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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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positive claim severities, independent of ",[1810,2459,2461,2480],{"className":2460},[1817],[1810,2462,2464],{"className":2463},[1821],[1823,2465,2466],{"xmlns":1825},[1828,2467,2468,2478],{},[1831,2469,2470,2472,2474,2476],{},[1834,2471,1886],{},[1838,2473,1841],{"stretchy":1840},[1834,2475,1844],{},[1838,2477,1847],{"stretchy":1840},[1906,2479,2324],{"encoding":1908},[1810,2481,2483],{"className":2482,"ariaHidden":1903},[1913],[1810,2484,2486,2489,2492,2495,2498],{"className":2485},[1917],[1810,2487],{"className":2488,"style":1922},[1921],[1810,2490,1886],{"className":2491,"style":1928},[1926,1927],[1810,2493,1841],{"className":2494},[1932],[1810,2496,1844],{"className":2497},[1926,1927],[1810,2499,1847],{"className":2500},[1939],[1793,2502,2503],{},"Between claims, surplus rises linearly; at claim times, it jumps downward.",[1793,2505,2506],{},"The time of ruin is",[1810,2508,2510],{"className":2509},[1813],[1810,2511,2513,2568],{"className":2512},[1817],[1810,2514,2516],{"className":2515},[1821],[1823,2517,2518],{"xmlns":1825,"display":1826},[1828,2519,2520,2565],{},[1831,2521,2522,2525,2527,2530,2533,2536,2538,2540,2542,2545,2547,2549,2551,2553,2556,2558,2561],{},[1834,2523,2524],{},"τ",[1838,2526,1850],{},[1834,2528,2529],{},"inf",[1838,2531,2532],{},"⁡",[1838,2534,2535],{"stretchy":1840},"{",[1834,2537,1844],{},[1838,2539,2196],{},[1879,2541,2199],{},[1838,2543,2544],{},":",[1834,2546,1836],{},[1838,2548,1841],{"stretchy":1840},[1834,2550,1844],{},[1838,2552,1847],{"stretchy":1840},[1838,2554,2555],{},"\u003C",[1879,2557,2199],{},[1838,2559,2560],{"stretchy":1840},"}",[1834,2562,2564],{"mathvariant":2563},"normal",".",[1906,2566,2567],{"encoding":1908},"\\tau=\\inf\\{t\\ge0:U(t)\u003C0\\}.",[1810,2569,2571,2591,2620,2638,2665],{"className":2570,"ariaHidden":1903},[1913],[1810,2572,2574,2578,2582,2585,2588],{"className":2573},[1917],[1810,2575],{"className":2576,"style":2577},[1921],"height:0.4306em;",[1810,2579,2524],{"className":2580,"style":2581},[1926,1927],"margin-right:0.1132em;",[1810,2583],{"className":2584,"style":1944},[1943],[1810,2586,1850],{"className":2587},[1948],[1810,2589],{"className":2590,"style":1944},[1943],[1810,2592,2594,2597,2605,2608,2611,2614,2617],{"className":2593},[1917],[1810,2595],{"className":2596,"style":1922},[1921],[1810,2598,2600,2601],{"className":2599},[2005],"in",[1810,2602,2604],{"style":2603},"margin-right:0.0778em;","f",[1810,2606,2535],{"className":2607},[1932],[1810,2609,1844],{"className":2610},[1926,1927],[1810,2612],{"className":2613,"style":1944},[1943],[1810,2615,2196],{"className":2616},[1948],[1810,2618],{"className":2619,"style":1944},[1943],[1810,2621,2623,2626,2629,2632,2635],{"className":2622},[1917],[1810,2624],{"className":2625,"style":2231},[1921],[1810,2627,2199],{"className":2628},[1926],[1810,2630],{"className":2631,"style":1944},[1943],[1810,2633,2544],{"className":2634},[1948],[1810,2636],{"className":2637,"style":1944},[1943],[1810,2639,2641,2644,2647,2650,2653,2656,2659,2662],{"className":2640},[1917],[1810,2642],{"className":2643,"style":1922},[1921],[1810,2645,1836],{"className":2646,"style":1928},[1926,1927],[1810,2648,1841],{"className":2649},[1932],[1810,2651,1844],{"className":2652},[1926,1927],[1810,2654,1847],{"className":2655},[1939],[1810,2657],{"className":2658,"style":1944},[1943],[1810,2660,2555],{"className":2661},[1948],[1810,2663],{"className":2664,"style":1944},[1943],[1810,2666,2668,2671,2674,2677],{"className":2667},[1917],[1810,2669],{"className":2670,"style":1922},[1921],[1810,2672,2199],{"className":2673},[1926],[1810,2675,2560],{"className":2676},[1939],[1810,2678,2564],{"className":2679},[1926],[1793,2681,2682],{},"Two probabilities must not be confused:",[1810,2684,2686],{"className":2685},[1813],[1810,2687,2689,2748],{"className":2688},[1817],[1810,2690,2692],{"className":2691},[1821],[1823,2693,2694],{"xmlns":1825,"display":1826},[1828,2695,2696,2745],{},[1831,2697,2698,2701,2703,2705,2707,2710,2712,2714,2717,2719,2721,2724,2726,2729,2731,2733,2735,2737,2739,2741,2743],{},[1834,2699,2700],{},"ψ",[1838,2702,1841],{"stretchy":1840},[1834,2704,1853],{},[1838,2706,1904],{"separator":1903},[1834,2708,2709],{},"T",[1838,2711,1847],{"stretchy":1840},[1838,2713,1850],{},[1834,2715,2716],{},"P",[1838,2718,1841],{"stretchy":1840},[1834,2720,2524],{},[1838,2722,2723],{},"≤",[1834,2725,2709],{},[1838,2727,2728],{},"∣",[1834,2730,1836],{},[1838,2732,1841],{"stretchy":1840},[1879,2734,2199],{},[1838,2736,1847],{"stretchy":1840},[1838,2738,1850],{},[1834,2740,1853],{},[1838,2742,1847],{"stretchy":1840},[1838,2744,1904],{"separator":1903},[1906,2746,2747],{"encoding":1908},"\\psi(u,T)=P(\\tau\\le T\\mid U(0)=u),",[1810,2749,2751,2789,2813,2831,2858],{"className":2750,"ariaHidden":1903},[1913],[1810,2752,2754,2757,2761,2764,2767,2770,2773,2777,2780,2783,2786],{"className":2753},[1917],[1810,2755],{"className":2756,"style":1922},[1921],[1810,2758,2700],{"className":2759,"style":2760},[1926,1927],"margin-right:0.0359em;",[1810,2762,1841],{"className":2763},[1932],[1810,2765,1853],{"className":2766},[1926,1927],[1810,2768,1904],{"className":2769},[2152],[1810,2771],{"className":2772,"style":2102},[1943],[1810,2774,2709],{"className":2775,"style":2776},[1926,1927],"margin-right:0.1389em;",[1810,2778,1847],{"className":2779},[1939],[1810,2781],{"className":2782,"style":1944},[1943],[1810,2784,1850],{"className":2785},[1948],[1810,2787],{"className":2788,"style":1944},[1943],[1810,2790,2792,2795,2798,2801,2804,2807,2810],{"className":2791},[1917],[1810,2793],{"className":2794,"style":1922},[1921],[1810,2796,2716],{"className":2797,"style":2776},[1926,1927],[1810,2799,1841],{"className":2800},[1932],[1810,2802,2524],{"className":2803,"style":2581},[1926,1927],[1810,2805],{"className":2806,"style":1944},[1943],[1810,2808,2723],{"className":2809},[1948],[1810,2811],{"className":2812,"style":1944},[1943],[1810,2814,2816,2819,2822,2825,2828],{"className":2815},[1917],[1810,2817],{"className":2818,"style":1922},[1921],[1810,2820,2709],{"className":2821,"style":2776},[1926,1927],[1810,2823],{"className":2824,"style":1944},[1943],[1810,2826,2728],{"className":2827},[1948],[1810,2829],{"className":2830,"style":1944},[1943],[1810,2832,2834,2837,2840,2843,2846,2849,2852,2855],{"className":2833},[1917],[1810,2835],{"className":2836,"style":1922},[1921],[1810,2838,1836],{"className":2839,"style":1928},[1926,1927],[1810,2841,1841],{"className":2842},[1932],[1810,2844,2199],{"className":2845},[1926],[1810,2847,1847],{"className":2848},[1939],[1810,2850],{"className":2851,"style":1944},[1943],[1810,2853,1850],{"className":2854},[1948],[1810,2856],{"className":2857,"style":1944},[1943],[1810,2859,2861,2864,2867,2870],{"className":2860},[1917],[1810,2862],{"className":2863,"style":1922},[1921],[1810,2865,1853],{"className":2866},[1926,1927],[1810,2868,1847],{"className":2869},[1939],[1810,2871,1904],{"className":2872},[2152],[1810,2874,2876],{"className":2875},[1813],[1810,2877,2879,2930],{"className":2878},[1817],[1810,2880,2882],{"className":2881},[1821],[1823,2883,2884],{"xmlns":1825,"display":1826},[1828,2885,2886,2927],{},[1831,2887,2888,2890,2892,2894,2896,2898,2900,2902,2904,2906,2909,2911,2913,2915,2917,2919,2921,2923,2925],{},[1834,2889,2700],{},[1838,2891,1841],{"stretchy":1840},[1834,2893,1853],{},[1838,2895,1847],{"stretchy":1840},[1838,2897,1850],{},[1834,2899,2716],{},[1838,2901,1841],{"stretchy":1840},[1834,2903,2524],{},[1838,2905,2555],{},[1834,2907,2908],{"mathvariant":2563},"∞",[1838,2910,2728],{},[1834,2912,1836],{},[1838,2914,1841],{"stretchy":1840},[1879,2916,2199],{},[1838,2918,1847],{"stretchy":1840},[1838,2920,1850],{},[1834,2922,1853],{},[1838,2924,1847],{"stretchy":1840},[1834,2926,2564],{"mathvariant":2563},[1906,2928,2929],{"encoding":1908},"\\psi(u)=P(\\tau\u003C\\infty\\mid U(0)=u).",[1810,2931,2933,2960,2984,3002,3029],{"className":2932,"ariaHidden":1903},[1913],[1810,2934,2936,2939,2942,2945,2948,2951,2954,2957],{"className":2935},[1917],[1810,2937],{"className":2938,"style":1922},[1921],[1810,2940,2700],{"className":2941,"style":2760},[1926,1927],[1810,2943,1841],{"className":2944},[1932],[1810,2946,1853],{"className":2947},[1926,1927],[1810,2949,1847],{"className":2950},[1939],[1810,2952],{"className":2953,"style":1944},[1943],[1810,2955,1850],{"className":2956},[1948],[1810,2958],{"className":2959,"style":1944},[1943],[1810,2961,2963,2966,2969,2972,2975,2978,2981],{"className":2962},[1917],[1810,2964],{"className":2965,"style":1922},[1921],[1810,2967,2716],{"className":2968,"style":2776},[1926,1927],[1810,2970,1841],{"className":2971},[1932],[1810,2973,2524],{"className":2974,"style":2581},[1926,1927],[1810,2976],{"className":2977,"style":1944},[1943],[1810,2979,2555],{"className":2980},[1948],[1810,2982],{"className":2983,"style":1944},[1943],[1810,2985,2987,2990,2993,2996,2999],{"className":2986},[1917],[1810,2988],{"className":2989,"style":1922},[1921],[1810,2991,2908],{"className":2992},[1926],[1810,2994],{"className":2995,"style":1944},[1943],[1810,2997,2728],{"className":2998},[1948],[1810,3000],{"className":3001,"style":1944},[1943],[1810,3003,3005,3008,3011,3014,3017,3020,3023,3026],{"className":3004},[1917],[1810,3006],{"className":3007,"style":1922},[1921],[1810,3009,1836],{"className":3010,"style":1928},[1926,1927],[1810,3012,1841],{"className":3013},[1932],[1810,3015,2199],{"className":3016},[1926],[1810,3018,1847],{"className":3019},[1939],[1810,3021],{"className":3022,"style":1944},[1943],[1810,3024,1850],{"className":3025},[1948],[1810,3027],{"className":3028,"style":1944},[1943],[1810,3030,3032,3035,3038,3041],{"className":3031},[1917],[1810,3033],{"className":3034,"style":1922},[1921],[1810,3036,1853],{"className":3037},[1926,1927],[1810,3039,1847],{"className":3040},[1939],[1810,3042,2564],{"className":3043},[1926],[1793,3045,3046],{},"Finite-horizon