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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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policies,",[1797,1844,1847],{"className":1845},[1846],"katex-display",[1797,1848,1850,1928],{"className":1849},[1800],[1797,1851,1853],{"className":1852},[1804],[1806,1854,1856],{"xmlns":1808,"display":1855},"block",[1810,1857,1858,1925],{},[1813,1859,1860,1863,1867,1886,1894,1897,1901,1907,1909,1916,1923],{},[1816,1861,1862],{},"S",[1864,1865,1866],"mo",{},"=",[1868,1869,1870,1873,1884],"munderover",{},[1864,1871,1872],{},"∑",[1813,1874,1875,1878,1880],{},[1816,1876,1877],{},"i",[1864,1879,1866],{},[1881,1882,1883],"mn",{},"1",[1816,1885,1818],{},[1887,1888,1889,1892],"msub",{},[1816,1890,1891],{},"Y",[1816,1893,1877],{},[1864,1895,1896],{"separator":1827},",",[1898,1899],"mspace",{"width":1900},"2em",[1887,1902,1903,1905],{},[1816,1904,1891],{},[1816,1906,1877],{},[1864,1908,1866],{},[1887,1910,1911,1914],{},[1816,1912,1913],{},"I",[1816,1915,1877],{},[1887,1917,1918,1921],{},[1816,1919,1920],{},"B",[1816,1922,1877],{},[1864,1924,1896],{"separator":1827},[1820,1926,1927],{"encoding":1822},"S=\\sum_{i=1}^{m}Y_i,\n\\qquad\nY_i=I_iB_i,",[1797,1929,1931,1953,2158],{"className":1930,"ariaHidden":1827},[1826],[1797,1932,1934,1938,1942,1946,1950],{"className":1933},[1831],[1797,1935],{"className":1936,"style":1937},[1835],"height:0.6833em;",[1797,1939,1862],{"className":1940,"style":1941},[1840,1841],"margin-right:0.0576em;",[1797,1943],{"className":1944,"style":1945},[1898],"margin-right:0.2778em;",[1797,1947,1866],{"className":1948},[1949],"mrel",[1797,1951],{"className":1952,"style":1945},[1898],[1797,1954,1956,1960,2048,2052,2098,2102,2106,2109,2149,2152,2155],{"className":1955},[1831],[1797,1957],{"className":1958,"style":1959},[1835],"height:2.9291em;vertical-align:-1.2777em;",[1797,1961,1965],{"className":1962},[1963,1964],"mop","op-limits",[1797,1966,1970,2039],{"className":1967},[1968,1969],"vlist-t","vlist-t2",[1797,1971,1974,2034],{"className":1972},[1973],"vlist-r",[1797,1975,1979,2006,2019],{"className":1976,"style":1978},[1977],"vlist","height:1.6514em;",[1797,1980,1982,1987],{"style":1981},"top:-1.8723em;margin-left:0em;",[1797,1983],{"className":1984,"style":1986},[1985],"pstrut","height:3.05em;",[1797,1988,1994],{"className":1989},[1990,1991,1992,1993],"sizing","reset-size6","size3","mtight",[1797,1995,1997,2000,2003],{"className":1996},[1840,1993],[1797,1998,1877],{"className":1999},[1840,1841,1993],[1797,2001,1866],{"className":2002},[1949,1993],[1797,2004,1883],{"className":2005},[1840,1993],[1797,2007,2009,2012],{"style":2008},"top:-3.05em;",[1797,2010],{"className":2011,"style":1986},[1985],[1797,2013,2014],{},[1797,2015,1872],{"className":2016},[1963,2017,2018],"op-symbol","large-op",[1797,2020,2022,2025],{"style":2021},"top:-4.3em;margin-left:0em;",[1797,2023],{"className":2024,"style":1986},[1985],[1797,2026,2028],{"className":2027},[1990,1991,1992,1993],[1797,2029,2031],{"className":2030},[1840,1993],[1797,2032,1818],{"className":2033},[1840,1841,1993],[1797,2035,2038],{"className":2036},[2037],"vlist-s","​",[1797,2040,2042],{"className":2041},[1973],[1797,2043,2046],{"className":2044,"style":2045},[1977],"height:1.2777em;",[1797,2047],{},[1797,2049],{"className":2050,"style":2051},[1898],"margin-right:0.1667em;",[1797,2053,2055,2059],{"className":2054},[1840],[1797,2056,1891],{"className":2057,"style":2058},[1840,1841],"margin-right:0.2222em;",[1797,2060,2063],{"className":2061},[2062],"msupsub",[1797,2064,2066,2089],{"className":2065},[1968,1969],[1797,2067,2069,2086],{"className":2068},[1973],[1797,2070,2073],{"className":2071,"style":2072},[1977],"height:0.3117em;",[1797,2074,2076,2080],{"style":2075},"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;",[1797,2077],{"className":2078,"style":2079},[1985],"height:2.7em;",[1797,2081,2083],{"className":2082},[1990,1991,1992,1993],[1797,2084,1877],{"className":2085},[1840,1841,1993],[1797,2087,2038],{"className":2088},[2037],[1797,2090,2092],{"className":2091},[1973],[1797,2093,2096],{"className":2094,"style":2095},[1977],"height:0.15em;",[1797,2097],{},[1797,2099,1896],{"className":2100},[2101],"mpunct",[1797,2103],{"className":2104,"style":2105},[1898],"margin-right:2em;",[1797,2107],{"className":2108,"style":2051},[1898],[1797,2110,2112,2115],{"className":2111},[1840],[1797,2113,1891],{"className":2114,"style":2058},[1840,1841],[1797,2116,2118],{"className":2117},[2062],[1797,2119,2121,2141],{"className":2120},[1968,1969],[1797,2122,2124,2138],{"className":2123},[1973],[1797,2125,2127],{"className":2126,"style":2072},[1977],[1797,2128,2129,2132],{"style":2075},[1797,2130],{"className":2131,"style":2079},[1985],[1797,2133,2135],{"className":2134},[1990,1991,1992,1993],[1797,2136,1877],{"className":2137},[1840,1841,1993],[1797,2139,2038],{"className":2140},[2037],[1797,2142,2144],{"className":2143},[1973],[1797,2145,2147],{"className":2146,"style":2095},[1977],[1797,2148],{},[1797,2150],{"className":2151,"style":1945},[1898],[1797,2153,1866],{"className":2154},[1949],[1797,2156],{"className":2157,"style":1945},[1898],[1797,2159,2161,2165,2207,2249],{"className":2160},[1831],[1797,2162],{"className":2163,"style":2164},[1835],"height:0.8778em;vertical-align:-0.1944em;",[1797,2166,2168,2172],{"className":2167},[1840],[1797,2169,1913],{"className":2170,"style":2171},[1840,1841],"margin-right:0.0785em;",[1797,2173,2175],{"className":2174},[2062],[1797,2176,2178,2199],{"className":2177},[1968,1969],[1797,2179,2181,2196],{"className":2180},[1973],[1797,2182,2184],{"className":2183,"style":2072},[1977],[1797,2185,2187,2190],{"style":2186},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[1797,2188],{"className":2189,"style":2079},[1985],[1797,2191,2193],{"className":2192},[1990,1991,1992,1993],[1797,2194,1877],{"className":2195},[1840,1841,1993],[1797,2197,2038],{"className