Convex Risk LLC

Sections and Slides by Module

Lettered headings are links to the PDF file for each module. Bold bullets are sections. Numbered bullets correspond to individual page(s) in each presentation. Sub-bullets highlight content on each slide.

A. Introduction

Table of Contents
1. Module A Contents
A.01. Course Overview
1. Objective
  - Understand different approaches to pricing risk
2. Methodology
3. Questions
  - Pricing and Return Related
  - Capital Structure and Reinsurance Related
4. Findings
  - Pricing
  - Pooling and reinsurance
5. Applications
  - Methodology has many applications
Appendix A.I. About the Author

B. Market Assumptions

Table of Contents
1. Module B Contents
B.01. Market Assumptions
1. All Models Are Wrong—Some Are Useful
  - Objective: build a useful simple model
2. Simple Market Model: Initial and Final Transactions in a One-Period Model
3. Simple Market Model: Initial and Final Transactions With an Insolvent Outcome
4. Simple Market Model
  - Three obvious questions
5. Accounting Definitions
6. Many Types of Risk Capital
  - On- and off-balance sheet capital
7. Why Equity is Special
  - Non-equity capital has a pre-negotiated cost
  - Non-equity cost characteristics
8. Cost of Capital vs. Cost of Equity
  - Weighted Average Cost of Capital (WACC)
  - Target ROE
9. What Is the Appropriate Level of Assets?
  - Regulators or rating agencies determine or heavily influence asset levels
10. Insurer Management: Conflicting Objectives
  - Four insurance market constraints
11. Two Paradigms of Insurance Pricing
  - Financial and actuarial pricing
12. Ingredients to Mix an Insurance Pricing Model
  - Three ingredients specify an insurance pricing model
  - Common variants
Appendix B.I. Accounting and Priority Details
1. Technical Loss Ratios vs. Combined Ratios
2. Accounting Definitions
  - Premium as Capital
  - Premium accounting
3. Default Priorities by State
4. Insurance Pricing: Three Degrees of Freedom