ruin is usually smaller than ultimate ruin and often closer to an operational planning question.",[1802,3048,3050],{"id":3049},"_2-net-profit-condition","2. Net profit condition",[1793,3052,3053,3054,3107],{},"Expected claim outflow per unit time is ",[1810,3055,3057,3082],{"className":3056},[1817],[1810,3058,3060],{"className":3059},[1821],[1823,3061,3062],{"xmlns":1825},[1828,3063,3064,3079],{},[1831,3065,3066,3068,3071,3074,3076],{},[1834,3067,2363],{},[1834,3069,3070],{},"E",[1838,3072,3073],{"stretchy":1840},"[",[1834,3075,1898],{},[1838,3077,3078],{"stretchy":1840},"]",[1906,3080,3081],{"encoding":1908},"\\lambda E[X]",[1810,3083,3085],{"className":3084,"ariaHidden":1903},[1913],[1810,3086,3088,3091,3094,3098,3101,3104],{"className":3087},[1917],[1810,3089],{"className":3090,"style":1922},[1921],[1810,3092,2363],{"className":3093},[1926,1927],[1810,3095,3070],{"className":3096,"style":3097},[1926,1927],"margin-right:0.0576em;",[1810,3099,3073],{"className":3100},[1932],[1810,3102,1898],{"className":3103,"style":2109},[1926,1927],[1810,3105,3078],{"className":3106},[1939],". For surplus to have positive drift,",[1810,3109,3111],{"className":3110},[1813],[1810,3112,3114,3142],{"className":3113},[1817],[1810,3115,3117],{"className":3116},[1821],[1823,3118,3119],{"xmlns":1825,"display":1826},[1828,3120,3121,3139],{},[1831,3122,3123,3125,3127,3129,3131,3133,3135,3137],{},[1834,3124,1859],{},[1838,3126,2258],{},[1834,3128,2363],{},[1834,3130,3070],{},[1838,3132,3073],{"stretchy":1840},[1834,3134,1898],{},[1838,3136,3078],{"stretchy":1840},[1834,3138,2564],{"mathvariant":2563},[1906,3140,3141],{"encoding":1908},"c>\\lambda E[X].",[1810,3143,3145,3163],{"className":3144,"ariaHidden":1903},[1913],[1810,3146,3148,3151,3154,3157,3160],{"className":3147},[1917],[1810,3149],{"className":3150,"style":2273},[1921],[1810,3152,1859],{"className":3153},[1926,1927],[1810,3155],{"className":3156,"style":1944},[1943],[1810,3158,2258],{"className":3159},[1948],[1810,3161],{"className":3162,"style":1944},[1943],[1810,3164,3166,3169,3172,3175,3178,3181,3184],{"className":3165},[1917],[1810,3167],{"className":3168,"style":1922},[1921],[1810,3170,2363],{"className":3171},[1926,1927],[1810,3173,3070],{"className":3174,"style":3097},[1926,1927],[1810,3176,3073],{"className":3177},[1932],[1810,3179,1898],{"className":3180,"style":2109},[1926,1927],[1810,3182,3078],{"className":3183},[1939],[1810,3185,2564],{"className":3186},[1926],[1793,3188,3189],{},"Write",[1810,3191,3193],{"className":3192},[1813],[1810,3194,3196,3235],{"className":3195},[1817],[1810,3197,3199],{"className":3198},[1821],[1823,3200,3201],{"xmlns":1825,"display":1826},[1828,3202,3203,3232],{},[1831,3204,3205,3207,3209,3211,3213,3215,3218,3220,3222,3224,3226,3228,3230],{},[1834,3206,1859],{},[1838,3208,1850],{},[1838,3210,1841],{"stretchy":1840},[1879,3212,1881],{},[1838,3214,1856],{},[1834,3216,3217],{},"θ",[1838,3219,1847],{"stretchy":1840},[1834,3221,2363],{},[1834,3223,3070],{},[1838,3225,3073],{"stretchy":1840},[1834,3227,1898],{},[1838,3229,3078],{"stretchy":1840},[1838,3231,1904],{"separator":1903},[1906,3233,3234],{"encoding":1908},"c=(1+\\theta)\\lambda E[X],",[1810,3236,3238,3256,3277],{"className":3237,"ariaHidden":1903},[1913],[1810,3239,3241,3244,3247,3250,3253],{"className":3240},[1917],[1810,3242],{"className":3243,"style":2577},[1921],[1810,3245,1859],{"className":3246},[1926,1927],[1810,3248],{"className":3249,"style":1944},[1943],[1810,3251,1850],{"className":3252},[1948],[1810,3254],{"className":3255,"style":1944},[1943],[1810,3257,3259,3262,3265,3268,3271,3274],{"className":3258},[1917],[1810,3260],{"className":3261,"style":1922},[1921],[1810,3263,1841],{"className":3264},[1932],[1810,3266,1881],{"className":3267},[1926],[1810,3269],{"className":3270,"style":1965},[1943],[1810,3272,1856],{"className":3273},[1969],[1810,3275],{"className":3276,"style":1965},[1943],[1810,3278,3280,3283,3287,3290,3293,3296,3299,3302,3305],{"className":3279},[1917],[1810,3281],{"className":3282,"style":1922},[1921],[1810,3284,3217],{"className":3285,"style":3286},[1926,1927],"margin-right:0.0278em;",[1810,3288,1847],{"className":3289},[1939],[1810,3291,2363],{"className":3292},[1926,1927],[1810,3294,3070],{"className":3295,"style":3097},[1926,1927],[1810,3297,3073],{"className":3298},[1932],[1810,3300,1898],{"className":3301,"style":2109},[1926,1927],[1810,3303,3078],{"className":3304},[1939],[1810,3306,1904],{"className":3307},[2152],[1793,3309,3310,3311,3363],{},"where ",[1810,3312,3314,3332],{"className":3313},[1817],[1810,3315,3317],{"className":3316},[1821],[1823,3318,3319],{"xmlns":1825},[1828,3320,3321,3329],{},[1831,3322,3323,3325,3327],{},[1834,3324,3217],{},[1838,3326,2258],{},[1879,3328,2199],{},[1906,3330,3331],{"encoding":1908},"\\theta>0",[1810,3333,3335,3354],{"className":3334,"ariaHidden":1903},[1913],[1810,3336,3338,3342,3345,3348,3351],{"className":3337},[1917],[1810,3339],{"className":3340,"style":3341},[1921],"height:0.7335em;vertical-align:-0.0391em;",[1810,3343,3217],{"className":3344,"style":3286},[1926,1927],[1810,3346],{"className":3347,"style":1944},[1943],[1810,3349,2258],{"className":3350},[1948],[1810,3352],{"className":3353,"style":1944},[1943],[1810,3355,3357,3360],{"className":3356},[1917],[1810,3358],{"className":3359,"style":2231},[1921],[1810,3361,2199],{"className":3362},[1926]," is premium loading in this simplified model.",[3365,3366,3368],"warning",{"title":3367},"Positive drift is not safety","Even when expected premium exceeds expected claims, one early large loss can cause ruin. The condition supports a non-trivial ultimate result; it does not imply a small finite-horizon probability.",[1802,3370,3372],{"id":3371},"_3-adjustment-coefficient-and-lundberg-bound","3. Adjustment coefficient and Lundberg bound",[1793,3374,3375,3376,3430],{},"If the severity MGF exists for positive arguments, the adjustment coefficient ",[1810,3377,3379,3398],{"className":3378},[1817],[1810,3380,3382],{"className":3381},[1821],[1823,3383,3384],{"xmlns":1825},[1828,3385,3386,3395],{},[1831,3387,3388,3391,3393],{},[1834,3389,3390],{},"R",[1838,3392,2258],{},[1879,3394,2199],{},[1906,3396,3397],{"encoding":1908},"R>0",[1810,3399,3401,3421],{"className":3400,"ariaHidden":1903},[1913],[1810,3402,3404,3408,3412,3415,3418],{"className":3403},[1917],[1810,3405],{"className":3406,"style":3407},[1921],"height:0.7224em;vertical-align:-0.0391em;",[1810,3409,3390],{"className":3410,"style":3411},[1926,1927],"margin-right:0.0077em;",[1810,3413],{"className":3414,"style":1944},[1943],[1810,3416,2258],{"className":3417},[1948],[1810,3419],{"className":3420,"style":1944},[1943],[1810,3422,3424,3427],{"className":3423},[1917],[1810,3425],{"className":3426,"style":2231},[1921],[1810,3428,2199],{"className":3429},[1926]," solves",[1810,3432,3434],{"className":3433},[1813],[1810,3435,3437,3480],{"className":3436},[1817],[1810,3438,3440],{"className":3439},[1821],[1823,3441,3442],{"xmlns":1825,"display":1826},[1828,3443,3444,3477],{},[1831,3445,3446,3448,3450,3457,3459,3461,3463,3465,3467,3469,3471,3473,3475],{},[1834,3447,2363],{},[1838,3449,2535],{"stretchy":1840},[1894,3451,3452,3455],{},[1834,3453,3454],{},"M",[1834,3456,1898],{},[1838,3458,1841],{"stretchy":1840},[1834,3460,3390],{},[1838,3462,1847],{"stretchy":1840},[1838,3464,1864],{},[1879,3466,1881],{},[1838,3468,2560],{"stretchy":1840},[1838,3470,1850],{},[1834,3472,1859],{},[1834,3474,3390],{},[1834,3476,2564],{"mathvariant":2563},[1906,3478,3479],{"encoding":1908},"\\lambda\\{M_X(R)-1\\}=cR.",[1810,3481,3483,3555,3576],{"className":3482,"ariaHidden":1903},[1913],[1810,3484,3486,3489,3492,3495,3537,3540,3543,3546,3549,3552],{"className":3485},[1917],[1810,3487],{"className":3488,"style":1922},[1921],[1810,3490,2363],{"className":3491},[1926,1927],[1810,3493,2535],{"className":3494},[1932],[1810,3496,3498,3501],{"className":3497},[1926],[1810,3499,3454],{"className":3500,"style":1928},[1926,1927],[1810,3502,3504],{"className":3503},[2113],[1810,3505,3507,3529],{"className":3506},[2010,2011],[1810,3508,3510,3526],{"className":3509},[2015],[1810,3511,3514],{"className":3512,"style":3513},[2019],"height:0.3283em;",[1810,3515,3517,3520],{"style":3516},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[1810,3518],{"className":3519,"style":2130},[2027],[1810,3521,3523],{"className":3522},[2032,2033,2034,2035],[1810,3524,1898],{"className":3525,"style":2109},[1926,1927,2035],[1810,3527,2089],{"className":3528},[2088],[1810,3530,3532],{"className":3531},[2015],[1810,3533,3535],{"className":3534,"style":2146},[2019],[1810,3536],{},[1810,3538,1841],{"className":3539},[1932],[1810,3541,3390],{"className":3542,"style":3411},[1926,1927],[1810,3544,1847],{"className":3545},[1939],[1810,3547],{"className":3548,"style":1965},[1943],[1810,3550,1864],{"className":3551},[1969],[1810,3553],{"className":3554,"style":1965},[1943],[1810,3556,3558,3561,3564,3567,3570,3573],{"className":3557},[1917],[1810,3559],{"className":3560,"style":1922},[1921],[1810,3562,1881],{"className":3563},[1926],[1810,3565,2560],{"className":3566},[1939],[1810,3568],{"className":3569,"style":1944},[1943],[1810,3571,1850],{"className":3572},[1948],[1810,3574],{"className":3575,"style":1944},[1943],[1810,3577,3579,3583,3586,3589],{"className":3578},[1917],[1810,3580],{"className":3581,"style":3582},[1921],"height:0.6833em;",[1810,3584,1859],{"className":3585},[1926,1927],[1810,3587,3390],{"className":3588,"style":3411},[1926,1927],[1810,3590,2564],{"className":3591},[1926],[1793,3593,3594],{},"Under