":2198},[2037],[1797,2200,2202],{"className":2201},[1973],[1797,2203,2205],{"className":2204,"style":2095},[1977],[1797,2206],{},[1797,2208,2210,2214],{"className":2209},[1840],[1797,2211,1920],{"className":2212,"style":2213},[1840,1841],"margin-right:0.0502em;",[1797,2215,2217],{"className":2216},[2062],[1797,2218,2220,2241],{"className":2219},[1968,1969],[1797,2221,2223,2238],{"className":2222},[1973],[1797,2224,2226],{"className":2225,"style":2072},[1977],[1797,2227,2229,2232],{"style":2228},"top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;",[1797,2230],{"className":2231,"style":2079},[1985],[1797,2233,2235],{"className":2234},[1990,1991,1992,1993],[1797,2236,1877],{"className":2237},[1840,1841,1993],[1797,2239,2038],{"className":2240},[2037],[1797,2242,2244],{"className":2243},[1973],[1797,2245,2247],{"className":2246,"style":2095},[1977],[1797,2248],{},[1797,2250,1896],{"className":2251},[2101],[1793,2253,2254,2255,2326,2327,2397],{},"where ",[1797,2256,2258,2276],{"className":2257},[1800],[1797,2259,2261],{"className":2260},[1804],[1806,2262,2263],{"xmlns":1808},[1810,2264,2265,2273],{},[1813,2266,2267],{},[1887,2268,2269,2271],{},[1816,2270,1913],{},[1816,2272,1877],{},[1820,2274,2275],{"encoding":1822},"I_i",[1797,2277,2279],{"className":2278,"ariaHidden":1827},[1826],[1797,2280,2282,2286],{"className":2281},[1831],[1797,2283],{"className":2284,"style":2285},[1835],"height:0.8333em;vertical-align:-0.15em;",[1797,2287,2289,2292],{"className":2288},[1840],[1797,2290,1913],{"className":2291,"style":2171},[1840,1841],[1797,2293,2295],{"className":2294},[2062],[1797,2296,2298,2318],{"className":2297},[1968,1969],[1797,2299,2301,2315],{"className":2300},[1973],[1797,2302,2304],{"className":2303,"style":2072},[1977],[1797,2305,2306,2309],{"style":2186},[1797,2307],{"className":2308,"style":2079},[1985],[1797,2310,2312],{"className":2311},[1990,1991,1992,1993],[1797,2313,1877],{"className":2314},[1840,1841,1993],[1797,2316,2038],{"className":2317},[2037],[1797,2319,2321],{"className":2320},[1973],[1797,2322,2324],{"className":2323,"style":2095},[1977],[1797,2325],{}," indicates whether a claim occurs and ",[1797,2328,2330,2348],{"className":2329},[1800],[1797,2331,2333],{"className":2332},[1804],[1806,2334,2335],{"xmlns":1808},[1810,2336,2337,2345],{},[1813,2338,2339],{},[1887,2340,2341,2343],{},[1816,2342,1920],{},[1816,2344,1877],{},[1820,2346,2347],{"encoding":1822},"B_i",[1797,2349,2351],{"className":2350,"ariaHidden":1827},[1826],[1797,2352,2354,2357],{"className":2353},[1831],[1797,2355],{"className":2356,"style":2285},[1835],[1797,2358,2360,2363],{"className":2359},[1840],[1797,2361,1920],{"className":2362,"style":2213},[1840,1841],[1797,2364,2366],{"className":2365},[2062],[1797,2367,2369,2389],{"className":2368},[1968,1969],[1797,2370,2372,2386],{"className":2371},[1973],[1797,2373,2375],{"className":2374,"style":2072},[1977],[1797,2376,2377,2380],{"style":2228},[1797,2378],{"className":2379,"style":2079},[1985],[1797,2381,2383],{"className":2382},[1990,1991,1992,1993],[1797,2384,1877],{"className":2385},[1840,1841,1993],[1797,2387,2038],{"className":2388},[2037],[1797,2390,2392],{"className":2391},[1973],[1797,2393,2395],{"className":2394,"style":2095},[1977],[1797,2396],{}," is its amount. The simplest form allows at most one claim per risk during the period.",[2399,2400,2402],"h2",{"id":2401},"_1-one-risk","1. One risk",[1793,2404,2405,2406,2571,2572,2641],{},"Let ",[1797,2407,2409,2450],{"className":2408},[1800],[1797,2410,2412],{"className":2411},[1804],[1806,2413,2414],{"xmlns":1808},[1810,2415,2416,2447],{},[1813,2417,2418,2424,2427,2431,2434,2438,2444],{},[1887,2419,2420,2422],{},[1816,2421,1913],{},[1816,2423,1877],{},[1864,2425,2426],{},"∼",[1816,2428,2430],{"mathvariant":2429},"normal","Bernoulli",[1864,2432,2433],{},"⁡",[1864,2435,2437],{"stretchy":2436},"false","(",[1887,2439,2440,2442],{},[1816,2441,1793],{},[1816,2443,1877],{},[1864,2445,2446],{"stretchy":2436},")",[1820,2448,2449],{"encoding":1822},"I_i\\sim\\operatorname{Bernoulli}(p_i)",[1797,2451,2453,2508],{"className":2452,"ariaHidden":1827},[1826],[1797,2454,2456,2459,2499,2502,2505],{"className":2455},[1831],[1797,2457],{"className":2458,"style":2285},[1835],[1797,2460,2462,2465],{"className":2461},[1840],[1797,2463,1913],{"className":2464,"style":2171},[1840,1841],[1797,2466,2468],{"className":2467},[2062],[1797,2469,2471,2491],{"className":2470},[1968,1969],[1797,2472,2474,2488],{"className":2473},[1973],[1797,2475,2477],{"className":2476,"style":2072},[1977],[1797,2478,2479,2482],{"style":2186},[1797,2480],{"className":2481,"style":2079},[1985],[1797,2483,2485],{"className":2484},[1990,1991,1992,1993],[1797,2486,1877],{"className":2487},[1840,1841,1993],[1797,2489,2038],{"className":2490},[2037],[1797,2492,2494],{"className":2493},[1973],[1797,2495,2497],{"className":2496,"style":2095},[1977],[1797,2498],{},[1797,2500],{"className":2501,"style":1945},[1898],[1797,2503,2426],{"className":2504},[1949],[1797,2506],{"className":2507,"style":1945},[1898],[1797,2509,2511,2515,2522,2526,2567],{"className":2510},[1831],[1797,2512],{"className":2513,"style":2514},[1835],"height:1em;vertical-align:-0.25em;",[1797,2516,2518],{"className":2517},[1963],[1797,2519,2430],{"className":2520},[1840,2521],"mathrm",[1797,2523,2437],{"className":2524},[2525],"mopen",[1797,2527,2529,2532],{"className":2528},[1840],[1797,2530,1793],{"className":2531},[1840,1841],[1797,2533,2535],{"className":2534},[2062],[1797,2536,2538,2559],{"className":2537},[1968,1969],[1797,2539,2541,2556],{"className":2540},[1973],[1797,2542,2544],{"className":2543,"style":2072},[1977],[1797,2545,2547,2550],{"style":2546},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[1797,2548],{"className":2549,"style":2079},[1985],[1797,2551,2553],{"className":2552},[1990,1991,1992,1993],[1797,2554,1877],{"className":2555},[1840,1841,1993],[1797,2557,2038],{"className":2558},[2037],[1797,2560,2562],{"className":2561},[1973],[1797,2563,2565],{"className":2564,"style":2095},[1977],[1797,2566],{},[1797,2568,2446],{"className":2569},[2570],"mclose",", independent