C. Historical US Property-Casualty Profitability and Volatility

Table of Contents
1. Module C Contents
C.01. Premium to GDP Ratio, 1923-2020e
1. Premium to GDP Ratio
2. Premium to GDP Ratio: Cyclical Growth Between 1947 and 1997
3. Premium to GDP Ratio: 1968 Watershed
4. Annual Change in Statutory Surplus, 1932 to 2020
5. Annual Change in Statutory Surplus, 1932 to 2020 With 10- And 25-Year Rolling SD
6. Multi-Year Change in Statutory Surplus, 1932 to 2020
7. Premium to Surplus Ratio, 1932 to 2020
8. Premium to GDP vs. Suprlus to GDP
9. Three Phases of the Market 1931-2020
10. Market Dynamics Since 1986 Explained by Prior Year Surplus Levels
C.02. Surplus to GDP Ratio, 1931-2020e
1. Surplus to GDP Ratio, 1931-2020e
2. The Surplus to GDP Ratio Has Increased During the Modern Period, 1968-2020
C.03. Industry Volatility and Profitability Metrics, 1985-2019
1. Industry Net Volatility and Profitability Metrics, 1985-2019
2. Insurance Volatility and Profitability Metrics: Underwriting and Operating
3. Insurance Volatility and Profitability Metrics: Investments and Returns
4. Insurance Volatility and Profitability Metrics
  - Gloss for rows
5. Time Series of Volatility and Profitability Metrics
6. Time Series of Volatility and Profitability Metrics
  - Title decoder: Net Earned Premium; SD=1.33e+11 linear fit, r2=0.967, rse0=2.46e+10
  - Line legend
  - Interpretation
C.04. Direct and Net Volatility and Profitability by Major Line, 1992-2019
1. Direct and Net Calendar Year Loss Ratio Average, Standard Deviation, Coefficient of Variation, Skewness, and Kurtosis by Major Line
2. Direct and Net Calendar Year Combined Ratio Average, Standard Deviation, and Coefficient of Variation by Major Line
3. Direct and Net Average Loss Ratio and Combined Ratio by Line
4. Direct and Net Average Combined Ratio vs. Standard Deviation Combined Ratio
5. Mapping Statutory Lines to Major Lines
  - US Statutory lines grouped as follows
6. Loss Ratio Correlation by Major Line
7. Loss Ratio, Premium Change, and Loss Change Correlation
8. Correlation Between Annual Change in Losses, by Major Line
9. Correlation Between Annual Change in Premium, by Major Line
10. Direct Loss Ratio Statistics by Major Line of Business
11. Net Loss Ratio Statistics by Major Line of Business
12. Loss Ratio Statistics by Major Line of Business
  - Key
13. Direct Loss Ratio Time Series by Major Line
14. Net Loss Ratio Time Series by Major Line
15. Loss Ratio Time Series by Major Line
  - Title decoder: CMP; SD=0.111 (0.0661), cor=0.915 ar fit, r2=0.268, rse0=0.089
  - Interpretation
  - Line Legend
16. Total Loss Ratio Time Series, Direct and Net
17. Direct Loss Ratio Time Series by Major Line, Loss Ratio Scale 0 to 150 Percent
18. Net Loss Ratio Time Series by Major Line, Loss Ratio Scale 0 to 150 Percent
19. Some Lines Are More Volatile Than Others…
20. Reinsurance More Effective for Some Lines Than Others…
21. Direct Premium and Loss Dynamics
22. Net Premium and Loss Dynamics
23. Direct Premium and Loss Dynamics
24. Net Direct Premium and Loss Dynamics
25. Premium and Loss Dynamics
26. Calm Surface Masks Inner Turmoil
27. The Underwriting Cycle Is Driven by Commercial Lines
28. Direct and Net Premium and Loss
29. Direct and Net Combined Ratio and Loss Ratio by Line by Year
30. Net Operating Results
31. Average Margins, All Lines Combined
  - Target average margins
C.05. Direct Premium Growth by Major Line, 1992-2019
1. Average Direct Premium Growth Rates by Line (Percent)
2. Annual Change in Direct Premium by Line
3. Annual Change in Direct Premium Detailed by Line
4. Annual Change in Direct Premium by Line Common Scale
5. Premium and GDP Growth Since 1992 (1992=1.0)
C.06. Implications
1. Implications
  - Loss ratio and margin expectations
  - Margin expectations

D. Model Specification and Properties

Table of Contents
1. Module D Contents
D.01. Modeling Assumptions
1. Modeling Assumptions
  - Standard simplifying assumptions
2. Modeling Output
  - Ratios
  - Gross margin vs. net margin
D.02. Model Portfolio
1. Model Portfolio
  - Summary
2. Aggregate Program: Gross
3. Aggregate Program: Net
4. Density and Log-Density by Line
5. Properties of Model Portfolio Loss Distributions
6. VaR, TVaR, EPD, and CV by Line
7. Detailed Gross Portfolio Statistics
8. Detailed Net Portfolio Statistics
D.03. Equal Priority
1. Description of Equal Priority
D.04. ASTIN/CAS Slides
1. Alpha and Kappa Description
D.05. Effective Diversification
1. Effective Diversification
  - Characteristics of effective diversification
  - Determining if diversification is effective
  - Application to capital structure
2. Testing Effective Diversification in a Portfolio
  - Three equivalent views
  - Examples
3. Effective Diversification: Quantile View
4. Effective Diversification: Expected Loss View
5. Effective Diversification: Alpha View
D.06. Measuring the Effects of Pooling
1. Measuring the Effects of Pooling
  - Which lines benefit from pooling?
2. Conditional Layer Effectiveness
  - Stand-alone basis
  - By line on pooled basis
3. Gross Loss CLE
4. Gross Loss CLE on Return Period Scale
5. Net Loss CLE
6. Net Loss CLE on Return Period Scale