the classical assumptions,",[1810,3596,3598],{"className":3597},[1813],[1810,3599,3601,3639],{"className":3600},[1817],[1810,3602,3604],{"className":3603},[1821],[1823,3605,3606],{"xmlns":1825,"display":1826},[1828,3607,3608,3636],{},[1831,3609,3610,3612,3614,3616,3618,3620,3634],{},[1834,3611,2700],{},[1838,3613,1841],{"stretchy":1840},[1834,3615,1853],{},[1838,3617,1847],{"stretchy":1840},[1838,3619,2723],{},[3621,3622,3623,3626],"msup",{},[1834,3624,3625],{},"e",[1831,3627,3628,3630,3632],{},[1838,3629,1864],{},[1834,3631,3390],{},[1834,3633,1853],{},[1834,3635,2564],{"mathvariant":2563},[1906,3637,3638],{"encoding":1908},"\\psi(u)\\le e^{-Ru}.",[1810,3640,3642,3669],{"className":3641,"ariaHidden":1903},[1913],[1810,3643,3645,3648,3651,3654,3657,3660,3663,3666],{"className":3644},[1917],[1810,3646],{"className":3647,"style":1922},[1921],[1810,3649,2700],{"className":3650,"style":2760},[1926,1927],[1810,3652,1841],{"className":3653},[1932],[1810,3655,1853],{"className":3656},[1926,1927],[1810,3658,1847],{"className":3659},[1939],[1810,3661],{"className":3662,"style":1944},[1943],[1810,3664,2723],{"className":3665},[1948],[1810,3667],{"className":3668,"style":1944},[1943],[1810,3670,3672,3676,3715],{"className":3671},[1917],[1810,3673],{"className":3674,"style":3675},[1921],"height:0.8913em;",[1810,3677,3679,3682],{"className":3678},[1926],[1810,3680,3625],{"className":3681},[1926,1927],[1810,3683,3685],{"className":3684},[2113],[1810,3686,3688],{"className":3687},[2010],[1810,3689,3691],{"className":3690},[2015],[1810,3692,3694],{"className":3693,"style":3675},[2019],[1810,3695,3697,3700],{"style":3696},"top:-3.113em;margin-right:0.05em;",[1810,3698],{"className":3699,"style":2130},[2027],[1810,3701,3703],{"className":3702},[2032,2033,2034,2035],[1810,3704,3706,3709,3712],{"className":3705},[1926,2035],[1810,3707,1864],{"className":3708},[1926,2035],[1810,3710,3390],{"className":3711,"style":3411},[1926,1927,2035],[1810,3713,1853],{"className":3714},[1926,1927,2035],[1810,3716,2564],{"className":3717},[1926],[1793,3719,3720,3748,3749,3777],{},[1810,3721,3723,3736],{"className":3722},[1817],[1810,3724,3726],{"className":3725},[1821],[1823,3727,3728],{"xmlns":1825},[1828,3729,3730,3734],{},[1831,3731,3732],{},[1834,3733,3390],{},[1906,3735,3390],{"encoding":1908},[1810,3737,3739],{"className":3738,"ariaHidden":1903},[1913],[1810,3740,3742,3745],{"className":3741},[1917],[1810,3743],{"className":3744,"style":3582},[1921],[1810,3746,3390],{"className":3747,"style":3411},[1926,1927]," summarises premium loading and severity-tail behaviour. A larger ",[1810,3750,3752,3765],{"className":3751},[1817],[1810,3753,3755],{"className":3754},[1821],[1823,3756,3757],{"xmlns":1825},[1828,3758,3759,3763],{},[1831,3760,3761],{},[1834,3762,3390],{},[1906,3764,3390],{"encoding":1908},[1810,3766,3768],{"className":3767,"ariaHidden":1903},[1913],[1810,3769,3771,3774],{"className":3770},[1917],[1810,3772],{"className":3773,"style":3582},[1921],[1810,3775,3390],{"className":3776,"style":3411},[1926,1927]," gives faster exponential decay of the bound with initial capital.",[1793,3779,3780,3781,3833],{},"This tool can fail for heavy-tailed severities whose MGF is infinite for all ",[1810,3782,3784,3802],{"className":3783},[1817],[1810,3785,3787],{"className":3786},[1821],[1823,3788,3789],{"xmlns":1825},[1828,3790,3791,3799],{},[1831,3792,3793,3795,3797],{},[1834,3794,1844],{},[1838,3796,2258],{},[1879,3798,2199],{},[1906,3800,3801],{"encoding":1908},"t>0",[1810,3803,3805,3824],{"className":3804,"ariaHidden":1903},[1913],[1810,3806,3808,3812,3815,3818,3821],{"className":3807},[1917],[1810,3809],{"className":3810,"style":3811},[1921],"height:0.6542em;vertical-align:-0.0391em;",[1810,3813,1844],{"className":3814},[1926,1927],[1810,3816],{"className":3817,"style":1944},[1943],[1810,3819,2258],{"className":3820},[1948],[1810,3822],{"className":3823,"style":1944},[1943],[1810,3825,3827,3830],{"className":3826},[1917],[1810,3828],{"className":3829,"style":2231},[1921],[1810,3831,2199],{"className":3832},[1926],". Failure of the equation is information about the model, not a numerical inconvenience to ignore.",[1802,3835,3837],{"id":3836},"_4-exact-exponential-example","4. Exact Exponential example",[1793,3839,3840],{},"Suppose:",[1810,3842,3844],{"className":3843},[1813],[1810,3845,3847,3907],{"className":3846},[1817],[1810,3848,3850],{"className":3849},[1821],[1823,3851,3852],{"xmlns":1825,"display":1826},[1828,3853,3854,3904],{},[1831,3855,3856,3858,3860,3862,3864,3867,3870,3872,3874,3877,3879,3881,3883,3886,3888,3890,3892,3894,3896,3899,3902],{},[1834,3857,1886],{},[1838,3859,1841],{"stretchy":1840},[1834,3861,1844],{},[1838,3863,1847],{"stretchy":1840},[1838,3865,3866],{},"∼",[1834,3868,3869],{"mathvariant":2563},"Poisson",[1838,3871,2532],{},[1838,3873,1841],{"stretchy":1840},[1879,3875,3876],{},"40",[1834,3878,1844],{},[1838,3880,1847],{"stretchy":1840},[1838,3882,1904],{"separator":1903},[1943,3884],{"width":3885},"2em",[1834,3887,3070],{},[1838,3889,3073],{"stretchy":1840},[1834,3891,1898],{},[1838,3893,3078],{"stretchy":1840},[1838,3895,1850],{},[1834,3897,3898],{"mathvariant":2563},"£",[1879,3900,3901],{},"5,000",[1838,3903,1904],{"separator":1903},[1906,3905,3906],{"encoding":1908},"N(t)\\sim\\operatorname{Poisson}(40t),\n\\qquad E[X]=£5{,}000,",[1810,3908,3910,3937,3993],{"className":3909,"ariaHidden":1903},[1913],[1810,3911,3913,3916,3919,3922,3925,3928,3931,3934],{"className":3912},[1917],[1810,3914],{"className":3915,"style":1922},[1921],[1810,3917,1886],{"className":3918,"style":1928},[1926,1927],[1810,3920,1841],{"className":3921},[1932],[1810,3923,1844],{"className":3924},[1926,1927],[1810,3926,1847],{"className":3927},[1939],[1810,3929],{"className":3930,"style":1944},[1943],[1810,3932,3866],{"className":3933},[1948],[1810,3935],{"className":3936,"style":1944},[1943],[1810,3938,3940,3943,3950,3953,3956,3959,3962,3965,3969,3972,3975,3978,3981,3984,3987,3990],{"className":3939},[1917],[1810,3941],{"className":3942,"style":1922},[1921],[1810,3944,3946],{"className":3945},[2005],[1810,3947,3869],{"className":3948},[1926,3949],"mathrm",[1810,3951,1841],{"className":3952},[1932],[1810,3954,3876],{"className":3955},[1926],[1810,3957,1844],{"className":3958},[1926,1927],[1810,3960,1847],{"className":3961},[1939],[1810,3963,1904],{"className":3964},[2152],[1810,3966],{"className":3967,"style":3968},[1943],"margin-right:2em;",[1810,3970],{"className":3971,"style":2102},[1943],[1810,3973,3070],{"className":3974,"style":3097},[1926,1927],[1810,3976,3073],{"className":3977},[1932],[1810,3979,1898],{"className":3980,"style":2109},[1926,1927],[1810,3982,3078],{"className":3983},[1939],[1810,3985],{"className":3986,"style":1944},[1943],[1810,3988,1850],{"className":3989},[1948],[1810,3991],{"className":3992,"style":1944},[1943],[1810,3994,3996,4000,4004,4010,4014],{"className":3995},[1917],[1810,3997],{"className":3998,"style":3999},[1921],"height:0.8889em;vertical-align:-0.1944em;",[1810,4001,4003],{"className":4002},[1926],"£5",[1810,4005,4007],{"className":4006},[1926],[1810,4008,1904],{"className":4009},[2152],[1810,4011,4013],{"className":4012},[1926],"000",[1810,4015,1904],{"className":4016},[2152],[1793,4018,4019],{},"and premium loading is 20%, so",[1810,4021,4023],{"className":4022},[1813],[1810,4024,4026,4070],{"className":4025},[1817],[1810,4027,4029],{"className":4028},[1821],[1823,4030,4031],{"xmlns":1825,"display":1826},[1828,4032,4033,4067],{},[1831,4034,4035,4037,4039,4042,4044,4046,4048,4050,4052,4054,4056,4058,4061,4065],{},[1834,4036,1859],{},[1838,4038,1850],{},[1879,4040,4041],{},"1.2",[1838,4043,1841],{"stretchy":1840},[1879,4045,3876],{},[1838,4047,1847],{"stretchy":1840},[1838,4049,1841],{"stretchy":1840},[1879,4051,3901],{},[1838,4053,1847],{"stretchy":1840},[1838,4055,1850],{},[1834,4057,3898],{"mathvariant":2563},[1879,4059,4060],{},"240,000",[4062,4063,4064],"mtext",{}," per year",[1834,4066,2564],{"mathvariant":2563},[1906,4068,4069],{"encoding":1908},"c=1.2(40)(5{,}000)=£240{,}000\\text{ per