of 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example",[4411,4412,4413,4508],"table",{},[4414,4415,4416],"thead",{},[4417,4418,4419,4423,4496,4499,4502,4505],"tr",{},[4420,4421,4422],"th",{},"Risk",[4420,4424,4426],{"align":4425},"right",[1797,4427,4429,4447],{"className":4428},[1800],[1797,4430,4432],{"className":4431},[1804],[1806,4433,4434],{"xmlns":1808},[1810,4435,4436,4444],{},[1813,4437,4438],{},[1887,4439,4440,4442],{},[1816,4441,1793],{},[1816,4443,1877],{},[1820,4445,4446],{"encoding":1822},"p_i",[1797,4448,4450],{"className":4449,"ariaHidden":1827},[1826],[1797,4451,4453,4456],{"className":4452},[1831],[1797,4454],{"className":4455,"style":3090},[1835],[1797,4457,4459,4462],{"className":4458},[1840],[1797,4460,1793],{"className":4461},[1840,1841],[1797,4463,4465],{"className":4464},[2062],[1797,4466,4468,4488],{"className":4467},[1968,1969],[1797,4469,4471,4485],{"className":4470},[1973],[1797,4472,4474],{"className":4473,"style":2072},[1977],[1797,4475,4476,4479],{"style":2546},[1797,4477],{"className":4478,"style":2079},[1985],[1797,4480,4482],{"className":4481},[1990,1991,1992,1993],[1797,4483,1877],{"className":4484},[1840,1841,1993],[1797,4486,2038],{"className":4487},[2037],[1797,4489,4491],{"className":4490},[1973],[1797,4492,4494],{"className":4493,"style":2095},[1977],[1797,4495],{},[4420,4497,4498],{"align":4425},"Mean claim",[4420,4500,4501],{"align":4425},"Claim SD",[4420,4503,4504],{"align":4425},"Expected loss",[4420,4506,4507],{"align":4425},"Loss variance (£²)",[4509,4510,4511,4624,4726,4830],"tbody",{},[4417,4512,4513,4517,4520,4523,4526,4529],{},[4514,4515,4516],"td",{},"A",[4514,4518,4519],{"align":4425},"2%",[4514,4521,4522],{"align":4425},"£10,000",[4514,4524,4525],{"align":4425},"£5,000",[4514,4527,4528],{"align":4425},"£200",[4514,4530,4531],{"align":4425},[1797,4532,4534,4561],{"className":4533},[1800],[1797,4535,4537],{"className":4536},[1804],[1806,4538,4539],{"xmlns":1808},[1810,4540,4541,4558],{},[1813,4542,4543,4546,4549],{},[1881,4544,4545],{},"2.460",[1864,4547,4548],{},"×",[4550,4551,4552,4555],"msup",{},[1881,4553,4554],{},"10",[1881,4556,4557],{},"6",[1820,4559,4560],{"encoding":1822},"2.460\\times10^6",[1797,4562,4564,4583],{"className":4563,"ariaHidden":1827},[1826],[1797,4565,4567,4571,4574,4577,4580],{"className":4566},[1831],[1797,4568],{"className":4569,"style":4570},[1835],"height:0.7278em;vertical-align:-0.0833em;",[1797,4572,4545],{"className":4573},[1840],[1797,4575],{"className":4576,"style":2058},[1898],[1797,4578,4548],{"className":4579},[3429],[1797,4581],{"className":4582,"style":2058},[1898],[1797,4584,4586,4590,4593],{"className":4585},[1831],[1797,4587],{"className":4588,"style":4589},[1835],"height:0.8141em;",[1797,4591,1883],{"className":4592},[1840],[1797,4594,4596,4600],{"className":4595},[1840],[1797,4597,4599],{"className":4598},[1840],"0",[1797,4601,4603],{"className":4602},[2062],[1797,4604,4606],{"className":4605},[1968],[1797,4607,4609],{"className":4608},[1973],[1797,4610,4612],{"className":4611,"style":4589},[1977],[1797,4613,4615,4618],{"style":4614},"top:-3.063em;margin-right:0.05em;",[1797,4616],{"className":4617,"style":2079},[1985],[1797,4619,4621],{"className":4620},[1990,1991,1992,1993],[1797,4622,4557],{"className":4623},[1840,1993],[4417,4625,4626,4628,4631,4633,4636,4639],{},[4514,4627,1920],{},[4514,4629,4630],{"align":4425},"5%",[4514,4632,4525],{"align":4425},[4514,4634,4635],{"align":4425},"£3,000",[4514,4637,4638],{"align":4425},"£250",[4514,4640,4641],{"align":4425},[1797,4642,4644,4667],{"className":4643},[1800],[1797,4645,4647],{"className":4646},[1804],[1806,4648,4649],{"xmlns":1808},[1810,4650,4651,4664],{},[1813,4652,4653,4656,4658],{},[1881,4654,4655],{},"1.638",[1864,4657,4548],{},[4550,4659,4660,4662],{},[1881,4661,4554],{},[1881,4663,4557],{},[1820,4665,4666],{"encoding":1822},"1.638\\times10^6",[1797,4668,4670,4688],{"className":4669,"ariaHidden":1827},[1826],[1797,4671,4673,4676,4679,4682,4685],{"className":4672},[1831],[1797,4674],{"className":4675,"style":4570},[1835],[1797,4677,4655],{"className":4678},[1840],[1797,4680],{"className":4681,"style":2058},[1898],[1797,4683,4548],{"className":4684},[3429],[1797,4686],{"className":4687,"style":2058},[1898],[1797,4689,4691,4694,4697],{"className":4690},[1831],[1797,4692],{"className":4693,"style":4589},[1835],[1797,4695,1883],{"className":4696},[1840],[1797,4698,4700,4703],{"className":4699},[1840],[1797,4701,4599],{"className":4702},[1840],[1797,4704,4706],{"className":4705},[2062],[1797,4707,4709],{"className":4708},[1968],[1797,4710,4712],{"className":4711},[1973],[1797,4713,4715],{"className":4714,"style":4589},[1977],[1797,4716,4717,4720],{"style":4614},[1797,4718],{"className":4719,"style":2079},[1985],[1797,4721,4723],{"className":4722},[1990,1991,1992,1993],[1797,4724,4557],{"className":4725},[1840,1993],[4417,4727,4728,4731,4734,4737,4740,4743],{},[4514,4729,4730],{},"C",[4514,4732,4733],{"align":4425},"1%",[4514,4735,4736],{"align":4425},"£50,000",[4514,4738,4739],{"align":4425},"£30,000",[4514,4741,4742],{"align":4425},"£500",[4514,4744,4745],{"align":4425},[1797,4746,4748,4771],{"className":4747},[1800],[1797,4749,4751],{"className":4750},[1804],[1806,4752,4753],{"xmlns":1808},[1810,4754,4755,4768],{},[1813,4756,4757,4760,4762],{},[1881,4758,4759],{},"33.750",[1864,4761,4548],{},[4550,4763,4764,4766],{},[1881,4765,4554],{},[1881,4767,4557],{},[1820,4769,4770],{"encoding":1822},"33.750\\times10^6",[1797,4772,4774,4792],{"className":4773,"ariaHidden":1827},[1826],[1797,4775,4777,4780,4783,4786,4789],{"className":4776},[1831],[1797,4778],{"className":4779,"style":4570},[1835],[1797,4781,4759],{"className":4782},[1840],[1797,4784],{"className