1867,2748,2601],{"stretchy":2600},[1816,2750,2031],{},[1867,2752,2615],{"stretchy":2600},[1867,2754,2034],{},[1816,2756,2597],{},[1867,2758,2601],{"stretchy":2600},[1816,2760,2597],{},[1867,2762,2601],{"stretchy":2600},[1816,2764,2031],{},[1867,2766,2606],{},[1816,2768,1818],{},[1867,2770,2615],{"stretchy":2600},[1867,2772,2615],{"stretchy":2600},[1867,2774,2034],{},[1816,2776,2597],{},[1867,2778,2601],{"stretchy":2600},[1816,2780,1818],{},[1867,2782,2615],{"stretchy":2600},[1816,2784,2597],{},[1867,2786,2601],{"stretchy":2600},[1816,2788,1861],{},[1867,2790,2615],{"stretchy":2600},[1816,2792,2326],{"mathvariant":2630},[1820,2794,2795],{"encoding":1822},"E[S]=E[E[S\\mid N]]=E[N]E[X].",[1797,2797,2799,2826,2856,2878],{"className":2798,"ariaHidden":1827},[1826],[1797,2800,2802,2805,2808,2811,2814,2817,2820,2823],{"className":2801},[1831],[1797,2803],{"className":2804,"style":2643},[1835],[1797,2806,2597],{"className":2807,"style":2075},[1840,1841],[1797,2809,2601],{"className":2810},[2650],[1797,2812,2031],{"className":2813,"style":2075},[1840,1841],[1797,2815,2615],{"className":2816},[2693],[1797,2818],{"className":2819,"style":2079},[1959],[1797,2821,2034],{"className":2822},[2083],[1797,2824],{"className":2825,"style":2079},[1959],[1797,2827,2829,2832,2835,2838,2841,2844,2847,2850,2853],{"className":2828},[1831],[1797,2830],{"className":2831,"style":2643},[1835],[1797,2833,2597],{"className":2834,"style":2075},[1840,1841],[1797,2836,2601],{"className":2837},[2650],[1797,2839,2597],{"className":2840,"style":2075},[1840,1841],[1797,2842,2601],{"className":2843},[2650],[1797,2845,2031],{"className":2846,"style":2075},[1840,1841],[1797,2848],{"className":2849,"style":2079},[1959],[1797,2851,2606],{"className":2852},[2083],[1797,2854],{"className":2855,"style":2079},[1959],[1797,2857,2859,2862,2865,2869,2872,2875],{"className":2858},[1831],[1797,2860],{"className":2861,"style":2643},[1835],[1797,2863,1818],{"className":2864,"style":1842},[1840,1841],[1797,2866,2868],{"className":2867},[2693],"]]",[1797,2870],{"className":2871,"style":2079},[1959],[1797,2873,2034],{"className":2874},[2083],[1797,2876],{"className":2877,"style":2079},[1959],[1797,2879,2881,2884,2887,2890,2893,2896,2899,2902,2905,2908],{"className":2880},[1831],[1797,2882],{"className":2883,"style":2643},[1835],[1797,2885,2597],{"className":2886,"style":2075},[1840,1841],[1797,2888,2601],{"className":2889},[2650],[1797,2891,1818],{"className":2892,"style":1842},[1840,1841],[1797,2894,2615],{"className":2895},[2693],[1797,2897,2597],{"className":2898,"style":2075},[1840,1841],[1797,2900,2601],{"className":2901},[2650],[1797,2903,1861],{"className":2904,"style":1901},[1840,1841],[1797,2906,2615],{"className":2907},[2693],[1797,2909,2326],{"className":2910},[1840],[1793,2912,2913],{},"If expected count is 40 and expected severity £5,000, expected annual aggregate is £200,000. Realised loss is not fixed at that amount.",[2328,2915,2917],{"id":2916},"_3-variance-by-conditioning","3. Variance by conditioning",[1793,2919,2526,2920,1869],{},[1797,2921,2923,2940],{"className":2922},[1800],[1797,2924,2926],{"className":2925},[1804],[1806,2927,2928],{"xmlns":1808},[1810,2929,2930,2938],{},[1813,2931,2932,2934,2936],{},[1816,2933,1818],{},[1867,2935,2034],{},[1816,2937,2545],{},[1820,2939,2548],{"encoding":1822},[1797,2941,2943,2961],{"className":2942,"ariaHidden":1827},[1826],[1797,2944,2946,2949,2952,2955,2958],{"className":2945},[1831],[1797,2947],{"className":2948,"style":1836},[1835],[1797,2950,1818],{"className":2951,"style":1842},[1840,1841],[1797,2953],{"className":2954,"style":2079},[1959],[1797,2956,2034],{"className":2957},[2083],[1797,2959],{"className":2960,"style":2079},[1959],[1797,2962,2964,2967],{"className":2963},[1831],[1797,2965],{"className":2966,"style":2576},[1835],[1797,2968,2545],{"className":2969},[1840,1841],[1797,2971,2973],{"className":2972},[2015],[1797,2974,2976,3026],{"className":2975},[1800],[1797,2977,2979],{"className":2978},[1804],[1806,2980,2981],{"xmlns":1808,"display":2024},[1810,2982,2983,3023],{},[1813,2984,2985,2988,2991,2994,2996,2998,3000,3002,3004,3007,3009,3011,3013,3015,3017,3019,3021],{},[1816,2986,2987],{"mathvariant":2630},"Var",[1867,2989,2990],{},"⁡",[1867,2992,2993],{"stretchy":2600},"(",[1816,2995,2031],{},[1867,2997,2606],{},[1816,2999,1818],{},[1867,3001,2034],{},[1816,3003,2545],{},[1867,3005,3006],{"stretchy":2600},")",[1867,3008,2034],{},[1816,3010,2545],{},[1816,3012,2987],{"mathvariant":2630},[1867,3014,2990],{},[1867,3016,2993],{"stretchy":2600},[1816,3018,1861],{},[1867,3020,3006],{"stretchy":2600},[1816,3022,2326],{"mathvariant":2630},[1820,3024,3025],{"encoding":1822},"\\operatorname{Var}(S\\mid