year}.",[1810,4071,4073,4091,4137],{"className":4072,"ariaHidden":1903},[1913],[1810,4074,4076,4079,4082,4085,4088],{"className":4075},[1917],[1810,4077],{"className":4078,"style":2577},[1921],[1810,4080,1859],{"className":4081},[1926,1927],[1810,4083],{"className":4084,"style":1944},[1943],[1810,4086,1850],{"className":4087},[1948],[1810,4089],{"className":4090,"style":1944},[1943],[1810,4092,4094,4097,4100,4103,4106,4109,4112,4116,4122,4125,4128,4131,4134],{"className":4093},[1917],[1810,4095],{"className":4096,"style":1922},[1921],[1810,4098,4041],{"className":4099},[1926],[1810,4101,1841],{"className":4102},[1932],[1810,4104,3876],{"className":4105},[1926],[1810,4107,1847],{"className":4108},[1939],[1810,4110,1841],{"className":4111},[1932],[1810,4113,4115],{"className":4114},[1926],"5",[1810,4117,4119],{"className":4118},[1926],[1810,4120,1904],{"className":4121},[2152],[1810,4123,4013],{"className":4124},[1926],[1810,4126,1847],{"className":4127},[1939],[1810,4129],{"className":4130,"style":1944},[1943],[1810,4132,1850],{"className":4133},[1948],[1810,4135],{"className":4136,"style":1944},[1943],[1810,4138,4140,4143,4147,4153,4156,4163],{"className":4139},[1917],[1810,4141],{"className":4142,"style":3999},[1921],[1810,4144,4146],{"className":4145},[1926],"£240",[1810,4148,4150],{"className":4149},[1926],[1810,4151,1904],{"className":4152},[2152],[1810,4154,4013],{"className":4155},[1926],[1810,4157,4160],{"className":4158},[1926,4159],"text",[1810,4161,4064],{"className":4162},[1926],[1810,4164,2564],{"className":4165},[1926],[1793,4167,4168,4169,4237],{},"For Exponential severity with rate ",[1810,4170,4172,4196],{"className":4171},[1817],[1810,4173,4175],{"className":4174},[1821],[1823,4176,4177],{"xmlns":1825},[1828,4178,4179,4193],{},[1831,4180,4181,4184,4186,4188,4191],{},[1834,4182,4183],{},"β",[1838,4185,1850],{},[1879,4187,1881],{},[1834,4189,4190],{"mathvariant":2563},"\u002F",[1879,4192,3901],{},[1906,4194,4195],{"encoding":1908},"\\beta=1\u002F5{,}000",[1810,4197,4199,4218],{"className":4198,"ariaHidden":1903},[1913],[1810,4200,4202,4205,4209,4212,4215],{"className":4201},[1917],[1810,4203],{"className":4204,"style":3999},[1921],[1810,4206,4183],{"className":4207,"style":4208},[1926,1927],"margin-right:0.0528em;",[1810,4210],{"className":4211,"style":1944},[1943],[1810,4213,1850],{"className":4214},[1948],[1810,4216],{"className":4217,"style":1944},[1943],[1810,4219,4221,4224,4228,4234],{"className":4220},[1917],[1810,4222],{"className":4223,"style":1922},[1921],[1810,4225,4227],{"className":4226},[1926],"1\u002F5",[1810,4229,4231],{"className":4230},[1926],[1810,4232,1904],{"className":4233},[2152],[1810,4235,4013],{"className":4236},[1926],", the adjustment coefficient is",[1810,4239,4241],{"className":4240},[1813],[1810,4242,4244,4289],{"className":4243},[1817],[1810,4245,4247],{"className":4246},[1821],[1823,4248,4249],{"xmlns":1825,"display":1826},[1828,4250,4251,4286],{},[1831,4252,4253,4255,4257,4259,4261,4268,4270,4273,4275,4281,4283],{},[1834,4254,3390],{},[1838,4256,1850],{},[1834,4258,4183],{},[1838,4260,1864],{},[4262,4263,4264,4266],"mfrac",{},[1834,4265,2363],{},[1834,4267,1859],{},[1838,4269,1850],{},[1879,4271,4272],{},"0.0002",[1838,4274,1864],{},[4262,4276,4277,4279],{},[1879,4278,3876],{},[1879,4280,4060],{},[1838,4282,1850],{},[1879,4284,4285],{},"0.00003333.",[1906,4287,4288],{"encoding":1908},"R=\\beta-\\frac{\\lambda}{c}\n=0.0002-\\frac{40}{240{,}000}\n=0.00003333.",[1810,4290,4292,4310,4328,4415,4434,4524],{"className":4291,"ariaHidden":1903},[1913],[1810,4293,4295,4298,4301,4304,4307],{"className":4294},[1917],[1810,4296],{"className":4297,"style":3582},[1921],[1810,4299,3390],{"className":4300,"style":3411},[1926,1927],[1810,4302],{"className":4303,"style":1944},[1943],[1810,4305,1850],{"className":4306},[1948],[1810,4308],{"className":4309,"style":1944},[1943],[1810,4311,4313,4316,4319,4322,4325],{"className":4312},[1917],[1810,4314],{"className":4315,"style":3999},[1921],[1810,4317,4183],{"className":4318,"style":4208},[1926,1927],[1810,4320],{"className":4321,"style":1965},[1943],[1810,4323,1864],{"className":4324},[1969],[1810,4326],{"className":4327,"style":1965},[1943],[1810,4329,4331,4335,4406,4409,4412],{"className":4330},[1917],[1810,4332],{"className":4333,"style":4334},[1921],"height:2.0574em;vertical-align:-0.686em;",[1810,4336,4338,4342,4403],{"className":4337},[1926],[1810,4339],{"className":4340},[1932,4341],"nulldelimiter",[1810,4343,4345],{"className":4344},[4262],[1810,4346,4348,4394],{"className":4347},[2010,2011],[1810,4349,4351,4391],{"className":4350},[2015],[1810,4352,4355,4368,4379],{"className":4353,"style":4354},[2019],"height:1.3714em;",[1810,4356,4358,4362],{"style":4357},"top:-2.314em;",[1810,4359],{"className":4360,"style":4361},[2027],"height:3em;",[1810,4363,4365],{"className":4364},[1926],[1810,4366,1859],{"className":4367},[1926,1927],[1810,4369,4371,4374],{"style":4370},"top:-3.23em;",[1810,4372],{"className":4373,"style":4361},[2027],[1810,4375],{"className":4376,"style":4378},[4377],"frac-line","border-bottom-width:0.04em;",[1810,4380,4382,4385],{"style":4381},"top:-3.677em;",[1810,4383],{"className":4384,"style":4361},[2027],[1810,4386,4388],{"className":4387},[1926],[1810,4389,2363],{"className":4390},[1926,1927],[1810,4392,2089],{"className":4393},[2088],[1810,4395,4397],{"className":4396},[2015],[1810,4398,4401],{"className":4399,"style":4400},[2019],"height:0.686em;",[1810,4402],{},[1810,4404],{"className":4405},[1939,4341],[1810,4407],{"className":4408,"style":1944},[1943],[1810,4410,1850],{"className":4411},[1948],[1810,4413],{"className":4414,"style":1944},[1943],[1810,4416,4418,4422,4425,4428,4431],{"className":4417},[1917],[1810,4419],{"className":4420,"style":4421},[1921],"height:0.7278em;vertical-align:-0.0833em;",[1810,4423,4272],{"className":4424},[1926],[1810,4426],{"className":4427,"style":1965},[1943],[1810,4429,1864],{"className":4430},[1969],[1810,4432],{"className":4433,"style":1965},[1943],[1810,4435,4437,4441,4515,4518,4521],{"className":4436},[1917],[1810,4438],{"className":4439,"style":4440},[1921],"height:2.2019em;vertical-align:-0.8804em;",[1810,4442,4444,4447,4512],{"className":4443},[1926],[1810,4445],{"className":4446},[1932,4341],[1810,4448,4450],{"className":4449},[4262],[1810,4451,4453,4503],{"className":4452},[2010,2011],[1810,4454,4456,4500],{"className":4455},[2015],[1810,4457,4460,4481,4489],{"className":4458,"style":4459},[2019],"height:1.3214em;",[1810,4461,4462,4465],{"style":4357},[1810,4463],{"className":4464,"style":4361},[2027],[1810,4466,4468,4472,4478],{"className":4467},[1926],[1810,4469,4471],{"className":4470},[1926],"240",[1810,4473,4475],{"className":4474},[1926],[1810,4476,1904],{"className":4477},[2152],[1810,4479,4013],{"className":4480},[1926],[1810,4482,4483,4486],{"style":4370},[1810,4484],{"className":4485,"style":4361},[2027],[1810,4487],{"className":4488,"style":4378},[4377],[1810,4490,4491,4494],{"style":4381},[1810,4492],{"className":4493,"style":4361},[2027],[1810,4495,4497],{"className":4496},[1926],[1810,4498,3876],{"className":4499},[1926],[1810,4501,2089],{"className":4502},[2088],[1810,4504,4506],{"className":4505},[2015],[1810,4507,4510],{"className":4508,"style":4509},[2019],"height:0.8804em;",[1810,4511],{},[1810,4513],{"className":4514},[1939,4341],[1810,4516],{"className":4517,"style":1944},[1943],[1810,4519,1850],{"className":4520},[1948],[1810,4522],{"className":4523,"style":1944},[1943],[1810,4525,4527,4530],{"className":4526},[1917],[1810,4528],{"className":4529,"style":2231},[1921],[1810,4531,4285],{"className":4532},[1926],[1793,4534,4535],{},"The exact ultimate ruin probability is",[1810,4537,4539],{"className":4538},[1813],[1810,4540,4542,4613],{"className":4541},[1817],[1810,4543,4545],{"className":4544},[1821],[1823,4546,4547],{"xmlns":1825,"display":1826},[1828,4548,4549,4610],{},[1831,4550,4551,4553,4555,4557,4559,4561,4564,4576,4578,4580,4582,4584,4600,4602,4608],{},[1834,4552,2700],{},[1838,4554,1841],{"stretchy":1840},[1834,4556,1853],{},[1838,4558,1847],{"stretchy":1840},[1838,4560,1850],{},[1834,4562,4563],{},"ρ",[3621,4565,4566,4568],{},[1834,4567,3625],{},[1831,4569,4570,4572,4574],{},[1838,4571,1864],{},[1834,4573,3390],{},[1834,4575,1853],{},[1838,4577,1904],{"separator":1903},[1943,4579],{"width":3885},[1834,4581,4563],{},[1838,4583,1850],{},[4262,4585,4586,4598],{},[1831,4587,4588,4590,4592,4594,4596],{},[1834,4589,2363],{},[1834,4591,3070],{},[1838,4593,3073],{"stretchy":1840},[1834,4595,1898],{},[1838,4597,3078],{"stretchy":1840},[1834,4599,1859],{},[1838,4601,1850],{},[4262,4603,4604,4606],{},[1879,4605,1881],{},[1879,4607,4041],{},[1834,4609,2564],{"mathvariant":2563},[1906,4611,4612],{"encoding":1908},"\\psi(u)=\\rho e^{-Ru},\n\\qquad\n\\rho=\\frac{\\lambda