":4785,"style":2058},[1898],[1797,4787,4548],{"className":4788},[3429],[1797,4790],{"className":4791,"style":2058},[1898],[1797,4793,4795,4798,4801],{"className":4794},[1831],[1797,4796],{"className":4797,"style":4589},[1835],[1797,4799,1883],{"className":4800},[1840],[1797,4802,4804,4807],{"className":4803},[1840],[1797,4805,4599],{"className":4806},[1840],[1797,4808,4810],{"className":4809},[2062],[1797,4811,4813],{"className":4812},[1968],[1797,4814,4816],{"className":4815},[1973],[1797,4817,4819],{"className":4818,"style":4589},[1977],[1797,4820,4821,4824],{"style":4614},[1797,4822],{"className":4823,"style":2079},[1985],[1797,4825,4827],{"className":4826},[1990,1991,1992,1993],[1797,4828,4557],{"className":4829},[1840,1993],[4417,4831,4832,4838,4840,4842,4844,4849],{},[4514,4833,4834],{},[4835,4836,4837],"strong",{},"Total",[4514,4839],{"align":4425},[4514,4841],{"align":4425},[4514,4843],{"align":4425},[4514,4845,4846],{"align":4425},[4835,4847,4848],{},"£950",[4514,4850,4851],{"align":4425},[4835,4852,4853],{},[1797,4854,4856,4879],{"className":4855},[1800],[1797,4857,4859],{"className":4858},[1804],[1806,4860,4861],{"xmlns":1808},[1810,4862,4863,4876],{},[1813,4864,4865,4868,4870],{},[1881,4866,4867],{},"37.848",[1864,4869,4548],{},[4550,4871,4872,4874],{},[1881,4873,4554],{},[1881,4875,4557],{},[1820,4877,4878],{"encoding":1822},"37.848\\times10^6",[1797,4880,4882,4900],{"className":4881,"ariaHidden":1827},[1826],[1797,4883,4885,4888,4891,4894,4897],{"className":4884},[1831],[1797,4886],{"className":4887,"style":4570},[1835],[1797,4889,4867],{"className":4890},[1840],[1797,4892],{"className":4893,"style":2058},[1898],[1797,4895,4548],{"className":4896},[3429],[1797,4898],{"className":4899,"style":2058},[1898],[1797,4901,4903,4906,4909],{"className":4902},[1831],[1797,4904],{"className":4905,"style":4589},[1835],[1797,4907,1883],{"className":4908},[1840],[1797,4910,4912,4915],{"className":4911},[1840],[1797,4913,4599],{"className":4914},[1840],[1797,4916,4918],{"className":4917},[2062],[1797,4919,4921],{"className":4920},[1968],[1797,4922,4924],{"className":4923},[1973],[1797,4925,4927],{"className":4926,"style":4589},[1977],[1797,4928,4929,4932],{"style":4614},[1797,4930],{"className":4931,"style":2079},[1985],[1797,4933,4935],{"className":4934},[1990,1991,1992,1993],[1797,4936,4557],{"className":4937},[1840,1993],[1793,4939,4940],{},"Portfolio SD is about £6,152. Risk C supplies more than half the expected loss and almost 89% of variance. Risk contribution, not policy count, identifies concentration.",[2399,4942,4944],{"id":4943},"_3-dependence-adds-covariance","3. Dependence adds covariance",[1793,4946,4947],{},"Without independence,",[1797,4949,4951],{"className":4950},[1846],[1797,4952,4954,5042],{"className":4953},[1800],[1797,4955,4957],{"className":4956},[1804],[1806,4958,4959],{"xmlns":1808,"display":1855},[1810,4960,4961,5039],{},[1813,4962,4963,4965,4967,4969,4971,4973,4975,4982,4984,4986,4988,4994,4996,4998,5000,5014,5017,5019,5021,5027,5029,5035,5037],{},[1816,4964,2687],{"mathvariant":2429},[1864,4966,2433],{},[1864,4968,2437],{"stretchy":2436},[1816,4970,1862],{},[1864,4972,2446],{"stretchy":2436},[1864,4974,1866],{},[4976,4977,4978,4980],"munder",{},[1864,4979,1872],{},[1816,4981,1877],{},[1816,4983,2687],{"mathvariant":2429},[1864,4985,2433],{},[1864,4987,2437],{"stretchy":2436},[1887,4989,4990,4992],{},[1816,4991,1891],{},[1816,4993,1877],{},[1864,4995,2446],{"stretchy":2436},[1864,4997,3221],{},[1881,4999,2712],{},[4976,5001,5002,5004],{},[1864,5003,1872],{},[1813,5005,5006,5008,5011],{},[1816,5007,1877],{},[1864,5009,5010],{},"\u003C",[1816,5012,5013],{},"j",[1816,5015,5016],{"mathvariant":2429},"Cov",[1864,5018,2433],{},[1864,5020,2437],{"stretchy":2436},[1887,5022,5023,5025],{},[1816,5024,1891],{},[1816,5026,1877],{},[1864,5028,1896],{"separator":1827},[1887,5030,5031,5033],{},[1816,5032,1891],{},[1816,5034,5013],{},[1864,5036,2446],{"stretchy":2436},[1816,5038,2715],{"mathvariant":2429},[1820,5040,5041],{"encoding":1822},"\\operatorname{Var}(S)\n=\\sum_i\\operatorname{Var}(Y_i)\n+2\\sum_{i\u003Cj}\\operatorname{Cov}(Y_i,Y_j).",[1797,5043,5045,5075,5191],{"className":5044,"ariaHidden":1827},[1826],[1797,5046,5048,5051,5057,5060,5063,5066,5069,5072],{"className":5047},[1831],[1797,5049],{"className":5050,"style":2514},[1835],[1797,5052,5054],{"className":5053},[1963],[1797,5055,2687],{"className":5056},[1840,2521],[1797,5058,2437],{"className":5059},[2525],[1797,5061,1862],{"className":5062,"style":1941},[1840,1841],[1797,5064,2446],{"className":5065},[2570],[1797,5067],{"className":5068,"style":1945},[1898],[1797,5070,1866],{"className":5071},[1949],[1797,5073],{"className":5074,"style":1945},[1898],[1797,5076,5078,5082,5127,5130,5136,5139,5179,5182,5185,5188],{"className":5077},[1831],[1797,5079],{"className":5080,"style":5081},[1835],"height:2.3277em;vertical-align:-1.2777em;",[1797,5083,5085],{"className":5084},[1963,1964],[1797,5086,5088,5119],{"className":5087},[1968,1969],[1797,5089,5091,5116],{"className":5090},[1973],[1797,5092,5095,5106],{"className":5093,"style":5094},[1977],"height:1.05em;",[1797,5096,5097,5100],{"style":1981},[1797,5098],{"className":5099,"style":1986},[1985],[1797,5101,5103],{"className":5102},[1990,1991,1992,1993],[1797,5104,1877],{"className":5105},[1840,1841,1993],[1797,5107,5108,5111],{"style":2008},[1797,5109],{"className":5110,"style":1986},[1985],[1797,5112,5113],{},[1797,5114,1872],{"className":5115},[1963,2017,2018],[1797,5117,2038],{"className":5118},[2037],[1797,5120,5122],{"className":5121},[1973],[1797,5123,5125],{"className":5124,"style":2045},[1977],[1797,5126],{},[1797,5128],{"className":5129,"style":2051},[1898],[1797,5131,5133],{"className":5132},[1963],[1797,5134,2687],{"className":5135},[1840,2521],[1797,5137,2437],{"className":5138},[2525],[1797,5140,5142,5145],{"className":5141},[1840],[1797,5143,1891],{"className