E. Distortions: Definition, Examples, and Properties

Table of Contents
1. Module E Contents
2. Summary
E.01. Distortions and Distortion Pricing Operators
1. Distortions
  - Distortion functions
  - Universal properties
2. Distortion Pricing Operator
3. Simulation Interpretation of Distortions
4. Astin / Cas

F. Cat Bonds, Their Pricing, and Its Implications for Pricing Non-Cat Lines

Table of Contents
1. Module F Contents
F.01. Catastrophe Bonds and Their Pricing
1. Catastrophe Bonds
  - Coverage, perils, geography, and triggers
2. Catastophe Bonds
  - Pricing statistics
  - Factors that influence pricing
3. Historical US Wind Cat Bond Pricing
4. Historical US Wind Cat Bond Pricing (2009-2020)
5. Cat Bond Statistics, All Years
6. Cat Bond Statistics, 2009-2020
F.02. Creating a Distortion From Cat Bond Prices in Theory
1. Creating a Distortion From Cat Bond Prices
  - Objective: use cat bond prices to calibrate a distortion that gives reasonable prices for all lines
  - Philosophies of modeling
2. Creating a Distortion From Cat Bond Prices: Difficulties
  - Data limitations
  - Equity returns
3. Creating a Distortion From Cat Bond Prices
  - Families of distortions
  - Explore four approaches, all using PLDs
4. Piecewise Linear Distortions (PLD)
  - A PLD is a weighted average of TVaRs at different probability levels
  - Advantages of PLDs
  - Gotcha
5. Point Distortions
  - A point distortion (PD) is a PLD defined by a single price point
6. Average of Points Distortions
  - Average individual prices
7. Convex Envelope Distortion
  - Convex envelope: join extreme points
8. Bagged Convex Envelope Distortion
  - Bootstrap Aggregation (bagging)
9. Regression Fits
  - One TVaR; Two TVaRs plus mean
  - Three TVaRs plus mean; bagged convex
F.03. Creating a Distortion From Cat Bond Prices in Practice
1. How Should General Insurance Market Prices Relate to Cat Bond Market Prices?
  - Cat bond market prices
  - General insurance market prices
2. Base Distortion
  - Base distortion: bagged convex envelope
3. Base Bagged Convex Envelope Distortion
4. Distortions Calibrated to Cat Bond Pricing
5. Distortions Calibrated to Cat Bond Pricing, Log Scale
6. Gross Portfolio Statistics by Cat Bond Distortion, 2009 to 2020
7. Net Portfolio Statistics by Cat Bond Distortion, 2009 to 2020
8. Gross and Net Loss Ratio by Cat Bond Distortion
9. Gross and Net Loss Ratio and Roe by Cat Bond Distortion
10. Pricing Non-Cat Business Using Cat Bond Data
11. Gross and Net Loss Ratio by Line and Distortion on Stand-Alone Basis
12. Gross and Net ROE by Line and Distortion on Stand-Alone Basis
13. Return Discount vs Exceeding Probability for Cat Bond Distortions