N=n)=n\\operatorname{Var}(X).",[1797,3027,3029,3057,3075,3096],{"className":3028,"ariaHidden":1827},[1826],[1797,3030,3032,3035,3042,3045,3048,3051,3054],{"className":3031},[1831],[1797,3033],{"className":3034,"style":2643},[1835],[1797,3036,3038],{"className":3037},[2097],[1797,3039,2987],{"className":3040},[1840,3041],"mathrm",[1797,3043,2993],{"className":3044},[2650],[1797,3046,2031],{"className":3047,"style":2075},[1840,1841],[1797,3049],{"className":3050,"style":2079},[1959],[1797,3052,2606],{"className":3053},[2083],[1797,3055],{"className":3056,"style":2079},[1959],[1797,3058,3060,3063,3066,3069,3072],{"className":3059},[1831],[1797,3061],{"className":3062,"style":1836},[1835],[1797,3064,1818],{"className":3065,"style":1842},[1840,1841],[1797,3067],{"className":3068,"style":2079},[1959],[1797,3070,2034],{"className":3071},[2083],[1797,3073],{"className":3074,"style":2079},[1959],[1797,3076,3078,3081,3084,3087,3090,3093],{"className":3077},[1831],[1797,3079],{"className":3080,"style":2643},[1835],[1797,3082,2545],{"className":3083},[1840,1841],[1797,3085,3006],{"className":3086},[2693],[1797,3088],{"className":3089,"style":2079},[1959],[1797,3091,2034],{"className":3092},[2083],[1797,3094],{"className":3095,"style":2079},[1959],[1797,3097,3099,3102,3105,3108,3114,3117,3120,3123],{"className":3098},[1831],[1797,3100],{"className":3101,"style":2643},[1835],[1797,3103,2545],{"className":3104},[1840,1841],[1797,3106],{"className":3107,"style":1960},[1959],[1797,3109,3111],{"className":3110},[2097],[1797,3112,2987],{"className":3113},[1840,3041],[1797,3115,2993],{"className":3116},[2650],[1797,3118,1861],{"className":3119,"style":1901},[1840,1841],[1797,3121,3006],{"className":3122},[2693],[1797,3124,2326],{"className":3125},[1840],[1793,3127,3128],{},"The law of total variance gives",[1797,3130,3132],{"className":3131},[2015],[1797,3133,3135,3209],{"className":3134},[1800],[1797,3136,3138],{"className":3137},[1804],[1806,3139,3140],{"xmlns":1808,"display":2024},[1810,3141,3142,3206],{},[1813,3143,3144,3146,3148,3150,3152,3154,3156,3158,3160,3162,3164,3166,3168,3170,3172,3174,3177,3179,3181,3183,3185,3187,3189,3191,3193,3195,3197,3204],{},[1816,3145,2987],{"mathvariant":2630},[1867,3147,2990],{},[1867,3149,2993],{"stretchy":2600},[1816,3151,2031],{},[1867,3153,3006],{"stretchy":2600},[1867,3155,2034],{},[1816,3157,2597],{},[1867,3159,2601],{"stretchy":2600},[1816,3161,1818],{},[1867,3163,2615],{"stretchy":2600},[1816,3165,2987],{"mathvariant":2630},[1867,3167,2990],{},[1867,3169,2993],{"stretchy":2600},[1816,3171,1861],{},[1867,3173,3006],{"stretchy":2600},[1867,3175,3176],{},"+",[1816,3178,2987],{"mathvariant":2630},[1867,3180,2990],{},[1867,3182,2993],{"stretchy":2600},[1816,3184,1818],{},[1867,3186,3006],{"stretchy":2600},[1867,3188,2993],{"stretchy":2600},[1816,3190,2597],{},[1867,3192,2601],{"stretchy":2600},[1816,3194,1861],{},[1867,3196,2615],{"stretchy":2600},[3198,3199,3200,3202],"msup",{},[1867,3201,3006],{"stretchy":2600},[1863,3203,1876],{},[1816,3205,2326],{"mathvariant":2630},[1820,3207,3208],{"encoding":1822},"\\operatorname{Var}(S)\n=E[N]\\operatorname{Var}(X)\n+\\operatorname{Var}(N)(E[X])^2.",[1797,3210,3212,3242,3289],{"className":3211,"ariaHidden":1827},[1826],[1797,3213,3215,3218,3224,3227,3230,3233,3236,3239],{"className":3214},[1831],[1797,3216],{"className":3217,"style":2643},[1835],[1797,3219,3221],{"className":3220},[2097],[1797,3222,2987],{"className":3223},[1840,3041],[1797,3225,2993],{"className":3226},[2650],[1797,3228,2031],{"className":3229,"style":2075},[1840,1841],[1797,3231,3006],{"className":3232},[2693],[1797,3234],{"className":3235,"style":2079},[1959],[1797,3237,2034],{"className":3238},[2083],[1797,3240],{"className":3241,"style":2079},[1959],[1797,3243,3245,3248,3251,3254,3257,3260,3263,3269,3272,3275,3278,3282,3286],{"className":3244},[1831],[1797,3246],{"className":3247,"style":2643},[1835],[1797,3249,2597],{"className":3250,"style":2075},[1840,1841],[1797,3252,2601],{"className":3253},[2650],[1797,3255,1818],{"className":3256,"style":1842},[1840,1841],[1797,3258,2615],{"className":3259},[2693],[1797,3261],{"className":3262,"style":1960},[1959],[1797,3264,3266],{"className":3265},[2097],[1797,3267,2987],{"className":3268},[1840,3041],[1797,3270,2993],{"className":3271},[2650],[1797,3273,1861],{"className":3274,"style":1901},[1840,1841],[1797,3276,3006],{"className":3277},[2693],[1797,3279],{"className":3280,"style":3281},[1959],"margin-right:0.2222em;",[1797,3283,3176],{"className":3284},[3285],"mbin",[1797,3287],{"className":3288,"style":3281},[1959],[1797,3290,3292,3296,3302,3305,3308,3311,3314,3317,3320,3323,3326,3357],{"className":3291},[1831],[1797,3293],{"className":3294,"style":3295},[1835],"height:1.1141em;vertical-align:-0.25em;",[1797,3297,3299],{"className":3298},[2097],[1797,3300,2987],{"className":3301},[1840,3041],[1797,3303,2993],{"className":3304},[2650],[1797,3306,1818],{"className":3307,"style":1842},[1840,1841],[1797,3309,3006],{"className":3310},[2693],[1797,3312,2993],{"className":3313},[2650],[1797,3315,2597],{"className":3316,"style":2075},[1840,1841],[1797,3318,2601],{"className":3319},[2650],[1797,3321,1861],{"className":3322,"style":1901},[1840,1841],[1797,3324,2615],{"className":3325},[2693],[1797,3327,3329,3332],{"className":3328},[2693],[1797,3330,3006],{"className":3331},[2693],[1797,3333,3335],{"className":3334},[1905],[1797,3336,3338],{"className":3337},[1909],[1797,3339,3341],{"className":3340},[1914],[1797,3342,3345],{"className":3343,"style":3344},[1918],"height:0.8641em;",[1797,3346,3348,3351],{"style":3347},"top:-3.113em;margin-right:0.05em;",[1797,3349],{"className":3350,"style":1927},[1926],[1797,3352,3354],{"className":3353},[1931,1932,1933,1934],[1797,3355,1876],{"className":3356},[1840,1934],[1797,3358,2326],{"className":3359},[1840],[1793,3361,3362],{},"The first term is severity variation for a given count; the second is count variation acting through mean severity.",[3364,3365,3367],"h3",{"id":3366},"compound-poisson-simplification","Compound Poisson