":5144,"style":2058},[1840,1841],[1797,5146,5148],{"className":5147},[2062],[1797,5149,5151,5171],{"className":5150},[1968,1969],[1797,5152,5154,5168],{"className":5153},[1973],[1797,5155,5157],{"className":5156,"style":2072},[1977],[1797,5158,5159,5162],{"style":2075},[1797,5160],{"className":5161,"style":2079},[1985],[1797,5163,5165],{"className":5164},[1990,1991,1992,1993],[1797,5166,1877],{"className":5167},[1840,1841,1993],[1797,5169,2038],{"className":5170},[2037],[1797,5172,5174],{"className":5173},[1973],[1797,5175,5177],{"className":5176,"style":2095},[1977],[1797,5178],{},[1797,5180,2446],{"className":5181},[2570],[1797,5183],{"className":5184,"style":2058},[1898],[1797,5186,3221],{"className":5187},[3429],[1797,5189],{"className":5190,"style":2058},[1898],[1797,5192,5194,5198,5201,5204,5259,5262,5269,5272,5312,5315,5318,5359,5362],{"className":5193},[1831],[1797,5195],{"className":5196,"style":5197},[1835],"height:2.4638em;vertical-align:-1.4138em;",[1797,5199,2712],{"className":5200},[1840],[1797,5202],{"className":5203,"style":2051},[1898],[1797,5205,5207],{"className":5206},[1963,1964],[1797,5208,5210,5250],{"className":5209},[1968,1969],[1797,5211,5213,5247],{"className":5212},[1973],[1797,5214,5216,5237],{"className":5215,"style":5094},[1977],[1797,5217,5218,5221],{"style":1981},[1797,5219],{"className":5220,"style":1986},[1985],[1797,5222,5224],{"className":5223},[1990,1991,1992,1993],[1797,5225,5227,5230,5233],{"className":5226},[1840,1993],[1797,5228,1877],{"className":5229},[1840,1841,1993],[1797,5231,5010],{"className":5232},[1949,1993],[1797,5234,5013],{"className":5235,"style":5236},[1840,1841,1993],"margin-right:0.0572em;",[1797,5238,5239,5242],{"style":2008},[1797,5240],{"className":5241,"style":1986},[1985],[1797,5243,5244],{},[1797,5245,1872],{"className":5246},[1963,2017,2018],[1797,5248,2038],{"className":5249},[2037],[1797,5251,5253],{"className":5252},[1973],[1797,5254,5257],{"className":5255,"style":5256},[1977],"height:1.4138em;",[1797,5258],{},[1797,5260],{"className":5261,"style":2051},[1898],[1797,5263,5265],{"className":5264},[1963],[1797,5266,5016],{"className":5267,"style":5268},[1840,2521],"margin-right:0.0139em;",[1797,5270,2437],{"className":5271},[2525],[1797,5273,5275,5278],{"className":5274},[1840],[1797,5276,1891],{"className":5277,"style":2058},[1840,1841],[1797,5279,5281],{"className":5280},[2062],[1797,5282,5284,5304],{"className":5283},[1968,1969],[1797,5285,5287,5301],{"className":5286},[1973],[1797,5288,5290],{"className":5289,"style":2072},[1977],[1797,5291,5292,5295],{"style":2075},[1797,5293],{"className":5294,"style":2079},[1985],[1797,5296,5298],{"className":5297},[1990,1991,1992,1993],[1797,5299,1877],{"className":5300},[1840,1841,1993],[1797,5302,2038],{"className":5303},[2037],[1797,5305,5307],{"className":5306},[1973],[1797,5308,5310],{"className":5309,"style":2095},[1977],[1797,5311],{},[1797,5313,1896],{"className":5314},[2101],[1797,5316],{"className":5317,"style":2051},[1898],[1797,5319,5321,5324],{"className":5320},[1840],[1797,5322,1891],{"className":5323,"style":2058},[1840,1841],[1797,5325,5327],{"className":5326},[2062],[1797,5328,5330,5350],{"className":5329},[1968,1969],[1797,5331,5333,5347],{"className":5332},[1973],[1797,5334,5336],{"className":5335,"style":2072},[1977],[1797,5337,5338,5341],{"style":2075},[1797,5339],{"className":5340,"style":2079},[1985],[1797,5342,5344],{"className":5343},[1990,1991,1992,1993],[1797,5345,5013],{"className":5346,"style":5236},[1840,1841,1993],[1797,5348,2038],{"className":5349},[2037],[1797,5351,5353],{"className":5352},[1973],[1797,5354,5357],{"className":5355,"style":5356},[1977],"height:0.2861em;",[1797,5358],{},[1797,5360,2446],{"className":5361},[2570],[1797,5363,2715],{"className":5364},[1840],[1793,5366,5367],{},"Small positive covariance across many pairs can dominate individual variances. Sources include weather, geography, supply-chain inflation, court decisions, and shared policyholders.",[4406,5369,5371],{"id":5370},"common-shock-construction","Common-shock construction",[1793,5373,2405,5374,5404,5405,5433,5434,5519,5520,5548],{},[1797,5375,5377,5391],{"className":5376},[1800],[1797,5378,5380],{"className":5379},[1804],[1806,5381,5382],{"xmlns":1808},[1810,5383,5384,5389],{},[1813,5385,5386],{},[1816,5387,5388],{},"Z",[1820,5390,5388],{"encoding":1822},[1797,5392,5394],{"className":5393,"ariaHidden":1827},[1826],[1797,5395,5397,5400],{"className":5396},[1831],[1797,5398],{"className":5399,"style":1937},[1835],[1797,5401,5388],{"className":5402,"style":5403},[1840,1841],"margin-right:0.0715em;"," indicate a catastrophe state. Conditional on ",[1797,5406,5408,5421],{"className":5407},[1800],[1797,5409,5411],{"className":5410},[1804],[1806,5412,5413],{"xmlns":1808},[1810,5414,5415,5419],{},[1813,5416,5417],{},[1816,5418,5388],{},[1820,5420,5388],{"encoding":1822},[1797,5422,5424],{"className":5423,"ariaHidden":1827},[1826],[1797,5425,5427,5430],{"className":5426},[1831],[1797,5428],{"className":5429,"style":1937},[1835],[1797,5431,5388],{"className":5432,"style":5403},[1840,1841],", risks may be independent with probabilities ",[1797,5435,5437,5461],{"className":5436},[1800],[1797,5438,5440],{"className":5439},[1804],[1806,5441,5442],{"xmlns":1808},[1810,5443,5444,5458],{},[1813,5445,5446,5452,5454,5456],{},[1887,5447,5448,5450],{},[1816,5449,1793],{},[1816,5451,1877],{},[1864,5453,2437],{"stretchy":2436},[1816,5455,5388],{},[1864,5457,2446],{"stretchy":2436},[1820,5459,5460],{"encoding":1822},"p_i(Z)",[1797,5462,5464],{"className":5463,"ariaHidden":1827},[1826],[1797,5465,5467,5470,5510,5513,5516],{"className":5466},[1831],[1797,5468],{"className":5469,"style":2514},[1835],[1797,5471,5473,5476],{"className":5472},[1840],[1797,5474,1793],{"className":5475},[1840,1841],[1797,5477,5479],{"className":5478},[2062],[1797,5480,5482,5502],{"className":5481},[1968,1969],[1797,5483,5485,5499],{"className":5484},[1973],[1797,5486,5488],{"className":5487,"style":2072},[1977],[1797,5489,5490,5493],{"style":2546},[1797,5491],{"className":5492,"style":2079},[1985],[1797,5494,5496],{"className":5495},[1990,1991,1992,1993],[1797,5497,1877],{"className":5498},[1840,1841,1993],[1797,5500,2038],{"className":5501},[2037],[1797,5503,5505],{"className":5504},[1973],[1797,5506,5508],{"className":5507,"style":2095},[1977],[1797,5509],{},[1797,5511,2437],{"className":5512},[2525],[1797,5514,5388],{"className":5515,"style":5403},[1840,1841],[1797,5517,2446],{"className":5518},[2570],"; marginally