E[X]}{c}=\\frac{1}{1.2}.",[1810,4614,4616,4643,4712,4803],{"className":4615,"ariaHidden":1903},[1913],[1810,4617,4619,4622,4625,4628,4631,4634,4637,4640],{"className":4618},[1917],[1810,4620],{"className":4621,"style":1922},[1921],[1810,4623,2700],{"className":4624,"style":2760},[1926,1927],[1810,4626,1841],{"className":4627},[1932],[1810,4629,1853],{"className":4630},[1926,1927],[1810,4632,1847],{"className":4633},[1939],[1810,4635],{"className":4636,"style":1944},[1943],[1810,4638,1850],{"className":4639},[1948],[1810,4641],{"className":4642,"style":1944},[1943],[1810,4644,4646,4650,4653,4691,4694,4697,4700,4703,4706,4709],{"className":4645},[1917],[1810,4647],{"className":4648,"style":4649},[1921],"height:1.0858em;vertical-align:-0.1944em;",[1810,4651,4563],{"className":4652},[1926,1927],[1810,4654,4656,4659],{"className":4655},[1926],[1810,4657,3625],{"className":4658},[1926,1927],[1810,4660,4662],{"className":4661},[2113],[1810,4663,4665],{"className":4664},[2010],[1810,4666,4668],{"className":4667},[2015],[1810,4669,4671],{"className":4670,"style":3675},[2019],[1810,4672,4673,4676],{"style":3696},[1810,4674],{"className":4675,"style":2130},[2027],[1810,4677,4679],{"className":4678},[2032,2033,2034,2035],[1810,4680,4682,4685,4688],{"className":4681},[1926,2035],[1810,4683,1864],{"className":4684},[1926,2035],[1810,4686,3390],{"className":4687,"style":3411},[1926,1927,2035],[1810,4689,1853],{"className":4690},[1926,1927,2035],[1810,4692,1904],{"className":4693},[2152],[1810,4695],{"className":4696,"style":3968},[1943],[1810,4698],{"className":4699,"style":2102},[1943],[1810,4701,4563],{"className":4702},[1926,1927],[1810,4704],{"className":4705,"style":1944},[1943],[1810,4707,1850],{"className":4708},[1948],[1810,4710],{"className":4711,"style":1944},[1943],[1810,4713,4715,4719,4794,4797,4800],{"className":4714},[1917],[1810,4716],{"className":4717,"style":4718},[1921],"height:2.113em;vertical-align:-0.686em;",[1810,4720,4722,4725,4791],{"className":4721},[1926],[1810,4723],{"className":4724},[1932,4341],[1810,4726,4728],{"className":4727},[4262],[1810,4729,4731,4783],{"className":4730},[2010,2011],[1810,4732,4734,4780],{"className":4733},[2015],[1810,4735,4738,4749,4757],{"className":4736,"style":4737},[2019],"height:1.427em;",[1810,4739,4740,4743],{"style":4357},[1810,4741],{"className":4742,"style":4361},[2027],[1810,4744,4746],{"className":4745},[1926],[1810,4747,1859],{"className":4748},[1926,1927],[1810,4750,4751,4754],{"style":4370},[1810,4752],{"className":4753,"style":4361},[2027],[1810,4755],{"className":4756,"style":4378},[4377],[1810,4758,4759,4762],{"style":4381},[1810,4760],{"className":4761,"style":4361},[2027],[1810,4763,4765,4768,4771,4774,4777],{"className":4764},[1926],[1810,4766,2363],{"className":4767},[1926,1927],[1810,4769,3070],{"className":4770,"style":3097},[1926,1927],[1810,4772,3073],{"className":4773},[1932],[1810,4775,1898],{"className":4776,"style":2109},[1926,1927],[1810,4778,3078],{"className":4779},[1939],[1810,4781,2089],{"className":4782},[2088],[1810,4784,4786],{"className":4785},[2015],[1810,4787,4789],{"className":4788,"style":4400},[2019],[1810,4790],{},[1810,4792],{"className":4793},[1939,4341],[1810,4795],{"className":4796,"style":1944},[1943],[1810,4798,1850],{"className":4799},[1948],[1810,4801],{"className":4802,"style":1944},[1943],[1810,4804,4806,4810,4872],{"className":4805},[1917],[1810,4807],{"className":4808,"style":4809},[1921],"height:2.0074em;vertical-align:-0.686em;",[1810,4811,4813,4816,4869],{"className":4812},[1926],[1810,4814],{"className":4815},[1932,4341],[1810,4817,4819],{"className":4818},[4262],[1810,4820,4822,4861],{"className":4821},[2010,2011],[1810,4823,4825,4858],{"className":4824},[2015],[1810,4826,4828,4839,4847],{"className":4827,"style":4459},[2019],[1810,4829,4830,4833],{"style":4357},[1810,4831],{"className":4832,"style":4361},[2027],[1810,4834,4836],{"className":4835},[1926],[1810,4837,4041],{"className":4838},[1926],[1810,4840,4841,4844],{"style":4370},[1810,4842],{"className":4843,"style":4361},[2027],[1810,4845],{"className":4846,"style":4378},[4377],[1810,4848,4849,4852],{"style":4381},[1810,4850],{"className":4851,"style":4361},[2027],[1810,4853,4855],{"className":4854},[1926],[1810,4856,1881],{"className":4857},[1926],[1810,4859,2089],{"className":4860},[2088],[1810,4862,4864],{"className":4863},[2015],[1810,4865,4867],{"className":4866,"style":4400},[2019],[1810,4868],{},[1810,4870],{"className":4871},[1939,4341],[1810,4873,2564],{"className":4874},[1926],[1793,4876,4877,4878,2544],{},"At initial capital ",[1810,4879,4881,4902],{"className":4880},[1817],[1810,4882,4884],{"className":4883},[1821],[1823,4885,4886],{"xmlns":1825},[1828,4887,4888,4899],{},[1831,4889,4890,4892,4894,4896],{},[1834,4891,1853],{},[1838,4893,1850],{},[1834,4895,3898],{"mathvariant":2563},[1879,4897,4898],{},"100,000",[1906,4900,4901],{"encoding":1908},"u=£100{,}000",[1810,4903,4905,4923],{"className":4904,"ariaHidden":1903},[1913],[1810,4906,4908,4911,4914,4917,4920],{"className":4907},[1917],[1810,4909],{"className":4910,"style":2577},[1921],[1810,4912,1853],{"className":4913},[1926,1927],[1810,4915],{"className":4916,"style":1944},[1943],[1810,4918,1850],{"className":4919},[1948],[1810,4921],{"className":4922,"style":1944},[1943],[1810,4924,4926,4929,4933,4939],{"className":4925},[1917],[1810,4927],{"className":4928,"style":3999},[1921],[1810,4930,4932],{"className":4931},[1926],"£100",[1810,4934,4936],{"className":4935},[1926],[1810,4937,1904],{"className":4938},[2152],[1810,4940,4013],{"className":4941},[1926],[1810,4943,4945],{"className":4944},[1813],[1810,4946,4948,4998],{"className":4947},[1817],[1810,4949,4951],{"className":4950},[1821],[1823,4952,4953],{"xmlns":1825,"display":1826},[1828,4954,4955,4995],{},[1831,4956,4957,4959,4961,4963,4965,4967,4973,4984,4987,4990,4993],{},[1834,4958,2700],{},[1838,4960,1841],{"stretchy":1840},[1879,4962,4898],{},[1838,4964,1847],{"stretchy":1840},[1838,4966,1850],{},[4262,4968,4969,4971],{},[1879,4970,1881],{},[1879,4972,4041],{},[3621,4974,4975,4977],{},[1834,4976,3625],{},[1831,4978,4979,4981],{},[1838,4980,1864],{},[1879,4982,4983],{},"3.3333",[1838,4985,4986],{},"≈",[1879,4988,4989],{},"2.97",[1834,4991,4992],{"mathvariant":2563},"%",[1834,4994,2564],{"mathvariant":2563},[1906,4996,4997],{"encoding":1908},"\\psi(100{,}000)\n=\\frac{1}{1.2}e^{-3.3333}\n\\approx2.97\\%.",[1810,4999,5001,5038,5151],{"className":5000,"ariaHidden":1903},[1913],[1810,5002,5004,5007,5010,5013,5017,5023,5026,5029,5032,5035],{"className":5003},[1917],[1810,5005],{"className":5006,"style":1922},[1921],[1810,5008,2700],{"className":5009,"style":2760},[1926,1927],[1810,5011,1841],{"className":5012},[1932],[1810,5014,5016],{"className":5015},[1926],"100",[1810,5018,5020],{"className":5019},[1926],[1810,5021,1904],{"className":5022},[2152],[1810,5024,4013],{"className":5025},[1926],[1810,5027,1847],{"className":5028},[1939],[1810,5030],{"className":5031,"style":1944},[1943],[1810,5033,1850],{"className":5034},[1948],[1810,5036],{"className":5037,"style":1944},[1943],[1810,5039,5041,5044,5106,5142,5145,5148],{"className":5040},[1917],[1810,5042],{"className":5043,"style":4809},[1921],[1810,5045,5047,5050,5103],{"className":5046},[1926],[1810,5048],{"className":5049},[1932,4341],[1810,5051,5053],{"className":5052},[4262],[1810,5054,5056,5095],{"className":5055},[2010,2011],[1810,5057,5059,5092],{"className":5058},[2015],[1810,5060,5062,5073,5081],{"className":5061,"style":4459},[2019],[1810,5063,5064,5067],{"style":4357},[1810,5065],{"className":5066,"style":4361},[2027],[1810,5068,5070],{"className":5069},[1926],[1810,5071,4041],{"className":5072},[1926],[1810,5074,5075,5078],{"style":4370},[1810,5076],{"className":5077,"style":4361},[2027],[1810,5079],{"className":5080,"style":4378},[4377],[1810,5082,5083,5086],{"style":4381},[1810,5084],{"className":5085,"style":4361},[2027],[1810,5087,5089],{"className":5088},[1926],[1810,5090,1881],{"className":5091},[1926],[1810,5093,2089],{"className":5094},[2088],[1810,5096,5098],{"className":5097},[2015],[1810,5099,5101],{"className":5100,"style":4400},[2019],[1810,5102],{},[1810,5104],{"className":5105},[1939,4341],[1810,5107,5109,5112],{"className":5108},[1926],[1810,5110,3625],{"className":5111},[1926,1927],[1810,5113,5115],{"className":5114},[2113],[1810,5116,5118],{"className":5117},[2010],[1810,5119,5121],{"className":5120},[2015],[1810,5122,5125],{"className":5123,"style":5124},[2019],"height:0.8641em;",[1810,5126,5127,5130],{"style":3696},[1810,5128],{"className":5129,"style":2130},[2027],[1810,5131,5133],{"className":5132},[2032,2033,2034,2035],[1810,5134,5136,5139],{"className":5135},[1926,2035],[1810,5137,1864],{"className":5138},[1926,2035],[1810,5140,4983],{"className":5141},[1926,2035],[1810,5143],{"className":5144,"style":1944},[1943],[1810,5146,4986],{"className":5147},[1948],[1810,5149],{"className":5150,"style":1944},[1943],[1810,5152,5154,5158],{"className":5153},[1917],[1810,5155],{"className":5156,"style":5157},[1921],"height:0.8056em;vertical-align:-0.0556em;",[1810,5159,5161],{"className":5160},[1926],"2.97%.",[1793,5163,5164,5165,5262,5263,2564],{},"The Lundberg bound is ",[1810,5166,5168,5197],{"className":5167},[1817],[1810,5169,5171],{"className":5170},[1821],[1823,5172,5173],{"xmlns":1825},[1828,5174,5175,5194],{},[1831,5176,5177,5187,5189,5192],{},[3621,5178,5179,5181],{},[1834,5180,3625],{},[1831,5182,5183,5185],{},[1838,5184,1864],{},[1879,5186,4983],{},[1838,5188,4986],{},[1879,5190,5191],{},"3.57",[1834,5193,4992],{"mathvariant":2563},[1906,5195,5196],{"encoding":1908},"e^{-3.3333}\\approx3.57\\%",[1810,5198,5200,5252],{"className":5199,"ariaHidden":1903},[1913],[1810,5201,5203,5207,5243,5246,5249],{"className":5202},[1917],[1810,5204],{"className":5205,"style":5206},[1921],"height:0.8141em;",[1810,5208,5210,5213],{"className":5209},[1926],[1810,5211,3625],{"className":5212},[1926,1927],[1810,5214,5216],{"className":5215},[2113],[1810,5217,5219],{"className":5218},[2010],[1810,5220,5222],{"className":5221},[2015],[1810,5223,5225],{"className":5224,"style":5206},[2019],[1810,5226,5228,5231],{"style":5227},"top:-3.063em;margin-right:0.05em;",[1810,5229],{"className":5230,"style":2130},[2027],[1810,5232,5234],{"className":5233},[2032,2033,2034,2035],[1810,5235,5237,5240],{"className":5236},[1926,2035],[1810,5238,1864],{"className":5239},[1926,2035],[1810,5241,4983],{"className":5242},[1926,2035],[1810,5244],{"className":5245,"style":1944},[1943],[1810,5247,4986],{"className":5248},[1948],[1810,5250],{"className":5251,"style":1944},[1943],[1810,5253,5255,5258],{"className":5254},[1917],[1810,5256],{"className":5257,"style":5157},[1921],[1810,5259,5261],{"className":5260},[1926],"3.57%",". The