simplification",[1793,3369,3370,3371,3445,3446,3555],{},"If ",[1797,3372,3374,3403],{"className":3373},[1800],[1797,3375,3377],{"className":3376},[1804],[1806,3378,3379],{"xmlns":1808},[1810,3380,3381,3400],{},[1813,3382,3383,3385,3388,3391,3393,3395,3398],{},[1816,3384,1818],{},[1867,3386,3387],{},"∼",[1816,3389,3390],{"mathvariant":2630},"Poisson",[1867,3392,2990],{},[1867,3394,2993],{"stretchy":2600},[1816,3396,3397],{},"λ",[1867,3399,3006],{"stretchy":2600},[1820,3401,3402],{"encoding":1822},"N\\sim\\operatorname{Poisson}(\\lambda)",[1797,3404,3406,3424],{"className":3405,"ariaHidden":1827},[1826],[1797,3407,3409,3412,3415,3418,3421],{"className":3408},[1831],[1797,3410],{"className":3411,"style":1836},[1835],[1797,3413,1818],{"className":3414,"style":1842},[1840,1841],[1797,3416],{"className":3417,"style":2079},[1959],[1797,3419,3387],{"className":3420},[2083],[1797,3422],{"className":3423,"style":2079},[1959],[1797,3425,3427,3430,3436,3439,3442],{"className":3426},[1831],[1797,3428],{"className":3429,"style":2643},[1835],[1797,3431,3433],{"className":3432},[2097],[1797,3434,3390],{"className":3435},[1840,3041],[1797,3437,2993],{"className":3438},[2650],[1797,3440,3397],{"className":3441},[1840,1841],[1797,3443,3006],{"className":3444},[2693],", then 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E[X],",[1797,3597,3599,3626],{"className":3598,"ariaHidden":1827},[1826],[1797,3600,3602,3605,3608,3611,3614,3617,3620,3623],{"className":3601},[1831],[1797,3603],{"className":3604,"style":2643},[1835],[1797,3606,2597],{"className":3607,"style":2075},[1840,1841],[1797,3609,2601],{"className":3610},[2650],[1797,3612,2031],{"className":3613,"style":2075},[1840,1841],[1797,3615,2615],{"className":3616},[2693],[1797,3618],{"className":3619,"style":2079},[1959],[1797,3621,2034],{"className":3622},[2083],[1797,3624],{"className":3625,"style":2079},[1959],[1797,3627,3629,3632,3635,3638,3641,3644,3647],{"className":3628},[1831],[1797,3630],{"className":3631,"style":2643},[1835],[1797,3633,3397],{"className":3634},[1840,1841],[1797,3636,2597],{"className":3637,"style":2075},[1840,1841],[1797,3639,2601],{"className":3640},[2650],[1797,3642,1861],{"className":3643,"style":1901},[1840,1841],[1797,3645,2615],{"className":3646},[2693],[1797,3648,1869],{"className":3649},[1955],[1797,3651,3653],{"className":3652},[2015],[1797,3654,3656,3696],{"className":3655},[1800],[1797,3657,3659],{"className":3658},[1804],[1806,3660,3661],{"xmlns":1808,"display":2024},[1810,3662,3663,3693],{},[1813,3664,3665,3667,3669,3671,3673,3675,3677,3679,3681,3683,3689,3691],{},[1816,3666,2987],{"mathvariant":2630},[1867,3668,2990],{},[1867,3670,2993],{"stretchy":2600},[1816,3672,2031],{},[1867,3674,3006],{"stretchy":2600},[1867,3676,2034],{},[1816,3678,3397],{},[1816,3680,2597],{},[1867,3682,2601],{"stretchy":2600},[3198,3684,3685,3687],{},[1816,3686,1861],{},[1863,3688,1876],{},[1867,3690,2615],{"stretchy":2600},[1816,3692,2326],{"mathvariant":2630},[1820,3694,3695],{"encoding":1822},"\\operatorname{Var}(S)=\\lambda 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identity supports moment derivation and analytic approximations. For heavy-tailed severities such as lognormal, the MGF is infinite for ",[1797,5025,5027,5046],{"className":5026},[1800],[1797,5028,5030],{"className":5029},[1804],[1806,5031,5032],{"xmlns":1808},[1810,5033,5034,5043],{},[1813,5035,5036,5038,5041],{},[1816,5037,4425],{},[1867,5039,5040],{},">",[1863,5042,2239],{},[1820,5044,5045],{"encoding":1822},"t>0",[1797,5047,5049,5068],{"className":5048,"ariaHidden":1827},[1826],[1797,5050,5052,5056,5059,5062,5065],{"className":5051},[1831],[1797,5053],{"className":5054,"style":5055},[1835],"height:0.6542em;vertical-align:-0.0391em;",[1797,5057,4425],{"className":5058},[1840,1841],[1797,5060],{"className":5061,"style":2079},[1959],[1797,5063,5040],{"className":5064},[2083],[1797,5066],{"className":5067,"style":2079},[1959],[1797,5069,5071,5074],{"className":5070},[1831],[1797,5072],{"className":5073,"style":2270},[1835],[1797,5075,2239],{"className":5076},[1840],"; use Laplace transforms, characteristic functions, recursion, or simulation instead of forcing an MGF argument.",[2328,5079,5081],{"id":5080},"_5-distribution-methods","5. Distribution methods",[5083,5084,5085,5101],"table",{},[5086,5087,5088],"thead",{},[5089,5090,5091,5095,5098],"tr",{},[5092,5093,5094],"th",{},"Method",[5092,5096,5097],{},"Best suited to",[5092,5099,5100],{},"Main caution",[5102,5103,5104,5116,5195,5206,5217],"tbody",{},[5089,5105,5106,5110,5113],{},[5107,5108,5109],"td",{},"exact convolution",[5107,5111,5112],{},"small discrete counts\u002Fsupport",[5107,5114,5115],{},"quickly becomes expensive",[5089,5117,5118,5121,5192],{},[5107,5119,5120],{},"Panjer recursion",[5107,5122,5123,5124,5191],{},"discrete severity and ",[1797,5125,5127,5155],{"className":5126},[1800],[1797,5128,5130],{"className":5129},[1804],[1806,5131,5132],{"xmlns":1808},[1810,5133,5134,5152],{},[1813,5135,5136,5138,5141,5143,5146,5148,5150],{},[1867,5137,2993],{"stretchy":2600},[1816,5139,5140],{},"a",[1867,5142,1869],{"separator":1827},[1816,5144,5145],{},"b",[1867,5147,1869],{"separator":1827},[1863,5149,2239],{},[1867,5151,3006],{"stretchy":2600},[1820,5153,5154],{"encoding":1822},"(a,b,0)",[1797,5156,5158],{"className":5157,"ariaHidden":1827},[1826],[1797,5159,5161,5164,5167,5170,5173,5176,5179,5182,5185,5188],{"className":5160},[1831],[1797,5162],{"className":5163,"style":2643},[1835],[1797,5165,2993],{"className":5166},[2650],[1797,5168,5140],{"className":5169},[1840,1841],[1797,5171,1869],{"className":5172},[1955],[1797,5174],{"className":5175,"style":1960},[1959],[1797,5177,5145],{"className":5178},[1840,1841],[1797,5180,1869],{"className":5181},[1955],[1797,5183],{"className":5184,"style":1960},[1959],[1797,5186,2239],{"className":5187},[1840],[1797,5189,3006],{"className":5190},[2693]," count