they are dependent because the same ",[1797,5521,5523,5536],{"className":5522},[1800],[1797,5524,5526],{"className":5525},[1804],[1806,5527,5528],{"xmlns":1808},[1810,5529,5530,5534],{},[1813,5531,5532],{},[1816,5533,5388],{},[1820,5535,5388],{"encoding":1822},[1797,5537,5539],{"className":5538,"ariaHidden":1827},[1826],[1797,5540,5542,5545],{"className":5541},[1831],[1797,5543],{"className":5544,"style":1937},[1835],[1797,5546,5388],{"className":5547,"style":5403},[1840,1841]," shifts many probabilities. This gives a mechanism and a scenario to validate, rather than inserting one arbitrary correlation coefficient.",[2399,5550,5552],{"id":5551},"_4-more-than-one-claim-per-policy","4. More than one claim per policy",[1793,5554,5555,5556,5625,5626,5699],{},"If a policy can have multiple claims, replace Bernoulli ",[1797,5557,5559,5576],{"className":5558},[1800],[1797,5560,5562],{"className":5561},[1804],[1806,5563,5564],{"xmlns":1808},[1810,5565,5566,5574],{},[1813,5567,5568],{},[1887,5569,5570,5572],{},[1816,5571,1913],{},[1816,5573,1877],{},[1820,5575,2275],{"encoding":1822},[1797,5577,5579],{"className":5578,"ariaHidden":1827},[1826],[1797,5580,5582,5585],{"className":5581},[1831],[1797,5583],{"className":5584,"style":2285},[1835],[1797,5586,5588,5591],{"className":5587},[1840],[1797,5589,1913],{"className":5590,"style":2171},[1840,1841],[1797,5592,5594],{"className":5593},[2062],[1797,5595,5597,5617],{"className":5596},[1968,1969],[1797,5598,5600,5614],{"className":5599},[1973],[1797,5601,5603],{"className":5602,"style":2072},[1977],[1797,5604,5605,5608],{"style":2186},[1797,5606],{"className":5607,"style":2079},[1985],[1797,5609,5611],{"className":5610},[1990,1991,1992,1993],[1797,5612,1877],{"className":5613},[1840,1841,1993],[1797,5615,2038],{"className":5616},[2037],[1797,5618,5620],{"className":5619},[1973],[1797,5621,5623],{"className":5622,"style":2095},[1977],[1797,5624],{}," with count ",[1797,5627,5629,5648],{"className":5628},[1800],[1797,5630,5632],{"className":5631},[1804],[1806,5633,5634],{"xmlns":1808},[1810,5635,5636,5645],{},[1813,5637,5638],{},[1887,5639,5640,5643],{},[1816,5641,5642],{},"N",[1816,5644,1877],{},[1820,5646,5647],{"encoding":1822},"N_i",[1797,5649,5651],{"className":5650,"ariaHidden":1827},[1826],[1797,5652,5654,5657],{"className":5653},[1831],[1797,5655],{"className":5656,"style":2285},[1835],[1797,5658,5660,5664],{"className":5659},[1840],[1797,5661,5642],{"className":5662,"style":5663},[1840,1841],"margin-right:0.109em;",[1797,5665,5667],{"className":5666},[2062],[1797,5668,5670,5691],{"className":5669},[1968,1969],[1797,5671,5673,5688],{"className":5672},[1973],[1797,5674,5676],{"className":5675,"style":2072},[1977],[1797,5677,5679,5682],{"style":5678},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[1797,5680],{"className":5681,"style":2079},[1985],[1797,5683,5685],{"className":5684},[1990,1991,1992,1993],[1797,5686,1877],{"className":5687},[1840,1841,1993],[1797,5689,2038],{"className":5690},[2037],[1797,5692,5694],{"className":5693},[1973],[1797,5695,5697],{"className":5696,"style":2095},[1977],[1797,5698],{},":",[1797,5701,5703],{"className":5702},[1846],[1797,5704,5706,5757],{"className":5705},[1800],[1797,5707,5709],{"className":5708},[1804],[1806,5710,5711],{"xmlns":1808,"display":1855},[1810,5712,5713,5754],{},[1813,5714,5715,5721,5723,5741,5752],{},[1887,5716,5717,5719],{},[1816,5718,1891],{},[1816,5720,1877],{},[1864,5722,1866],{},[1868,5724,5725,5727,5735],{},[1864,5726,1872],{},[1813,5728,5729,5731,5733],{},[1816,5730,5013],{},[1864,5732,1866],{},[1881,5734,1883],{},[1887,5736,5737,5739],{},[1816,5738,5642],{},[1816,5740,1877],{},[1887,5742,5743,5746],{},[1816,5744,5745],{},"X",[1813,5747,5748,5750],{},[1816,5749,1877],{},[1816,5751,5013],{},[1816,5753,2715],{"mathvariant":2429},[1820,5755,5756],{"encoding":1822},"Y_i=\\sum_{j=1}^{N_i}X_{ij}.",[1797,5758,5760,5815],{"className":5759,"ariaHidden":1827},[1826],[1797,5761,5763,5766,5806,5809,5812],{"className":5762},[1831],[1797,5764],{"className":5765,"style":2285},[1835],[1797,5767,5769,5772],{"className":5768},[1840],[1797,5770,1891],{"className":5771,"style":2058},[1840,1841],[1797,5773,5775],{"className":5774},[2062],[1797,5776,5778,5798],{"className":5777},[1968,1969],[1797,5779,5781,5795],{"className":5780},[1973],[1797,5782,5784],{"className":5783,"style":2072},[1977],[1797,5785,5786,5789],{"style":2075},[1797,5787],{"className":5788,"style":2079},[1985],[1797,5790,5792],{"className":5791},[1990,1991,1992,1993],[1797,5793,1877],{"className":5794},[1840,1841,1993],[1797,5796,2038],{"className":5797},[2037],[1797,5799,5801],{"className":5800},[1973],[1797,5802,5804],{"className":5803,"style":2095},[1977],[1797,5805],{},[1797,5807],{"className":5808,"style":1945},[1898],[1797,5810,1866],{"className":5811},[1949],[1797,5813],{"className":5814,"style":1945},[1898],[1797,5816,5818,5822,5933,5936,5980],{"className":5817},[1831],[1797,5819],{"className":5820,"style":5821},[1835],"height:3.2532em;vertical-align:-1.4138em;",[1797,5823,5825],{"className":5824},[1963,1964],[1797,5826,5828,5925],{"className":5827},[1968,1969],[1797,5829,5831,5922],{"className":5830},[1973],[1797,5832,5835,5855,5865],{"className":5833,"style":5834},[1977],"height:1.8394em;",[1797,5836,5837,5840],{"style":1981},[1797,5838],{"className