G. Comparative Pricing Across Different Methods and Lines

Table of Contents
1. Module G Contents
G.01. Stand-Alone by Line and Portfolio Pricing
1. Stand-Alone by Line and Portfolio Pricing
  - Stand-Alone By Line Pricing
  - Amount of Capital
  - No Allocation
2. Stand-Alone by Line and Portfolio Pricing for Constant ROE, Bagged Bond Base Distortion, and Traditional Methods
  - Cost of Capital
  - Distortions
3. Gross Stand-Alone Traditional and Distortion
  - Data elements
  - Loss Ratio and ROE by line using stand-alone distortion pricing
4. Net Traditional and Distortion Stand-Alone
5. Net Traditional and Distortion Stand Alone
6. Gross Minus Net Statistics
7. Concluding Comments
  - Gross minus net does not equal ceded
  - Market premiums support optimal capital structure
G.02. Gross and Net Line-Level and Portfolio Pricing
1. Roadmap: Multiline Pricing
  - Analysis proceeds in three steps
2. Multiline Pricing
  - Allocation
  - Losses
  - Distortion methods
3. The Natural Allocation
  - The natural allocation is a conditional measure approach to premium allocation that applies to distortion methods
4. Gross Traditional Stand-Alone vs Natural Allocation Margin by Line
5. Net Traditional Stand-Alone vs Natural Allocation Margin by Line
6. Summary of Gross and Net Margin and Assets: Natural Allocation, Base
7. Summary of Gross and Net Margin and Assets: Natural Allocation, Constant Roe
8. Gross and Net Margin and Assets: Natural Allocation
  - Offsetting Dynamics
9. Gross and Net Loss Ratios by Method by Line
10. Gross to Net Transition: Summary
  - In total, gross to net shows…
  - By line, gross to net shows…
11. Gross Loss Ratio by Cat Bond Distortion
12. Net Loss Ratio by Cat Bond Distortion
G.03. Line and Portfolio Multiline Pricing Across Multiple Methods
1. Road Map: Multiline Pricing
  - Analysis proceeds in three steps
2. Overview
  - Recall objective: understand different approaches to pricing risk
  - Common calibration
3. Pricing Methods
  - Distortion methods
  - What are traditional methods?
  - Traditional methods
4. Calibrated Standard Distortions
5. Calibrated Standard Distortions, Log Scale
6. Roe vs Exceeding Probability for Standard Distortions
7. ROE vs. Exceeding Probability
  - Distinguishing distortion behavior
8. Gross Loss Ratio, by Method by Line
9. Net Loss Ratio, by Method by Line
10. Gross Margin, by Method by Line
11. Net Margin, by Method by Line
12. Gross Allocated Capital, by Method by Line
13. Net Allocated Capital, by Method by Line
14. Gross Leverage, by Method by Line
15. Net Leverage, by Method by Line
16. Gross ROE, by Method by Line
17. Net Roe, by Method by Line
18. ROE by Method by Line
  - Hard to interpret
19. Gross and Net Pricing, by Method by Line
20. Loss Ratio Ranges, by Method by Line
G.04. Conclusions and Answers
1. Questions & Answers
  - 1. How does price vary with risk? With volatility vs. tail risk?
  - 1. Can reasonable risk loads for non-cat lines be estimated from cat bond price data?
  - 1. Are negative risk loads ever appropriate? Why? When?
  - 1. Should all lines target the same return on capital (ROE)?
  - 1. Should target ROE be calibrated to gross or net returns? Are the two the same?
  - 1. Can the same distortion be used to price gross and net? All lines?
  - 1. Is pricing in one line independent of reinsurance on other lines?