families",[5107,5193,5194],{},"discretisation and grid choice",[5089,5196,5197,5200,5203],{},[5107,5198,5199],{},"FFT",[5107,5201,5202],{},"discretised aggregate distribution",[5107,5204,5205],{},"aliasing, truncation, numerical setup",[5089,5207,5208,5211,5214],{},[5107,5209,5210],{},"Normal\u002FGamma approximation",[5107,5212,5213],{},"high-frequency, moderate-tail screening",[5107,5215,5216],{},"can miss skew and extreme tail",[5089,5218,5219,5222,5225],{},[5107,5220,5221],{},"Monte Carlo",[5107,5223,5224],{},"complex treaties and dependence",[5107,5226,5227],{},"simulation error and rare-event inefficiency",[1793,5229,5230],{},"Panjer recursion is not a new risk model; it is an algorithm for computing the compound distribution under eligible count laws and discretised severity.",[2328,5232,5234],{"id":5233},"_6-reinsurance-enters-at-the-right-level","6. Reinsurance enters at the right level",[1793,5236,5237,5238,5283],{},"For a per-loss treaty ",[1797,5239,5241,5262],{"className":5240},[1800],[1797,5242,5244],{"className":5243},[1804],[1806,5245,5246],{"xmlns":1808},[1810,5247,5248,5259],{},[1813,5249,5250,5253,5255,5257],{},[1816,5251,5252],{},"g",[1867,5254,2993],{"stretchy":2600},[1816,5256,1861],{},[1867,5258,3006],{"stretchy":2600},[1820,5260,5261],{"encoding":1822},"g(X)",[1797,5263,5265],{"className":5264,"ariaHidden":1827},[1826],[1797,5266,5268,5271,5274,5277,5280],{"className":5267},[1831],[1797,5269],{"className":5270,"style":2643},[1835],[1797,5272,5252],{"className":5273,"style":3816},[1840,1841],[1797,5275,2993],{"className":5276},[2650],[1797,5278,1861],{"className":5279,"style":1901},[1840,1841],[1797,5281,3006],{"className":5282},[2693]," retained by the insurer,",[1797,5285,5287],{"className":5286},[2015],[1797,5288,5290,5340],{"className":5289},[1800],[1797,5291,5293],{"className":5292},[1804],[1806,5294,5295],{"xmlns":1808,"display":2024},[1810,5296,5297,5337],{},[1813,5298,5299,5307,5309,5323,5325,5327,5333,5335],{},[3198,5300,5301,5303],{},[1816,5302,2031],{},[5304,5305,5306],"mtext",{},"ret",[1867,5308,2034],{},[2036,5310,5311,5313,5321],{},[1867,5312,2040],{},[1813,5314,5315,5317,5319],{},[1816,5316,2045],{},[1867,5318,2034],{},[1863,5320,1865],{},[1816,5322,1818],{},[1816,5324,5252],{},[1867,5326,2993],{"stretchy":2600},[1857,5328,5329,5331],{},[1816,5330,1861],{},[1816,5332,2045],{},[1867,5334,3006],{"stretchy":2600},[1816,5336,2326],{"mathvariant":2630},[1820,5338,5339],{"encoding":1822},"S^{\\text{ret}}=\\sum_{i=1}^{N}g(X_i).",[1797,5341,5343,5395],{"className":5342,"ariaHidden":1827},[1826],[1797,5344,5346,5350,5386,5389,5392],{"className":5345},[1831],[1797,5347],{"className":5348,"style":5349},[1835],"height:0.8436em;",[1797,5351,5353,5356],{"className":5352},[1840],[1797,5354,2031],{"className":5355,"style":2075},[1840,1841],[1797,5357,5359],{"className":5358},[1905],[1797,5360,5362],{"className":5361},[1909],[1797,5363,5365],{"className":5364},[1914],[1797,5366,5368],{"className":5367,"style":5349},[1918],[1797,5369,5370,5373],{"style":3347},[1797,5371],{"className":5372,"style":1927},[1926],[1797,5374,5376],{"className":5375},[1931,1932,1933,1934],[1797,5377,5379],{"className":5378},[1840,1934],[1797,5380,5383],{"className":5381},[1840,5382,1934],"text",[1797,5384,5306],{"className":5385},[1840,1934],[1797,5387],{"className":5388,"style":2079},[1959],[1797,5390,2034],{"className":5391},[2083],[1797,5393],{"className":5394,"style":2079},[1959],[1797,5396,5398,5401,5468,5471,5474,5477,5517,5520],{"className":5397},[1831],[1797,5399],{"className":5400,"style":2093},[1835],[1797,5402,5404],{"className":5403},[2097,2098],[1797,5405,5407,5460],{"className":5406},[1909,1910],[1797,5408,5410,5457],{"className":5409},[1914],[1797,5411,5413,5433,5443],{"className":5412,"style":2108},[1918],[1797,5414,5415,5418],{"style":2111},[1797,5416],{"className":5417,"style":2115},[1926],[1797,5419,5421],{"className":5420},[1931,1932,1933,1934],[1797,5422,5424,5427,5430],{"className":5423},[1840,1934],[1797,5425,2045],{"className":5426},[1840,1841,1934],[1797,5428,2034],{"className":5429},[2083,1934],[1797,5431,1865],{"className":5432},[1840,1934],[1797,5434,5435,5438],{"style":2133},[1797,5436],{"className":5437,"style":2115},[1926],[1797,5439,5440],{},[1797,5441,2040],{"className":5442},[2097,2142,2143],[1797,5444,5445,5448],{"style":2146},[1797,5446],{"className":5447,"style":2115},[1926],[1797,5449,5451],{"className":5450},[1931,1932,1933,1934],[1797,5452,5454],{"className":5453},[1840,1934],[1797,5455,1818],{"className":5456,"style":1842},[1840,1841,1934],[1797,5458,1942],{"className":5459},[1941],[1797,5461,5463],{"className":5462},[1914],[1797,5464,5466],{"className":5465,"style":2168},[1918],[1797,5467],{},[1797,5469],{"className":5470,"style":1960},[1959],[1797,5472,5252],{"className":5473,"style":3816},[1840,1841],[1797,5475,2993],{"className":5476},[2650],[1797,5478