":5839,"style":1986},[1985],[1797,5841,5843],{"className":5842},[1990,1991,1992,1993],[1797,5844,5846,5849,5852],{"className":5845},[1840,1993],[1797,5847,5013],{"className":5848,"style":5236},[1840,1841,1993],[1797,5850,1866],{"className":5851},[1949,1993],[1797,5853,1883],{"className":5854},[1840,1993],[1797,5856,5857,5860],{"style":2008},[1797,5858],{"className":5859,"style":1986},[1985],[1797,5861,5862],{},[1797,5863,1872],{"className":5864},[1963,2017,2018],[1797,5866,5868,5871],{"style":5867},"top:-4.3111em;margin-left:0em;",[1797,5869],{"className":5870,"style":1986},[1985],[1797,5872,5874],{"className":5873},[1990,1991,1992,1993],[1797,5875,5877],{"className":5876},[1840,1993],[1797,5878,5880,5883],{"className":5879},[1840,1993],[1797,5881,5642],{"className":5882,"style":5663},[1840,1841,1993],[1797,5884,5886],{"className":5885},[2062],[1797,5887,5889,5913],{"className":5888},[1968,1969],[1797,5890,5892,5910],{"className":5891},[1973],[1797,5893,5896],{"className":5894,"style":5895},[1977],"height:0.3281em;",[1797,5897,5899,5903],{"style":5898},"top:-2.357em;margin-left:-0.109em;margin-right:0.0714em;",[1797,5900],{"className":5901,"style":5902},[1985],"height:2.5em;",[1797,5904,5907],{"className":5905},[1990,5906,4143,1993],"reset-size3",[1797,5908,1877],{"className":5909},[1840,1841,1993],[1797,5911,2038],{"className":5912},[2037],[1797,5914,5916],{"className":5915},[1973],[1797,5917,5920],{"className":5918,"style":5919},[1977],"height:0.143em;",[1797,5921],{},[1797,5923,2038],{"className":5924},[2037],[1797,5926,5928],{"className":5927},[1973],[1797,5929,5931],{"className":5930,"style":5256},[1977],[1797,5932],{},[1797,5934],{"className":5935,"style":2051},[1898],[1797,5937,5939,5942],{"className":5938},[1840],[1797,5940,5745],{"className":5941,"style":2171},[1840,1841],[1797,5943,5945],{"className":5944},[2062],[1797,5946,5948,5972],{"className":5947},[1968,1969],[1797,5949,5951,5969],{"className":5950},[1973],[1797,5952,5954],{"className":5953,"style":2072},[1977],[1797,5955,5956,5959],{"style":2186},[1797,5957],{"className":5958,"style":2079},[1985],[1797,5960,5962],{"className":5961},[1990,1991,1992,1993],[1797,5963,5965],{"className":5964},[1840,1993],[1797,5966,5968],{"className":5967,"style":5236},[1840,1841,1993],"ij",[1797,5970,2038],{"className":5971},[2037],[1797,5973,5975],{"className":5974},[1973],[1797,5976,5978],{"className":5977,"style":5356},[1977],[1797,5979],{},[1797,5981,2715],{"className":5982},[1840],[1793,5984,5985],{},"The portfolio becomes a sum of policy-level collective models. The individual\u002Fcollective distinction is therefore about retained detail, not mutually exclusive theories.",[2399,5987,5989],{"id":5988},"_5-approximation-by-a-collective-model","5. Approximation by a collective model",[1793,5991,5992,5993,6021,6022,6091,6092,6172],{},"When ",[1797,5994,5996,6009],{"className":5995},[1800],[1797,5997,5999],{"className":5998},[1804],[1806,6000,6001],{"xmlns":1808},[1810,6002,6003,6007],{},[1813,6004,6005],{},[1816,6006,1818],{},[1820,6008,1818],{"encoding":1822},[1797,6010,6012],{"className":6011,"ariaHidden":1827},[1826],[1797,6013,6015,6018],{"className":6014},[1831],[1797,6016],{"className":6017,"style":1836},[1835],[1797,6019,1818],{"className":6020},[1840,1841]," is large and each ",[1797,6023,6025,6042],{"className":6024},[1800],[1797,6026,6028],{"className":6027},[1804],[1806,6029,6030],{"xmlns":1808},[1810,6031,6032,6040],{},[1813,6033,6034],{},[1887,6035,6036,6038],{},[1816,6037,1793],{},[1816,6039,1877],{},[1820,6041,4446],{"encoding":1822},[1797,6043,6045],{"className":6044,"ariaHidden":1827},[1826],[1797,6046,6048,6051],{"className":6047},[1831],[1797,6049],{"className":6050,"style":3090},[1835],[1797,6052,6054,6057],{"className":6053},[1840],[1797,6055,1793],{"className":6056},[1840,1841],[1797,6058,6060],{"className":6059},[2062],[1797,6061,6063,6083],{"className":6062},[1968,1969],[1797,6064,6066,6080],{"className":6065},[1973],[1797,6067,6069],{"className":6068,"style":2072},[1977],[1797,6070,6071,6074],{"style":2546},[1797,6072],{"className":6073,"style":2079},[1985],[1797,6075,6077],{"className":6076},[1990,1991,1992,1993],[1797,6078,1877],{"className":6079},[1840,1841,1993],[1797,6081,2038],{"className":6082},[2037],[1797,6084,6086],{"className":6085},[1973],[1797,6087,6089],{"className":6088,"style":2095},[1977],[1797,6090],{}," is small, total claim count ",[1797,6093,6095,6115],{"className":6094},[1800],[1797,6096,6098],{"className":6097},[1804],[1806,6099,6100],{"xmlns":1808},[1810,6101,6102,6112],{},[1813,6103,6104,6106],{},[1864,6105,1872],{},[1887,6107,6108,6110],{},[1816,6109,1913],{},[1816,6111,1877],{},[1820,6113,6114],{"encoding":1822},"\\sum I_i",[1797,6116,6118],{"className":6117,"ariaHidden":1827},[1826],[1797,6119,6121,6124,6129,6132],{"className":6120},[1831],[1797,6122],{"className":6123,"style":2514},[1835],[1797,6125,1872],{"className":6126,"style":6128},[1963,2017,6127],"small-op","position:relative;top:0em;",[1797,6130],{"className":6131,"style":2051},[1898],[1797,6133,6135,6138],{"className":6134},[1840],[1797,6136,1913],{"className":6137,"style":2171},[1840,1841],[1797,6139,6141],{"className":6140},[2062],[1797,6142,6144,6164],{"className":6143},[1968,1969],[1797,6145,6147,6161],{"className":6146},[1973],[1797,6148,6150],{"className":6149,"style":2072},[1977],[1797,6151,6152,6155],{"style":2186},[1797,6153],{"className":6154,"style":2079},[1985],[1797,6156,6158],{"className":6157},[1990,1991,1992,1993],[1797,6159,1877],{"className":6160},[1840,1841,1993],[1797,6162,2038],{"className":6163},[2037],[1797,6165,6167],{"className":6166},[1973],[1797,6168,6170],{"className":6169,"style":2095},[1977],[1797,6171],{}," may be approximated by Poisson