H. Convex Envelopes (file does not exist)

I. Theory in Practice and the Cost of Capital for Insurance Risk

Table of Contents
1. Module I Contents
I.01. Introduction and Purpose
1. What Is the Cost of Capital for Insurance Risk?
2. Equity vs. Capital: Two Measures of Net Assets
  - Capital
  - Equity
3. More Accounting Basics
  - Accounting assets and liabilities
  - Accounting conventions
4. Fundamental Equation of Insurance Accounting
5. Traditional Accounting Presentation Distinguishes Insurance From Reinsurance
  - Example, assuming assets a
  - Traditional reinsurance accounting
  - Economic reinsurance accounting
6. Traditional Reinsurance Accounting
  - Assets
  - Liabilities
7. Unified Insurance and Reinsurance Accounting Presentation
  - Insurance transaction
  - Reinsurance transaction
8. Economic Reinsurance Accounting
  - Assets
  - Liabilities
9. Capital Structure
  - Simple capital (liability) structure
  - Illustrative capital structure
  - Equity is unique: it is the only residual value
  - Non-equity capital has an explicit cost
  - Non-equity cost characteristics
  - Dollar cost of capital vs. unit cost
  - Impact of pooling on cost of capital
10. Cost of Capital
  - Why the complicated capital structure?
  - Weighted Average Cost of Capital (WACC)
11. Plan From Hereon
I.02. Notional Portfolios
1. Notional Portfolios
  - Apply theory to four notional portfolios
  - Portfolio characteristics
2. Four Notional Portfolios, 1 and 2
  - 1. discrete
  - 1. balanced
3. Four Notional Portfolios, 3 and 4
  - 1. mixed
  - 1. cat
4. Notional Portfolio Statistics
5. Notional Portfolio Densities, Log Densities, and Survival Functions
I.03. Insurance Pricing and Insurer Capital Structure
1. Pricing Statistics for Each Notional Portfolio
2. All-Equity Capital (Liability) Structures
  - discrete
  - balanced
3. All-Equity Capital Structures
  - mixed
  - cat (different height scale)
I.04. Comparative Pricing: Traditional, Stand-Alone
1. Traditional Pricing
  - Stand-alone traditional pricing
  - All-or-nothing layer pricing
  - Limited Liability
  - Premium under limited liability
2. Traditional Pricing Comparative Analysis Across Portfolios
I.05. Comparative Pricing: Distortion, Stand-Alone
1. Distortion Functions: Rate on Line vs. Expected Loss for an All-Or-Nothing Layer
2. More General Distortion Functions
3. Proportional Hazard Distortions Reproducing Total Pricing for Each Portfolios
4. Pricing With General Distortion Functions
  - Distortion Pricing Operator
  - What distortion should we use?
5. Cat Bond Price + POYA ROE Calibrated Distortion
  - Constant ROE
  - Fit to cat bond prices
  - Cat with POYA ROE factor
6. Tail Behavior and Return on Capital by Distortion
7. Why Should Insurance Equity Returns Be Higher Than Cat Bond Returns?
  - Cat Bonds Returns
  - Insurance Equity Returns
8. Stand-Alone Pricing Comparison: Stunner—Returns Vary by Line!
I.06. Comparative Pricing: Distortion, Multiline With Allocation
1. The Natural Allocation
2. Detailed Pricing Statistics for
3. Detailed Pricing Statistics for
4. Detailed Pricing Statistics for
5. Detailed Pricing Statistics for
6. Efficient Capital Structure
  - Integrating alternative capital
  - Comparing three pricing distortions
7. Blended Distortion
8. Structure Diagram Key
  - Left: Assets
  - Middle: Details by Line
  - Right: Liabitites
9. Capital Structure for With All Equity Financing
  - Capital structure and tranche returns for roe distortion
10. Capital Structure for With Blended Equity and Alternative Capital Financing
  - Capital structure and tranche returns for blended distortion
11. Capital Structure for With All Equity Financing
  - Capital structure and tranche returns for roe distortion
12. Capital Structure for With Blended Equity and Alternative Capital Financing
  - Capital structure and tranche returns for blended distortion
13. Capital Structure for With All Equity Financing
  - Capital structure and tranche returns for roe distortion
14. Capital Structure for With Blended Equity and Alternative Capital Financing
  - Capital structure and tranche returns for blended distortion
15. Capital Structure for With All Equity Financing
  - Capital structure and tranche returns for roe distortion
16. Capital Structure for With Blended Equity and Alternative Capital Financing
  - Capital structure and tranche returns for blended distortion
17. Capital Structure: AAA Capital Standard, All Equity Capital
  - Structure diagram
18. Capital Structure: AAA Capital Standard, All Equity
19. Capital Structure: AAA Capital Standard, All Equity Capital
  - Structure diagram
20. Capital Structure: AAA Capital Standard, All Equity, Blended Distortion
21. Standard and Poor’s Default Probabilties by Bond Rating
I.07. Comparative Pricing: Distortion, Multiline With Allocation With Stricter Capital Standard
1. Distortion and Traditional Pricing: Details With Sticter Capital Standard
I.08. Conclusions and Next Steps
1. Conclusions and Next Steps
  - Conclusions
  - Next Steps
Appendix I.I. About

K. Gradients: Ibrgimov, Jaffee and Walden vs. Aumann-Shapley (file does not exist)

L. Simple Nine Scenario Example

1. Q2 
2. Q245 
3. Q 
4. Q3 
5. Q4

Pricing Insurance Risk Course (PIRC)

Sections by Module