,5480,5483],{"className":5479},[1840],[1797,5481,1861],{"className":5482,"style":1901},[1840,1841],[1797,5484,5486],{"className":5485},[1905],[1797,5487,5489,5509],{"className":5488},[1909,1910],[1797,5490,5492,5506],{"className":5491},[1914],[1797,5493,5495],{"className":5494,"style":2192},[1918],[1797,5496,5497,5500],{"style":1922},[1797,5498],{"className":5499,"style":1927},[1926],[1797,5501,5503],{"className":5502},[1931,1932,1933,1934],[1797,5504,2045],{"className":5505},[1840,1841,1934],[1797,5507,1942],{"className":5508},[1941],[1797,5510,5512],{"className":5511},[1914],[1797,5513,5515],{"className":5514,"style":1949},[1918],[1797,5516],{},[1797,5518,3006],{"className":5519},[2693],[1797,5521,2326],{"className":5522},[1840],[1793,5524,5525,5526,5569,5570,5615,5616,5644,5645,3555],{},"Replace severity moments by moments of ",[1797,5527,5529,5548],{"className":5528},[1800],[1797,5530,5532],{"className":5531},[1804],[1806,5533,5534],{"xmlns":1808},[1810,5535,5536,5546],{},[1813,5537,5538,5540,5542,5544],{},[1816,5539,5252],{},[1867,5541,2993],{"stretchy":2600},[1816,5543,1861],{},[1867,5545,3006],{"stretchy":2600},[1820,5547,5261],{"encoding":1822},[1797,5549,5551],{"className":5550,"ariaHidden":1827},[1826],[1797,5552,5554,5557,5560,5563,5566],{"className":5553},[1831],[1797,5555],{"className":5556,"style":2643},[1835],[1797,5558,5252],{"className":5559,"style":3816},[1840,1841],[1797,5561,2993],{"className":5562},[2650],[1797,5564,1861],{"className":5565,"style":1901},[1840,1841],[1797,5567,3006],{"className":5568},[2693],". For an annual aggregate treaty ",[1797,5571,5573,5594],{"className":5572},[1800],[1797,5574,5576],{"className":5575},[1804],[1806,5577,5578],{"xmlns":1808},[1810,5579,5580,5591],{},[1813,5581,5582,5585,5587,5589],{},[1816,5583,5584],{},"h",[1867,5586,2993],{"stretchy":2600},[1816,5588,2031],{},[1867,5590,3006],{"stretchy":2600},[1820,5592,5593],{"encoding":1822},"h(S)",[1797,5595,5597],{"className":5596,"ariaHidden":1827},[1826],[1797,5598,5600,5603,5606,5609,5612],{"className":5599},[1831],[1797,5601],{"className":5602,"style":2643},[1835],[1797,5604,5584],{"className":5605},[1840,1841],[1797,5607,2993],{"className":5608},[2650],[1797,5610,2031],{"className":5611,"style":2075},[1840,1841],[1797,5613,3006],{"className":5614},[2693],", first construct gross ",[1797,5617,5619,5632],{"className":5618},[1800],[1797,5620,5622],{"className":5621},[1804],[1806,5623,5624],{"xmlns":1808},[1810,5625,5626,5630],{},[1813,5627,5628],{},[1816,5629,2031],{},[1820,5631,2031],{"encoding":1822},[1797,5633,5635],{"className":5634,"ariaHidden":1827},[1826],[1797,5636,5638,5641],{"className":5637},[1831],[1797,5639],{"className":5640,"style":1836},[1835],[1797,5642,2031],{"className":5643,"style":2075},[1840,1841],", then apply ",[1797,5646,5648,5661],{"className":5647},[1800],[1797,5649,5651],{"className":5650},[1804],[1806,5652,5653],{"xmlns":1808},[1810,5654,5655,5659],{},[1813,5656,5657],{},[1816,5658,5584],{},[1820,5660,5584],{"encoding":1822},[1797,5662,5664],{"className":5663,"ariaHidden":1827},[1826],[1797,5665,5667,5670],{"className":5666},[1831],[1797,5668],{"className":5669,"style":3551},[1835],[1797,5671,5584],{"className":5672},[1840,1841],[1797,5674,5676],{"className":5675},[2015],[1797,5677,5679,5709],{"className":5678},[1800],[1797,5680,5682],{"className":5681},[1804],[1806,5683,5684],{"xmlns":1808,"display":2024},[1810,5685,5686,5706],{},[1813,5687,5688,5694,5696,5698,5700,5702,5704],{},[3198,5689,5690,5692],{},[1816,5691,2031],{},[5304,5693,5306],{},[1867,5695,2034],{},[1816,5697,5584],{},[1867,5699,2993],{"stretchy":2600},[1816,5701,2031],{},[1867,5703,3006],{"stretchy":2600},[1816,5705,2326],{"mathvariant":2630},[1820,5707,5708],{"encoding":1822},"S^{\\text{ret}}=h(S).",[1797,5710,5712,5762],{"className":5711,"ariaHidden":1827},[1826],[1797,5713,5715,5718,5753,5756,5759],{"className":5714},[1831],[1797,5716],{"className":5717,"style":5349},[1835],[1797,5719,5721,5724],{"className":5720},[1840],[1797,5722,2031],{"className":5723,"style":2075},[1840,1841],[1797,5725,5727],{"className":5726},[1905],[1797,5728,5730],{"className":5729},[1909],[1797,5731,5733],{"className":5732},[1914],[1797,5734,5736],{"className":5735,"style":5349},[1918],[1797,5737,5738,5741],{"style":3347},[1797,5739],{"className":5740,"style":1927},[1926],[1797,5742,5744],{"className":5743},[1931,1932,1933,1934],[1797,5745,5747],{"className":5746},[1840,1934],[1797,5748,5750],{"className":5749},[1840,5382,1934],[1797,5751,5306],{"className":5752},[1840,1934],[1797,5754],{"className":5755,"style":2079},[1959],[1797,5757,2034],{"className":5758},[2083],[1797,5760],{"className":5761,"style":2079},[1959],[1797,5763,5765,5768,5771,5774,5777,5780],{"className":5764},[1831],[1797,5766],{"className":5767,"style":2643},[1835],[1797,5769,5584],{"className":5770},[1840,1841],[1797,5772,2993],{"className":5773},[2650],[1797,5775,2031],{"className":5776,"style":2075},[1840,1841],[1797,5778,3006],{"className":5779},[2693],[1797,5781,2326],{"className":5782},[1840],[1793,5784,5785],{},"Applying an aggregate treaty claim by claim answers a different contract.",[2328,5787,5789],{"id":5788},"_7-dependence-breaks-the-simple-product","7. Dependence breaks the simple product",[1793,5791,5792,5793,1869],{},"If count and severity share a catastrophe state ",[1797,5794,5796,5810],{"className":5795},[1800],[1797,5797,5799],{"className":5798},[1804],[1806,5800,5801],{"xmlns":1808},[1810,5802,5803,5808],{},[1813,5804,5805],{},[1816,5806,5807],{},"Z",[1820,5809,5807],{"encoding":1822},[1797,5811,5813],{"className":5812,"ariaHidden":1827},[1826],[1797,5814,5816,5819],{"className":5815},[1831],[1797,5817],{"className":5818,"style":1836},[1835],[1797,5820,5807],{"className":5821,"style":5822},[1840,1841],"margin-right:0.0715em;",[1797,5824,5826],{"className":5825},[2015],[1797,5827,5829,5885],{"className":5828},[1800],[1797,5830,5832],{"className":5831},[1804],[1806,5833,5834],{"xmlns":1808,"display":2024},[1810,5835,5836,5882],{},[1813,5837,5838,5840,5842,5844,5846,5848,5850,5880],{},[1816,5839,2597],{},[1867,5841,2601],{"stretchy":2600},[1816,5843,2031],{},[1867,5845,2615],{"stretchy":2600},[1867,5847,2034],{},[1816,5849,2597],{},[1813,5851,5852,5854,5856,5858,5860,5862,5864,5866,5868,5870,5872,5874,5876,5878],{},[1867,5853,2601],{"fence":1827},[1816,5855,2597],{},[1867,5857,2601],{"stretchy":2600},[1816,5859,1818],{},[1867,5861,2606],{},[1816,5863,5807],{},[1867,5865,2615],{"stretchy":2600},[1816,5867,2597],{},[1867,5869,2601],{"stretchy":2600},[1816,5871,1861],{},[1867,5873,2606],{},[1816,5875,5807],{},[1867,5877,2615],{"stretchy":2600},[1867,5879,2615],{"fence":1827},[1867,5881,1869],{"separator":1827},[1820,5883,5884],{"encoding":1822},"E[S]=E\\left[E[N\\mid Z]E[X\\mid Z]\\right],",[1797,5886,5888,5915],{"className":5887,"ariaHidden":1827},[1826],[1797,5889,5891,5894,5897,5900,5903,5906,5909,5912],{"className":5890},[1831],[1797,5892],{"className":5893,"style":2643},[1835],[1797,5895,2597],{"className":5896,"style":2075},[1840,1841],[1797,5898,2601],{"className":5899},[2650],[1797,5901,2031],{"className":5902,"style":2075},[1840,1841],[1797,5904,2615],{"className":5905},[2693],[1797,5907],{"className":5908,"style":2079},[1959],[1797,5910,2034],{"className":5911},[2083],[1797,5913],{"className":5914,"style":2079},[1959],[1797,5916,5918,5921,5924,5927,5986,5989],{"className":5917},[1831],[1797,5919],{"className":5920,"style":2643},[1835],[1797,5922,2597],{"className":5923,"style":2075},[1840,1841],[1797,5925],{"className":5926,"style":1960},[1959],[1797,5928,5930,5935,5938,5941,5944,5947,5950,5953,5956,5959,5962,5965,5968,5971,5974,5977,5980,5983],{"className":5929},[2010],[1797,5931,2601],{"className":5932,"style":5934},[2650,5933],"delimcenter","top:0em;",[1797,5936,2597],{"className":5937,"style":2075},[1840,1841],[1797,5939,2601],{"className":5940},[2650],[1797,5942,1818],{"className":5943,"style":1842},[1840,1841],[1797,5945],{"className":5946,"style":2079},[1959],[1797,5948,2606],{"className":5949},[2083],[1797,5951],{"className":5952,"style":2079},[1959],[1797,5954,5807],{"className":5955,"style":5822},[1840,1841],[1797,5957,2615],{"className":5958},[2693],[1797,5960,2597],{"className":5961,"style":2075},[1840,1841],[1797,5963,2601],{"className":5964},[2650],[1797,5966,1861],{"className":5967,"style":1901},[1840,1841],[1797,5969],{"className":5970,"style":2079},[1959],[1797,5972,2606],{"className":5973},[2083],[1797,5975],{"className":5976,"style":2079},[1959],[1797,5978,5807],{"className":5979,"style":5822},[1840,1841],[1797,5981,2615],{"className":5982},[2693],[1797,5984,2615],{"className":5985,"style":5934},[2693,5933],[1797,5987],{"className":5988,"style":1960},[1959],[1797,5990,1869],{"className":5991},[1955],[1793,5993,5994,5995,6059],{},"which need not equal ",[1797,5996,5998,6026],{"className":5997},[1800],[1797,5999,6001],{"className":6000},[1804],[1806,6002,6003],{"xmlns":1808},[1810,6004,6005,6023],{},[1813,6006,6007,6009,6011,6013,6015,6017,6019,6021],{},[1816,6008,2597],{},[1867,6010,2601],{"stretchy":2600},[1816,6012,1818],{},[1867,6014,2615],{"stretchy":2600},[1816,6016,2597],{},[1867,6018,2601],{"stretchy":2600},[1816,6020,1861],{},[1867,6022,2615],{"stretchy":2600},[1820,6024,6025],{"encoding":1822},"E[N]E[X]",[1797,6027,6029],{"className":6028,"ariaHidden":1827},[1826],[1797,6030,6032,6035,6038,6041,6044,6047,6050,6053,6056],{"className":6031},[1831],[1797,6033],{"className":6034,"style":2643},[1835],[1797,6036,2597],{"className":6037,"style":2075},[1840,1841],[1797,6039,2601],{"className":6040},[2650],[1797,6042,1818],{"className":6043,"style":1842},[1840,1841],[1797,6045,2615],{"className":6046},[2693],[1797,6048,2597],{"className":6049,"style":2075},[1840,1841],[1797,6051,2601],{"className":6052},[2650],[1797,6054,1861],{"className":6055,"style":1901},[1840,1841],[1797,6057,2615],{"className":6058},[2693],". If high-count states also have high severity, the independence formula understates expected loss and usually understates tail