with",[1797,6174,6176],{"className":6175},[1846],[1797,6177,6179,6210],{"className":6178},[1800],[1797,6180,6182],{"className":6181},[1804],[1806,6183,6184],{"xmlns":1808,"display":1855},[1810,6185,6186,6207],{},[1813,6187,6188,6191,6193,6199,6205],{},[1816,6189,6190],{},"λ",[1864,6192,1866],{},[4976,6194,6195,6197],{},[1864,6196,1872],{},[1816,6198,1877],{},[1887,6200,6201,6203],{},[1816,6202,1793],{},[1816,6204,1877],{},[1816,6206,2715],{"mathvariant":2429},[1820,6208,6209],{"encoding":1822},"\\lambda=\\sum_i p_i.",[1797,6211,6213,6232],{"className":6212,"ariaHidden":1827},[1826],[1797,6214,6216,6220,6223,6226,6229],{"className":6215},[1831],[1797,6217],{"className":6218,"style":6219},[1835],"height:0.6944em;",[1797,6221,6190],{"className":6222},[1840,1841],[1797,6224],{"className":6225,"style":1945},[1898],[1797,6227,1866],{"className":6228},[1949],[1797,6230],{"className":6231,"style":1945},[1898],[1797,6233,6235,6238,6282,6285,6325],{"className":6234},[1831],[1797,6236],{"className":6237,"style":5081},[1835],[1797,6239,6241],{"className":6240},[1963,1964],[1797,6242,6244,6274],{"className":6243},[1968,1969],[1797,6245,6247,6271],{"className":6246},[1973],[1797,6248,6250,6261],{"className":6249,"style":5094},[1977],[1797,6251,6252,6255],{"style":1981},[1797,6253],{"className":6254,"style":1986},[1985],[1797,6256,6258],{"className":6257},[1990,1991,1992,1993],[1797,6259,1877],{"className":6260},[1840,1841,1993],[1797,6262,6263,6266],{"style":2008},[1797,6264],{"className":6265,"style":1986},[1985],[1797,6267,6268],{},[1797,6269,1872],{"className":6270},[1963,2017,2018],[1797,6272,2038],{"className":6273},[2037],[1797,6275,6277],{"className":6276},[1973],[1797,6278,6280],{"className":6279,"style":2045},[1977],[1797,6281],{},[1797,6283],{"className":6284,"style":2051},[1898],[1797,6286,6288,6291],{"className":6287},[1840],[1797,6289,1793],{"className":6290},[1840,1841],[1797,6292,6294],{"className":6293},[2062],[1797,6295,6297,6317],{"className":6296},[1968,1969],[1797,6298,6300,6314],{"className":6299},[1973],[1797,6301,6303],{"className":6302,"style":2072},[1977],[1797,6304,6305,6308],{"style":2546},[1797,6306],{"className":6307,"style":2079},[1985],[1797,6309,6311],{"className":6310},[1990,1991,1992,1993],[1797,6312,1877],{"className":6313},[1840,1841,1993],[1797,6315,2038],{"className":6316},[2037],[1797,6318,6320],{"className":6319},[1973],[1797,6321,6323],{"className":6322,"style":2095},[1977],[1797,6324],{},[1797,6326,2715],{"className":6327},[1840],[1793,6329,6330],{},"Severity becomes a probability-weighted mixture of risk severities. This simplifies calculation but can lose concentration and dependence information. Validate the aggregate tail, not only the mean.",[2399,6332,6334],{"id":6333},"_6-portfolio-decisions","6. Portfolio decisions",[1793,6336,6337],{},"The individual model is useful for:",[6339,6340,6341,6345,6348,6351,6354],"ul",{},[6342,6343,6344],"li",{},"pricing\u002Frisk classification with policy covariates;",[6342,6346,6347],{},"concentration by geography or sum insured;",[6342,6349,6350],{},"policy-specific deductibles and limits;",[6342,6352,6353],{},"accumulation and scenario analysis;",[6342,6355,6356],{},"explaining which segments drive capital.",[1793,6358,6359],{},"It is less useful when risk-level identifiers are unstable or the available data only support aggregate counts and severities.",[2399,6361,6363],{"id":6362},"practice","Practice",[6365,6366,6367,6424,6427],"ol",{},[6342,6368,6369,6370,6423],{},"One risk has ",[1797,6371,6373,6392],{"className":6372},[1800],[1797,6374,6376],{"className":6375},[1804],[1806,6377,6378],{"xmlns":1808},[1810,6379,6380,6389],{},[1813,6381,6382,6384,6386],{},[1816,6383,1793],{},[1864,6385,1866],{},[1881,6387,6388],{},"0.1",[1820,6390,6391],{"encoding":1822},"p=0.1",[1797,6393,6395,6413],{"className":6394,"ariaHidden":1827},[1826],[1797,6396,6398,6401,6404,6407,6410],{"className":6397},[1831],[1797,6399],{"className":6400,"style":3090},[1835],[1797,6402,1793],{"className":6403},[1840,1841],[1797,6405],{"className":6406,"style":1945},[1898],[1797,6408,1866],{"className":6409},[1949],[1797,6411],{"className":6412,"style":1945},[1898],[1797,6414,6416,6420],{"className":6415},[1831],[1797,6417],{"className":6418,"style":6419},[1835],"height:0.6444em;",[1797,6421,6388],{"className":6422},[1840],", claim mean £2,000, and claim SD £1,000. Find its expected loss and variance.",[6342,6425,6426],{},"Two risks each have variance 4 and covariance 1. Find variance of their sum.",[6342,6428,6429],{},"Why might a Poisson approximation preserve expected count but understate catastrophe tail?",[6431,6432,6434],"legacy-details",{"title":6433},"Answers",[6365,6435,6436,6717,6830],{},[6342,6437,6438,6439,2715],{},"Mean £200; variance 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can discard shared-event dependence, treating many claims from one catastrophe as independent arrivals.",[1793,6834,6835,6836,2715],{},"Verify both models in the ",[6837,6838,6840],"a",{"href":6839},".\u002F03-examples","Computation Lab",{"title":10,"searchDepth":6842,"depth":6842,"links":6843},2,[6844,6845,6849,6852,6853,6854,6855],{"id":2401,"depth":6842,"text":2402},{"id":3604,"depth":6842,"text":3605,"children":6846},[6847],{"id":4408,"depth":6848,"text":4409},3,{"id":4943,"depth":6842,"text":4944,"children":6850},[6851],{"id":5370,"depth":6848,"text":5371},{"id":5551,"depth":6842,"text":5552},{"id":5988,"depth":6842,"text":5989},{"id":6333,"depth":6842,"text":6334},{"id":6362,"depth":6842,"text":6363},"Aggregate heterogeneous policy risks and expose the effect of common dependence.","md",{"sidebar":6859},{"order":6842},true,{"title":916,"description":6856},"WanJgmgiICGzB69LwKrAyvuZE3pBabrPB599y2QGQKM",[6864,6866],{"title":912,"path":913,"stem":914,"description":6865,"children":-1},"Derive aggregate-loss moments and transforms, then see exactly where independence enters.",{"title":920,"path":921,"stem":922,"description":6867,"children":-1},"Reconcile analytical moments with transparent R and Python simulations.",1785754736705]