exact result depends on Exponential severity; the bound depends on the broader classical assumptions and existence of ",[1810,5264,5266,5279],{"className":5265},[1817],[1810,5267,5269],{"className":5268},[1821],[1823,5270,5271],{"xmlns":1825},[1828,5272,5273,5277],{},[1831,5274,5275],{},[1834,5276,3390],{},[1906,5278,3390],{"encoding":1908},[1810,5280,5282],{"className":5281,"ariaHidden":1903},[1913],[1810,5283,5285,5288],{"className":5284},[1917],[1810,5286],{"className":5287,"style":3582},[1921],[1810,5289,3390],{"className":5290,"style":3411},[1926,1927],[1802,5292,5294],{"id":5293},"_5-how-assumptions-change-the-answer","5. How assumptions change the answer",[2157,5296,5297,5310],{},[2160,5298,5299],{},[2163,5300,5301,5304,5307],{},[2166,5302,5303],{},"Classical assumption",[2166,5305,5306],{},"Practical complication",[2166,5308,5309],{},"Direction is not automatic",[2173,5311,5312,5323,5334,5345,5356,5367],{},[2163,5313,5314,5317,5320],{},[2178,5315,5316],{},"constant premium rate",[2178,5318,5319],{},"renewals, rate changes, expenses",[2178,5321,5322],{},"higher gross premium may accompany higher exposure",[2163,5324,5325,5328,5331],{},[2178,5326,5327],{},"Poisson independent arrivals",[2178,5329,5330],{},"catastrophe clustering and seasonality",[2178,5332,5333],{},"clustering usually worsens short-horizon paths",[2163,5335,5336,5339,5342],{},[2178,5337,5338],{},"iid severity",[2178,5340,5341],{},"inflation, limits, mix, trend",[2178,5343,5344],{},"changing tail can dominate average trend",[2163,5346,5347,5350,5353],{},[2178,5348,5349],{},"no investment return",[2178,5351,5352],{},"stochastic assets and liquidity",[2178,5354,5355],{},"return can help or introduce market dependence",[2163,5357,5358,5361,5364],{},[2178,5359,5360],{},"immediate known payment",[2178,5362,5363],{},"reporting and settlement delays",[2178,5365,5366],{},"accounting insolvency and cash ruin differ",[2163,5368,5369,5372,5375],{},[2178,5370,5371],{},"no reinsurance default",[2178,5373,5374],{},"delayed\u002Fdisputed recoveries",[2178,5376,5377],{},"nominal cover can overstate available liquidity",[1802,5379,5381],{"id":5380},"_6-reinsurance-in-the-surplus-process","6. Reinsurance in the surplus process",[1793,5383,5384,5385,5471,5472,1904],{},"For retained claim ",[1810,5386,5388,5413],{"className":5387},[1817],[1810,5389,5391],{"className":5390},[1821],[1823,5392,5393],{"xmlns":1825},[1828,5394,5395,5410],{},[1831,5396,5397,5400,5402,5408],{},[1834,5398,5399],{},"g",[1838,5401,1841],{"stretchy":1840},[1894,5403,5404,5406],{},[1834,5405,1898],{},[1834,5407,1875],{},[1838,5409,1847],{"stretchy":1840},[1906,5411,5412],{"encoding":1908},"g(X_i)",[1810,5414,5416],{"className":5415,"ariaHidden":1903},[1913],[1810,5417,5419,5422,5425,5428,5468],{"className":5418},[1917],[1810,5420],{"className":5421,"style":1922},[1921],[1810,5423,5399],{"className":5424,"style":2760},[1926,1927],[1810,5426,1841],{"className":5427},[1932],[1810,5429,5431,5434],{"className":5430},[1926],[1810,5432,1898],{"className":5433,"style":2109},[1926,1927],[1810,5435,5437],{"className":5436},[2113],[1810,5438,5440,5460],{"className":5439},[2010,2011],[1810,5441,5443,5457],{"className":5442},[2015],[1810,5444,5446],{"className":5445,"style":2123},[2019],[1810,5447,5448,5451],{"style":2126},[1810,5449],{"className":5450,"style":2130},[2027],[1810,5452,5454],{"className":5453},[2032,2033,2034,2035],[1810,5455,1875],{"className":5456},[1926,1927,2035],[1810,5458,2089],{"className":5459},[2088],[1810,5461,5463],{"className":5462},[2015],[1810,5464,5466],{"className":5465,"style":2146},[2019],[1810,5467],{},[1810,5469,1847],{"className":5470},[1939]," and net premium rate ",[1810,5473,5475,5500],{"className":5474},[1817],[1810,5476,5478],{"className":5477},[1821],[1823,5479,5480],{"xmlns":1825},[1828,5481,5482,5497],{},[1831,5483,5484],{},[1894,5485,5486,5488],{},[1834,5487,1859],{},[1831,5489,5490,5493,5495],{},[1834,5491,5492],{},"n",[1834,5494,3625],{},[1834,5496,1844],{},[1906,5498,5499],{"encoding":1908},"c_{net}",[1810,5501,5503],{"className":5502,"ariaHidden":1903},[1913],[1810,5504,5506,5510],{"className":5505},[1917],[1810,5507],{"className":5508,"style":5509},[1921],"height:0.5806em;vertical-align:-0.15em;",[1810,5511,5513,5516],{"className":5512},[1926],[1810,5514,1859],{"className":5515},[1926,1927],[1810,5517,5519],{"className":5518},[2113],[1810,5520,5522,5553],{"className":5521},[2010,2011],[1810,5523,5525,5550],{"className":5524},[2015],[1810,5526,5529],{"className":5527,"style":5528},[2019],"height:0.2806em;",[1810,5530,5532,5535],{"style":5531},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[1810,5533],{"className":5534,"style":2130},[2027],[1810,5536,5538],{"className":5537},[2032,2033,2034,2035],[1810,5539,5541,5544,5547],{"className":5540},[1926,2035],[1810,5542,5492],{"className":5543},[1926,1927,2035],[1810,5545,3625],{"className":5546},[1926,1927,2035],[1810,5548,1844],{"className":5549},[1926,1927,2035],[1810,5551,2089],{"className":5552},[2088],[1810,5554,5556],{"className":5555},[2015],[1810,5557,5559],{"className":5558,"style":2146},[2019],[1810,5560],{},[1810,5562,5564],{"className":5563},[1813],[1810,5565,5567,5655],{"className":5566},[1817],[1810,5568,5570],{"className":5569},[1821],[1823,5571,5572],{"xmlns":1825,"display":1826},[1828,5573,5574,5652],{},[1831,5575,5576,5588,5590,5592,5594,5596,5598,5600,5612,5614,5616,5638,5640,5642,5648,5650],{},[3621,5577,5578,5580],{},[1834,5579,1836],{},[1831,5581,5582,5584,5586],{},[1834,5583,5492],{},[1834,5585,3625],{},[1834,5587,1844],{},[1838,5589,1841],{"stretchy":1840},[1834,5591,1844],{},[1838,5593,1847],{"stretchy":1840},[1838,5595,1850],{},[1834,5597,1853],{},[1838,5599,1856],{},[1894,5601,5602,5604],{},[1834,5603,1859],{},[1831,5605,5606,5608,5610],{},[1834,5607,5492],{},[1834,5609,3625],{},[1834,5611,1844],{},[1834,5613,1844],{},[1838,5615,1864],{},[1866,5617,5618,5620,5628],{},[1838,5619,1870],{},[1831,5621,5622,5624,5626],{},[1834,5623,1875],{},[1838,5625,1850],{},[1879,5627,1881],{},[1831,5629,5630,5632,5634,5636],{},[1834,5631,1886],{},[1838,5633,1841],{"stretchy":1840},[1834,5635,1844],{},[1838,5637,1847],{"stretchy":1840},[1834,5639,5399],{},[1838,5641,1841],{"stretchy":1840},[1894,5643,5644,5646],{},[1834,5645,1898],{},[1834,5647,1875],{},[1838,5649,1847],{"stretchy":1840},[1834,5651,2564],{"mathvariant":2563},[1906,5653,5654],{"encoding":1908},"U^{net}(t)=u+c_{net}t-\\sum_{i=1}^{N(t)}g(X_i).",[1810,5656,5658,5722,5740,5808],{"className":5657,"ariaHidden":1903},[1913],[1810,5659,5661,5665,5704,5707,5710,5713,5716,5719],{"className":5660},[1917],[1810,5662],{"className":5663,"style":5664},[1921],"height:1.0936em;vertical-align:-0.25em;",[1810,5666,5668,5671],{"className":5667},[1926],[1810,5669,1836],{"className":5670,"style":1928},[1926,1927],[1810,5672,5674],{"className":5673},[2113],[1810,5675,5677],{"className":5676},[2010],[1810,5678,5680],{"className":5679},[2015],[1810,5681,5684],{"className":5682,"style":5683},[2019],"height:0.8436em;",[1810,5685,5686,5689],{"style":3696},[1810,5687],{"className":5688,"style":2130},[2027],[1810,5690,5692],{"className":5691},[2032,2033,2034,2035],[1810,5693,5695,5698,5701],{"className":5694},[1926,2035],[1810,5696,5492],{"className":5697},[1926,1927,2035],[1810,5699,3625],{"className":5700},[1926,1927,2035],[1810,5702,1844],{"className":5703},[1926,1927,2035],[1810,5705,1841],{"className":5706},[1932],[1810,5708,1844],{"className":5709},[1926,1927],[1810,5711,1847],{"className":5712},[1939],[1810,5714],{"className":5715,"style":1944},[1943],[1810,5717,1850],{"className":5718},[1948],[1810,5720],{"className":5721,"style":1944},[1943],[1810,5723,5725,5728,5731,5734,5737],{"className":5724},[1917],[1810,5726],{"className":5727,"style":1958},[1921],[1810,5729,1853],{"className":5730},[1926,1927],[1810,5732],{"className":5733,"style":1965},[1943],[1810,5735,1856],{"className":5736},[1969],[1810,5738],{"className":5739,"style":1965},[1943],[1810,5741,5743,5747,5796,5799,5802,5805],{"className":5742},[1917],[1810,5744],{"className":5745,"style":5746},[1921],"height:0.7651em;vertical-align:-0.15em;",[1810,5748,5750,5753],{"className":5749},[1926],[1810,5751,1859],{"className":5752},[1926,1927],[1810,5754,5756],{"className":5755},[2113],[1810,5757,5759,5788],{"className":5758},[2010,2011],[1810,5760,5762,5785],{"className":5761},[2015],[1810,5763,5765],{"className":5764,"style":5528},[2019],[1810,5766,5767,5770],{"style":5531},[1810,5768],{"className":5769,"style":2130},[2027],[1810,5771,5773],{"className":5772},[2032,2033,2034,2035],[1810,5774,5776,5779,5782],{"className":5775},[1926,2035],[1810,