risk.",[2328,6061,6063],{"id":6062},"practice","Practice",[2336,6065,6066,6442,6445],{},[2339,6067,6068,3835,6135,3835,6207,6274,6275,6347,6348,6392,6393,2326],{},[1797,6069,6071,6096],{"className":6070},[1800],[1797,6072,6074],{"className":6073},[1804],[1806,6075,6076],{"xmlns":1808},[1810,6077,6078,6093],{},[1813,6079,6080,6082,6084,6086,6088,6090],{},[1816,6081,2597],{},[1867,6083,2601],{"stretchy":2600},[1816,6085,1818],{},[1867,6087,2615],{"stretchy":2600},[1867,6089,2034],{},[1863,6091,6092],{},"10",[1820,6094,6095],{"encoding":1822},"E[N]=10",[1797,6097,6099,6126],{"className":6098,"ariaHidden":1827},[1826],[1797,6100,6102,6105,6108,6111,6114,6117,6120,6123],{"className":6101},[1831],[1797,6103],{"className":6104,"style":2643},[1835],[1797,6106,2597],{"className":6107,"style":2075},[1840,1841],[1797,6109,2601],{"className":6110},[2650],[1797,6112,1818],{"className":6113,"style":1842},[1840,1841],[1797,6115,2615],{"className":6116},[2693],[1797,6118],{"className":6119,"style":2079},[1959],[1797,6121,2034],{"className":6122},[2083],[1797,6124],{"className":6125,"style":2079},[1959],[1797,6127,6129,6132],{"className":6128},[1831],[1797,6130],{"className":6131,"style":2270},[1835],[1797,6133,6092],{"className":6134},[1840],[1797,6136,6138,6165],{"className":6137},[1800],[1797,6139,6141],{"className":6140},[1804],[1806,6142,6143],{"xmlns":1808},[1810,6144,6145,6162],{},[1813,6146,6147,6149,6151,6153,6155,6157,6159],{},[1816,6148,2987],{"mathvariant":2630},[1867,6150,2990],{},[1867,6152,2993],{"stretchy":2600},[1816,6154,1818],{},[1867,6156,3006],{"stretchy":2600},[1867,6158,2034],{},[1863,6160,6161],{},"15",[1820,6163,6164],{"encoding":1822},"\\operatorname{Var}(N)=15",[1797,6166,6168,6198],{"className":6167,"ariaHidden":1827},[1826],[1797,6169,6171,6174,6180,6183,6186,6189,6192,6195],{"className":6170},[1831],[1797,6172],{"className":6173,"style":2643},[1835],[1797,6175,6177],{"className":6176},[2097],[1797,6178,2987],{"className":6179},[1840,3041],[1797,6181,2993],{"className":6182},[2650],[1797,6184,1818],{"className":6185,"style":1842},[1840,1841],[1797,6187,3006],{"className":6188},[2693],[1797,6190],{"className":6191,"style":2079},[1959],[1797,6193,2034],{"className":6194},[2083],[1797,6196],{"className":6197,"style":2079},[1959],[1797,6199,6201,6204],{"className":6200},[1831],[1797,6202],{"className":6203,"style":2270},[1835],[1797,6205,6161],{"className":6206},[1840],[1797,6208,6210,6235],{"className":6209},[1800],[1797,6211,6213],{"className":6212},[1804],[1806,6214,6215],{"xmlns":1808},[1810,6216,6217,6232],{},[1813,6218,6219,6221,6223,6225,6227,6229],{},[1816,6220,2597],{},[1867,6222,2601],{"stretchy":2600},[1816,6224,1861],{},[1867,6226,2615],{"stretchy":2600},[1867,6228,2034],{},[1863,6230,6231],{},"3",[1820,6233,6234],{"encoding":1822},"E[X]=3",[1797,6236,6238,6265],{"className":6237,"ariaHidden":1827},[1826],[1797,6239,6241,6244,6247,6250,6253,6256,6259,6262],{"className":6240},[1831],[1797,6242],{"className":6243,"style":2643},[1835],[1797,6245,2597],{"className":6246,"style":2075},[1840,1841],[1797,6248,2601],{"className":6249},[2650],[1797,6251,1861],{"className":6252,"style":1901},[1840,1841],[1797,6254,2615],{"className":6255},[2693],[1797,6257],{"className":6258,"style":2079},[1959],[1797,6260,2034],{"className":6261},[2083],[1797,6263],{"className":6264,"style":2079},[1959],[1797,6266,6268,6271],{"className":6267},[1831],[1797,6269],{"className":6270,"style":2270},[1835],[1797,6272,6231],{"className":6273},[1840],", and ",[1797,6276,6278,6305],{"className":6277},[1800],[1797,6279,6281],{"className":6280},[1804],[1806,6282,6283],{"xmlns":1808},[1810,6284,6285,6302],{},[1813,6286,6287,6289,6291,6293,6295,6297,6299],{},[1816,6288,2987],{"mathvariant":2630},[1867,6290,2990],{},[1867,6292,2993],{"stretchy":2600},[1816,6294,1861],{},[1867,6296,3006],{"stretchy":2600},[1867,6298,2034],{},[1863,6300,6301],{},"20",[1820,6303,6304],{"encoding":1822},"\\operatorname{Var}(X)=20",[1797,6306,6308,6338],{"className":6307,"ariaHidden":1827},[1826],[1797,6309,6311,6314,6320,6323,6326,6329,6332,6335],{"className":6310},[1831],[1797,6312],{"className":6313,"style":2643},[1835],[1797,6315,6317],{"className":6316},[2097],[1797,6318,2987],{"className":6319},[1840,3041],[1797,6321,2993],{"className":6322},[2650],[1797,6324,1861],{"className":6325,"style":1901},[1840,1841],[1797,6327,3006],{"className":6328},[2693],[1797,6330],{"className":6331,"style":2079},[1959],[1797,6333,2034],{"className":6334},[2083],[1797,6336],{"className":6337,"style":2079},[1959],[1797,6339,6341,6344],{"className":6340},[1831],[1797,6342],{"className":6343,"style":2270},[1835],[1797,6345,6301],{"className":6346},[1840],". Find ",[1797,6349,6351,6371],{"className":6350},[1800],[1797,6352,6354],{"className":6353},[1804],[1806,6355,6356],{"xmlns":1808},[1810,6357,6358,6368],{},[1813,6359,6360,6362,6364,6366],{},[1816,6361,2597],{},[1867,6363,2601],{"stretchy":2600},[1816,6365,2031],{},[1867,6367,2615],{"stretchy":2600},[1820,6369,6370],{"encoding":1822},"E[S]",[1797,6372,6374],{"className":6373,"ariaHidden":1827},[1826],[1797,6375,6377,6380,6383,6386,6389],{"className":6376},[1831],[1797,6378],{"className":6379,"style":2643},[1835],[1797,6381,2597],{"className":6382,"style":2075},[1840,1841],[1797,6384,2601],{"className":6385},[2650],[1797,6387,2031],{"className":6388,"style":2075},[1840,1841],[1797,6390,2615],{"className":6391},[2693]," and ",[1797,6394,6396,6418],{"className":6395},[1800],[1797,6397,6399],{"className":6398},[1804],[1806,6400,6401],{"xmlns":1808},[1810,6402,6403,6415],{},[1813,6404,6405,6407,6409,6411,6413],{},[1816,6406,2987],{"mathvariant":2630},[1867,6408,2990],{},[1867,6410,2993],{"stretchy":2600},[1816,6412,2031],{},[1867,6414,3006],{"stretchy":2600},[1820,6416,6417],{"encoding":1822},"\\operatorname{Var}(S)",[1797,6419,6421],{"className":6420,"ariaHidden":1827},[1826],[1797,6422,6424,6427,6433,6436,6439],{"className":6423},[1831],[1797,6425],{"className":6426,"style":2643},[1835],[1797,6428,6430],{"className":6429},[2097],[1797,6431,2987],{"className":6432},[1840,3041],[1797,6434,2993],{"className":6435},[2650],[1797,6437,2031],{"className":6438,"style":2075},[1840,1841],[1797,6440,3006],{"className":6441},[2693],[2339,6443,6444],{},"Why