5777,5492],{"className":5778},[1926,1927,2035],[1810,5780,3625],{"className":5781},[1926,1927,2035],[1810,5783,1844],{"className":5784},[1926,1927,2035],[1810,5786,2089],{"className":5787},[2088],[1810,5789,5791],{"className":5790},[2015],[1810,5792,5794],{"className":5793,"style":2146},[2019],[1810,5795],{},[1810,5797,1844],{"className":5798},[1926,1927],[1810,5800],{"className":5801,"style":1965},[1943],[1810,5803,1864],{"className":5804},[1969],[1810,5806],{"className":5807,"style":1965},[1943],[1810,5809,5811,5814,5890,5893,5896,5899,5939,5942],{"className":5810},[1917],[1810,5812],{"className":5813,"style":2001},[1921],[1810,5815,5817],{"className":5816},[2005,2006],[1810,5818,5820,5882],{"className":5819},[2010,2011],[1810,5821,5823,5879],{"className":5822},[2015],[1810,5824,5826,5846,5856],{"className":5825,"style":2020},[2019],[1810,5827,5828,5831],{"style":2023},[1810,5829],{"className":5830,"style":2028},[2027],[1810,5832,5834],{"className":5833},[2032,2033,2034,2035],[1810,5835,5837,5840,5843],{"className":5836},[1926,2035],[1810,5838,1875],{"className":5839},[1926,1927,2035],[1810,5841,1850],{"className":5842},[1948,2035],[1810,5844,1881],{"className":5845},[1926,2035],[1810,5847,5848,5851],{"style":2050},[1810,5849],{"className":5850,"style":2028},[2027],[1810,5852,5853],{},[1810,5854,1870],{"className":5855},[2005,2059,2060],[1810,5857,5858,5861],{"style":2063},[1810,5859],{"className":5860,"style":2028},[2027],[1810,5862,5864],{"className":5863},[2032,2033,2034,2035],[1810,5865,5867,5870,5873,5876],{"className":5866},[1926,2035],[1810,5868,1886],{"className":5869,"style":1928},[1926,1927,2035],[1810,5871,1841],{"className":5872},[1932,2035],[1810,5874,1844],{"className":5875},[1926,1927,2035],[1810,5877,1847],{"className":5878},[1939,2035],[1810,5880,2089],{"className":5881},[2088],[1810,5883,5885],{"className":5884},[2015],[1810,5886,5888],{"className":5887,"style":2096},[2019],[1810,5889],{},[1810,5891],{"className":5892,"style":2102},[1943],[1810,5894,5399],{"className":5895,"style":2760},[1926,1927],[1810,5897,1841],{"className":5898},[1932],[1810,5900,5902,5905],{"className":5901},[1926],[1810,5903,1898],{"className":5904,"style":2109},[1926,1927],[1810,5906,5908],{"className":5907},[2113],[1810,5909,5911,5931],{"className":5910},[2010,2011],[1810,5912,5914,5928],{"className":5913},[2015],[1810,5915,5917],{"className":5916,"style":2123},[2019],[1810,5918,5919,5922],{"style":2126},[1810,5920],{"className":5921,"style":2130},[2027],[1810,5923,5925],{"className":5924},[2032,2033,2034,2035],[1810,5926,1875],{"className":5927},[1926,1927,2035],[1810,5929,2089],{"className":5930},[2088],[1810,5932,5934],{"className":5933},[2015],[1810,5935,5937],{"className":5936,"style":2146},[2019],[1810,5938],{},[1810,5940,1847],{"className":5941},[1939],[1810,5943,2564],{"className":5944},[1926],[1793,5946,5947,5948,6014,6015,6059,6060,6088],{},"Reinsurance reduces claim severity but also costs premium. Comparing gross ",[1810,5949,5951,5975],{"className":5950},[1817],[1810,5952,5954],{"className":5953},[1821],[1823,5955,5956],{"xmlns":1825},[1828,5957,5958,5972],{},[1831,5959,5960,5962,5964,5966,5968,5970],{},[1834,5961,5399],{},[1838,5963,1841],{"stretchy":1840},[1834,5965,1898],{},[1838,5967,1847],{"stretchy":1840},[1838,5969,1850],{},[1834,5971,1898],{},[1906,5973,5974],{"encoding":1908},"g(X)=X",[1810,5976,5978,6005],{"className":5977,"ariaHidden":1903},[1913],[1810,5979,5981,5984,5987,5990,5993,5996,5999,6002],{"className":5980},[1917],[1810,5982],{"className":5983,"style":1922},[1921],[1810,5985,5399],{"className":5986,"style":2760},[1926,1927],[1810,5988,1841],{"className":5989},[1932],[1810,5991,1898],{"className":5992,"style":2109},[1926,1927],[1810,5994,1847],{"className":5995},[1939],[1810,5997],{"className":5998,"style":1944},[1943],[1810,6000,1850],{"className":6001},[1948],[1810,6003],{"className":6004,"style":1944},[1943],[1810,6006,6008,6011],{"className":6007},[1917],[1810,6009],{"className":6010,"style":3582},[1921],[1810,6012,1898],{"className":6013,"style":2109},[1926,1927]," with net ",[1810,6016,6018,6038],{"className":6017},[1817],[1810,6019,6021],{"className":6020},[1821],[1823,6022,6023],{"xmlns":1825},[1828,6024,6025,6035],{},[1831,6026,6027,6029,6031,6033],{},[1834,6028,5399],{},[1838,6030,1841],{"stretchy":1840},[1834,6032,1898],{},[1838,6034,1847],{"stretchy":1840},[1906,6036,6037],{"encoding":1908},"g(X)",[1810,6039,6041],{"className":6040,"ariaHidden":1903},[1913],[1810,6042,6044,6047,6050,6053,6056],{"className":6043},[1917],[1810,6045],{"className":6046,"style":1922},[1921],[1810,6048,5399],{"className":6049,"style":2760},[1926,1927],[1810,6051,1841],{"className":6052},[1932],[1810,6054,1898],{"className":6055,"style":2109},[1926,1927],[1810,6057,1847],{"className":6058},[1939]," while leaving ",[1810,6061,6063,6076],{"className":6062},[1817],[1810,6064,6066],{"className":6065},[1821],[1823,6067,6068],{"xmlns":1825},[1828,6069,6070,6074],{},[1831,6071,6072],{},[1834,6073,1859],{},[1906,6075,1859],{"encoding":1908},[1810,6077,6079],{"className":6078,"ariaHidden":1903},[1913],[1810,6080,6082,6085],{"className":6081},[1917],[1810,6083],{"className":6084,"style":2577},[1921],[1810,6086,1859],{"className":6087},[1926,1927]," unchanged overstates the economic benefit.",[1793,6090,6091,6092,2564],{},"For occurrence cover, the process must group claims into events before applying ",[1810,6093,6095,6108],{"className":6094},[1817],[1810,6096,6098],{"className":6097},[1821],[1823,6099,6100],{"xmlns":1825},[1828,6101,6102,6106],{},[1831,6103,6104],{},[1834,6105,5399],{},[1906,6107,5399],{"encoding":1908},[1810,6109,6111],{"className":6110,"ariaHidden":1903},[1913],[1810,6112,6114,6118],{"className":6113},[1917],[1810,6115],{"className":6116,"style":6117},[1921],"height:0.625em;vertical-align:-0.1944em;",[1810,6119,5399],{"className":6120,"style":2760},[1926,1927],[1802,6122,6124],{"id":6123},"_7-what-classical-ruin-isand-is-not","7. What classical ruin is—and is not",[1793,6126,6127],{},"Ruin theory is valuable for:",[6129,6130,6131,6135,6138,6141],"ul",{},[6132,6133,6134],"li",{},"understanding path dependence and timing;",[6132,6136,6137],{},"studying the interaction of premium loading, capital, and claim tails;",[6132,6139,6140],{},"comparing finite and ultimate horizons;",[6132,6142,6143],{},"designing stress tests and simulation checks.",[1793,6145,6146],{},"It is not by itself IFRS 17 measurement, a Solvency UK SCR calculation, liquidity regulation, or proof of firm viability. Real decisions add reserving, expenses, taxes, assets, management actions, legal entities, and governance.",[1793,6148,6149,6150,6155],{},"Proceed to the ",[6151,6152,6154],"a",{"href":6153},".\u002F01-finite-time-simulation","Finite-Time Simulation Lab"," to estimate path probabilities directly.",[1802,6157,6159],{"id":6158},"practice","Practice",[6161,6162,6163,6392,6544],"ol",{},[6132,6164,6165,6217,6218,6297,6298,6362,6363,2564],{},[1810,6166,6168,6187],{"className":6167},[1817],[1810,6169,6171],{"className":6170},[1821],[1823,6172,6173],{"xmlns":1825},[1828,6174,6175,6184],{},[1831,6176,6177,6179,6181],{},[1834,6178,2363],{},[1838,6180,1850],{},[1879,6182,6183],{},"10",[1906,6185,6186],{"encoding":1908},"\\lambda=10",[1810,6188,6190,6208],{"className":6189,"ariaHidden":1903},[1913],[1810,6191,6193,6196,6199,6202,6205],{"className":6192},[1917],[1810,6194],{"className":6195,"style":2376},[1921],[1810,6197,2363],{"className":6198},[1926,1927],[1810,6200],{"className":6201,"style":1944},[1943],[1810,6203,1850],{"className":6204},[1948],[1810,6206],{"className":6207,"style":1944},[1943],[1810,6209,6211,6214],{"className":6210},[1917],[1810,6212],{"className":6213,"style":2231},[1921],[1810,6215,6183],{"className":6216},[1926],", ",[1810,6219,6221,6248],{"className":6220},[1817],[1810,6222,6224],{"className":6223},[1821],[1823,6225,6226],{"xmlns":1825},[1828,6227,6228,6245],{},[1831,6229,6230,6232,6234,6236,6238,6240,6242],{},[1834,6231,3070],{},[1838,6233,3073],{"stretchy":1840},[1834,6235,1898],{},[1838,6237,3078],{"stretchy":1840},[1838,6239,1850],{},[1834,6241,3898],{"mathvariant":2563},[1879,6243,6244],{},"4,000",[1906,6246,6247],{"encoding":1908},"E[X]=£4{,}000",[1810,6249,6251,6278],{"className":6250,"ariaHidden":1903},[1913],[1810,6252,6254,6257,6260,6263,6266,6269,6272,6275],{"className":6253},[1917],[1810,6255],{"className":6256,"style":1922},[1921],[1810,6258,3070],{"className":6259,"style":3097},[1926,1927],[1810,6261,3073],{"className":6262},[1932],[1810,6264,1898],{"className":6265,"style":2109},[1926,1927],[1810,6267,3078],{"className":6268},[1939],[1810,6270],{"className":6271,"style":1944},[1943],[1810,6273,1850],{"className":6274},[1948],[1810,6276],{"className":6277,"style":1944},[1943],[1810,6279,6281,6284,6288,6294],{"className":6280},[1917],[1810,6282],{"className":6283,"style":3999},[1921],[1810,6285,6287],{"className":6286},[1926],"£4",[1810,6289,6291],{"className":6290},[1926],[1810,6292,1904],{"className":6293},[2152],[1810,6295,4013],{"className":6296},[1926],", and ",[1810,6299,6301,6322],{"className":6300},[1817],[1810,6302,6304],{"className":6303},[1821],[1823,6305,6306],{"xmlns":1825},[1828,6307,6308,6319],{},[1831,6309,6310,6312,6314,6316],{},[1834,6311,1859],{},[1838,6313,1850],{},[1834,6315,3898],{"mathvariant":2563},[1879,6317,6318],{},"44,000",[1906,6320,6321],{"encoding":1908},"c=£44{,}000",[1810,6323,6325,6343],{"className":6324,"ariaHidden":1903},[1913],[1810,6326,6328,6331,6334,6337,6340],{"className":6327},[1917],[1810,6329],{"className":6330,"style":2577},[1921],[1810,6332,1859],{"className":6333},[1926,1927],[1810,6335],{"className":6336,"style":1944},[1943],[1810,6338,1850],{"className":6339},[1948],[1810,6341],{"className":6342,"style":1944},[1943],[1810,6344,6346,6349,6353,6359],{"className":6345},[1917],[1810,6347],{"className":6348,"style":3999},[1921],[1810,6350,6352],{"className":6351},[1926],"£44",[1810,6354,6356],{"className":6355},[1926],[1810,6357,1904],{"className":6358},[2152],[1810,6360,4013],{"className":6361},[1926]," per year. Find loading ",[1810,6364,6366,6380],{"className":6365},[1817],[1810,6367,6369],{"className":6368},[1821],[1823,6370,6371],{"xmlns":1825},[1828,6372,6373,6377],{},[1831,6374,6375],{},[1834,6376,3217],{},[1906,6378,6379],{"encoding":1908},"\\theta",[1810,6381,6383],{"className":6382,"ariaHidden":1903},[1913],[1810,6384,6386,6389],{"className":6385},[1917],[1810,6387],{"className":6388,"style":2376},[1921],[1810,6390,3217],{"className":6391,"style":3286},[1926,1927],[6132,6393,6394,6395,6476,6477,6543],{},"Why is ",[1810,6396,6398,6428],{"className":6397},[1817],[1810,6399,6401],{"className":6400},[1821],[1823,6402,6403],{"xmlns":1825},[1828,6404,6405,6425],{},[1831,6406,6407,6409,6411,6413,6415,6417,6419,6421,6423],{},[1834,6408,2716],{},[1838,6410,1841],{"stretchy":1840},[1834,6412,1836],{},[1838,6414,1841],{"stretchy":1840},[1879,6416,6183],{},[1838,6418,1847],{"stretchy":1840},[1838,6420,2555],{},[1879,6422,2199],{},[1838,6424,1847],{"stretchy":1840},[1906,6426,6427],{"encoding":1908},"P(U(10)\u003C0)",[1810,6429,6431,6464],{"className":6430,"ariaHidden":1903},[1913],[1810,6432,6434,6437,6440,6443,6446,6449,6452,6455,6458,6461],{"className":6433},[1917],[1810,6435],{"className":6436,"style":1922},[1921],[1810,6438,2716],{"className":6439,"style":2776},[1926,1927],[1810,6441,1841],{"className":6442},[1932],[1810,6444,1836],{"className":6445,"style":1928},[1926,1927],[1810,6447,1841],{"className":6448},[1932],[1810,6450,6183],{"className":6451},[1926],[1810,6453,1847],{"className":6454},[1939],[1810,6456],{"className":6457,"style":1944},[1943],[1810,6459,2555],{"className":6460},[1948],[1810,6462],{"className":6463,"style":1944},[1943],[1810,6465,6467,6470,6473],{"className":6466},[1917],[1810,6468],{"className":6469,"style":1922},[1921],[1810,6471,2199],{"className":6472},[1926],[1810,6474,1847],{"className":6475},[1939]," not generally equal to ",[1810,6478,6480,6504],{"className":6479},[1817],[1810,6481,6483],{"className":6482},[1821],[1823,6484,6485],{"xmlns":1825},[1828,6486,6487,6501],{},[1831,6488,6489,6491,6493,6495,6497,6499],{},[1834,6490,2716],{},[1838,6492,1841],{"stretchy":1840},[1834,6494,2524],{},[1838,6496,2723],{},[1879,6498,6183],{},[1838,6500,1847],{"stretchy":1840},[1906,6502,6503],{"encoding":1908},"P(\\tau\\le10)",[1810,6505,6507,6531],{"className":6506,"ariaHidden":1903},[1913],[1810,6508,6510,6513,6516,6519,6522,6525,6528],{"className":6509},[1917],[1810,6511],{"className":6512,"style":1922},[1921],[1810,6514,2716],{"className":6515,"style":2776},[1926,1927],[1810,6517,1841],{"className":6518},[1932],[1810,6520,2524],{"className":6521,"style":2581},[1926,1927],[1810,6523],{"className":6524,"style":1944},[1943],[1810,6526,2723],{"className":6527},[1948],[1810,6529],{"className":6530,"style":1944},[1943],[1810,6532,6534,6537,6540],{"className":6533},[1917],[1810,6535],{"className":6536,"style":1922},[1921],[1810,6538,6183],{"className":6539},[1926],[1810,6541,1847],{"className":6542},[1939],"?",[6132,6545,6546,6547,6618],{},"If ",[1810,6548,6550,6576],{"className":6549},[1817],[1810,6551,6553],{"className":6552},[1821],[1823,6554,6555],{"xmlns":1825},[1828,6556,6557,6573],{},[1831,6558,6559,6561,6563,6565,6567,6569,6571],{},[1834,6560,1859],{},[1838,6562,2723],{},[1834,6564,2363],{},[1834,6566,3070],{},[1838,6568,3073],{"stretchy":1840},[1834,6570,1898],{},[1838,6572,3078],{"stretchy":1840},[1906,6574,6575],{"encoding":1908},"c\\le\\lambda E[X]",[1810,6577,6579,6597],{"className":6578,"ariaHidden":1903},[1913],[1810,6580,6582,6585,6588,6591,6594],{"className":6581},[1917],[1810,6583],{"className":6584,"style":2212},[1921],[1810,6586,1859],{"className":6587},[1926,1927],[1810,6589],{"className":6590,"style":1944},[1943],[1810,6592,2723],{"className":6593},[1948],[1810,6595],{"className":6596,"style":1944},[1943],[1810,6598,6600,6603,6606,6609,6612,6615],{"className":6599},[1917],[1810,6601],{"className":6602,"style":1922},[1921],[1810,6604,2363],{"className":6605},[1926,1927],[1810,6607,3070],{"className":6608,"style":3097},[1926,1927],[1810,6610,3073],{"className":6611},[1932],[1810,6613,1898],{"className":6614,"style":2109},[1926,1927],[1810,6616,3078],{"className":6617},[1939],", what does the long-run drift imply in the classical model?",[6620,6621,6623],"legacy-details",{"title":6622},"Answers",[6161,6624,6625,6781,6784],{},[6132,6626,6627,6628,6726,6727,2564],{},"Expected claims are £40,000; ",[1810,6629,6631,6660],{"className":6630},[1817],[1810,6632,6634],{"className":6633},[1821],[1823,6635,6636],{"xmlns":1825},[1828,6637,6638,6657],{},[1831,6639,6640,6642,6644,6646,6648,6650,6652,6654],{},[1834,6641,1859],{},[1838,6643,1850],{},[1838,6645,1841],{"stretchy":1840},[1879,6647,1881],{},[1838,6649,1856],{},[1834,6651,3217],{},[1838,6653,1847],{"stretchy":1840},[1879,6655,6656],{},"40,000",[1906,6658,6659],{"encoding":1908},"c=(1+\\theta)40{,}000",[1810,6661,6663,6681,6702],{"className":6662,"ariaHidden":1903},[1913],[1810,6664,6666,6669,6672,6675,6678],{"className":6665},[1917],[1810,6667],{"className":6668,"style":2577},[1921],[1810,6670,1859],{"className":6671},[1926,1927],[1810,6673],{"className":6674,"style":1944},[1943],[1810,6676,1850],{"className":6677},[1948],[1810,6679],{"className":6680,"style":1944},[1943],[1810,6682,6684,6687,6690,6693,6696,6699],{"className":6683},[1917],[1810,6685],{"className":6686,"style":1922},[1921],[1810,6688,1841],{"className":6689},[1932],[1810,6691,1881],{"className":6692},[1926],[1810,6694],{"className":6695,"style":1965},[1943],[1810,6697,1856],{"className":6698},[1969],[1810,6700],{"className":6701,"style":1965},[1943],[1810,6703,6705,6708,6711,6714,6717,6723],{"className":6704},[1917],[1810,6706],{"className":6707,"style":1922},[1921],[1810,6709,3217],{"className":6710,"style":3286},[1926,1927],[1810,6712,1847],{"className":6713},[1939],[1810,6715,3876],{"className":6716},[1926],[1810,6718,6720],{"className":6719},[1926],[1810,6721,1904],{"className":6722},[2152],[1810,6724,4013],{"className":6725},[1926],", so ",[1810,6728,6730,6750],{"className":6729},[1817],[1810,6731,6733],{"className":6732},[1821],[1823,6734,6735],{"xmlns":1825},[1828,6736,6737,6747],{},[1831,6738,6739,6741,6743,6745],{},[1834,6740,3217],{},[1838,6742,1850],{},[1879,6744,6183],{},[1834,6746,4992],{"mathvariant":2563},[1906,6748,6749],{"encoding":1908},"\\theta=10\\%",[1810,6751,6753,6771],{"className":6752,"ariaHidden":1903},[1913],[1810,6754,6756,6759,6762,6765,6768],{"className":6755},[1917],[1810,6757],{"className":6758,"style":2376},[1921],[1810,6760,3217],{"className":6761,"style":3286},[1926,1927],[1810,6763],{"className":6764,"style":1944},[1943],[1810,6766,1850],{"className":6767},[1948],[1810,6769],{"className":6770,"style":1944},[1943],[1810,6772,6774,6777],{"className":6773},[1917],[1810,6775],{"className":6776,"style":5157},[1921],[1810,6778,6780],{"className":6779},[1926],"10%",[6132,6782,6783],{},"Surplus may cross below zero before year 10 and later recover; ruin records the first crossing, not only the endpoint.",[6132,6785,6786],{},"Drift is non-positive; under the usual conditions ultimate ruin occurs with probability one.",[1802,6788,6790],{"id":6789},"further-reading","Further reading",[6129,6792,6793,6801],{},[6132,6794,6795,6796,6800],{},"Asmussen, S. and Albrecher, H. (2010), ",[6797,6798,6799],"em",{},"Ruin Probabilities",", 2nd ed.",[6132,6802,6803,6804,6800],{},"Dickson, D. C. M. (2016), ",[6797,6805,6806],{},"Insurance Risk and Ruin",{"title":10,"searchDepth":6808,"depth":6808,"links":6809},2,[6810,6811,6812,6813,6814,6815,6816,6817,6818],{"id":1804,"depth":6808,"text":1805},{"id":3049,"depth":6808,"text":3050},{"id":3371,"depth":6808,"text":3372},{"id":3836,"depth":6808,"text":3837},{"id":5293,"depth":6808,"text":5294},{"id":5380,"depth":6808,"text":5381},{"id":6123,"depth":6808,"text":6124},{"id":6158,"depth":6808,"text":6159},{"id":6789,"depth":6808,"text":6790},"Analyse when a classical insurer surplus process crosses zero and preserve the model's practical boundaries.","md",{"sidebar":6822},{"order":6823},0,true,{"title":928,"description":6819},"0HEnfjXB11VvWIZteAydyvGsz2DSvpg3vAaz1un4J1E",[6828,6830],{"title":924,"path":925,"stem":926,"description":6829,"children":-1},"Calculate VaR and TVaR, model common shocks, and preserve the boundary between risk theory and regulation.",{"title":934,"path":935,"stem":936,"description":6831,"children":-1},"Simulate claim arrival times, estimate confidence intervals, and compare gross with economically charged reinsurance.",1785754737234]