can a Normal approximation produce impossible results for a low-frequency portfolio?",[2339,6446,6447],{},"Where should a per-occurrence catastrophe treaty be applied?",[6449,6450,6452],"legacy-details",{"title":6451},"Answers",[2336,6453,6454,6673,6676],{},[2339,6455,6456,6523,6524,2326],{},[1797,6457,6459,6484],{"className":6458},[1800],[1797,6460,6462],{"className":6461},[1804],[1806,6463,6464],{"xmlns":1808},[1810,6465,6466,6481],{},[1813,6467,6468,6470,6472,6474,6476,6478],{},[1816,6469,2597],{},[1867,6471,2601],{"stretchy":2600},[1816,6473,2031],{},[1867,6475,2615],{"stretchy":2600},[1867,6477,2034],{},[1863,6479,6480],{},"30",[1820,6482,6483],{"encoding":1822},"E[S]=30",[1797,6485,6487,6514],{"className":6486,"ariaHidden":1827},[1826],[1797,6488,6490,6493,6496,6499,6502,6505,6508,6511],{"className":6489},[1831],[1797,6491],{"className":6492,"style":2643},[1835],[1797,6494,2597],{"className":6495,"style":2075},[1840,1841],[1797,6497,2601],{"className":6498},[2650],[1797,6500,2031],{"className":6501,"style":2075},[1840,1841],[1797,6503,2615],{"className":6504},[2693],[1797,6506],{"className":6507,"style":2079},[1959],[1797,6509,2034],{"className":6510},[2083],[1797,6512],{"className":6513,"style":2079},[1959],[1797,6515,6517,6520],{"className":6516},[1831],[1797,6518],{"className":6519,"style":2270},[1835],[1797,6521,6480],{"className":6522},[1840],"; variance ",[1797,6525,6527,6568],{"className":6526},[1800],[1797,6528,6530],{"className":6529},[1804],[1806,6531,6532],{"xmlns":1808},[1810,6533,6534,6565],{},[1813,6535,6536,6538,6540,6542,6544,6546,6548,6550,6552,6558,6560,6562],{},[1867,6537,2034],{},[1863,6539,6092],{},[1867,6541,2993],{"stretchy":2600},[1863,6543,6301],{},[1867,6545,3006],{"stretchy":2600},[1867,6547,3176],{},[1863,6549,6161],{},[1867,6551,2993],{"stretchy":2600},[3198,6553,6554,6556],{},[1863,6555,6231],{},[1863,6557,1876],{},[1867,6559,3006],{"stretchy":2600},[1867,6561,2034],{},[1863,6563,6564],{},"335",[1820,6566,6567],{"encoding":1822},"=10(20)+15(3^2)=335",[1797,6569,6571,6584,6611,6664],{"className":6570,"ariaHidden":1827},[1826],[1797,6572,6574,6578,6581],{"className":6573},[1831],[1797,6575],{"className":6576,"style":6577},[1835],"height:0.3669em;",[1797,6579,2034],{"className":6580},[2083],[1797,6582],{"className":6583,"style":2079},[1959],[1797,6585,6587,6590,6593,6596,6599,6602,6605,6608],{"className":6586},[1831],[1797,6588],{"className":6589,"style":2643},[1835],[1797,6591,6092],{"className":6592},[1840],[1797,6594,2993],{"className":6595},[2650],[1797,6597,6301],{"className":6598},[1840],[1797,6600,3006],{"className":6601},[2693],[1797,6603],{"className":6604,"style":3281},[1959],[1797,6606,3176],{"className":6607},[3285],[1797,6609],{"className":6610,"style":3281},[1959],[1797,6612,6614,6617,6620,6623,6652,6655,6658,6661],{"className":6613},[1831],[1797,6615],{"className":6616,"style":3891},[1835],[1797,6618,6161],{"className":6619},[1840],[1797,6621,2993],{"className":6622},[2650],[1797,6624,6626,6629],{"className":6625},[1840],[1797,6627,6231],{"className":6628},[1840],[1797,6630,6632],{"className":6631},[1905],[1797,6633,6635],{"className":6634},[1909],[1797,6636,6638],{"className":6637},[1914],[1797,6639,6641],{"className":6640,"style":3916},[1918],[1797,6642,6643,6646],{"style":3919},[1797,6644],{"className":6645,"style":1927},[1926],[1797,6647,6649],{"className":6648},[1931,1932,1933,1934],[1797,6650,1876],{"className":6651},[1840,1934],[1797,6653,3006],{"className":6654},[2693],[1797,6656],{"className":6657,"style":2079},[1959],[1797,6659,2034],{"className":6660},[2083],[1797,6662],{"className":6663,"style":2079},[1959],[1797,6665,6667,6670],{"className":6666},[1831],[1797,6668],{"className":6669,"style":2270},[1835],[1797,6671,6564],{"className":6672},[1840],[2339,6674,6675],{},"It is symmetric and has support on negative values, while aggregate loss is non-negative and may have a large point mass at zero plus strong skew.",[2339,6677,6678],{},"Aggregate all claims belonging to an occurrence, apply the occurrence layer, then aggregate retained occurrences over the year.",[1793,6680,6681,6682,2326],{},"Next, retain risk-specific structure in the ",[5140,6683,916],{"href":6684},".\u002F02-individual",{"title":10,"searchDepth":6686,"depth":6686,"links":6687},2,[6688,6689,6690,6694,6695,6696,6697,6698],{"id":2330,"depth":6686,"text":2331},{"id":2522,"depth":6686,"text":2523},{"id":2916,"depth":6686,"text":2917,"children":6691},[6692],{"id":3366,"depth":6693,"text":3367},3,{"id":4237,"depth":6686,"text":4238},{"id":5080,"depth":6686,"text":5081},{"id":5233,"depth":6686,"text":5234},{"id":5788,"depth":6686,"text":5789},{"id":6062,"depth":6686,"text":6063},"Derive aggregate-loss moments and transforms, then see exactly where independence enters.","md",{"sidebar":6702},{"order":6703},1,true,{"title":912,"description":6699},"N2If9SnNKFMmK82rPk8HCS4LasiuffO6-PifP7XxIos",[6708,6710],{"title":906,"path":907,"stem":908,"description":6709,"children":-1},"Combine frequency, severity, heterogeneity, dependence, and reinsurance into portfolio loss distributions.",{"title":916,"path":917,"stem":918,"description":6711,"children":-1},"Aggregate heterogeneous policy risks and expose the effect of common dependence.",1785754736546]