Stephen J. Mildenhall - Pricing Insurance Risk

11 Part IV Advanced Topics 16 Asset Risk 16.1 Background16.2 Adding Asset Risk to Ins Co.16.3 Learning Objectives 17 Reserves 17.1 Time Periods and Notation17.2 Liability for Ultimate Losses17.3 The Solvency II Risk Margin17.4 Learning Objectives 18 Going Concern Franchise Value 18.1 Optimal Dividends18.2 The Firm Life Annuity18.3 Learning Objectives 19 Reinsurance Optimization 19.1 Background19.2 Evaluating Ceded Reinsurance19.3 Learning Objectives 20 Portfolio Optimization 20.1 Strategic Framework20.2 Market Regulation20.3 Dynamic Capital Allocation and Marginal Cost20.4 Marginal Cost and Marginal Revenue20.5 Performance Management and Regulatory Rigidities20.6 Practical Implications20.7 Learning Objectives

12 A Background Material A.1 Interest Rate, Discount Rate, and Discount FactorA.2 Actuarial vs. Accounting Sign ConventionsA.3 Probability TheoryA.4 Additional Mathematical Terminology

13 B Notation

14 References

15 Index

16 End User License Agreement

1 Chapter 1Figure 1.1 Players and their roles. The regulator...Figure 1.2 The different roles of capital and pricing risk measures.

2 Chapter 2Figure 2.1 The eight variables that control...Figure 2.2 Tame Case Study, gross (top...Figure 2.3 Tame Case Study, bivariate densities:...Figure 2.4 Cat/Non-Cat Case Study, gross...Figure 2.5 Cat/Non-Cat Case Study, bivariate...Figure 2.6 Hu/SCS Case Study, gross (top)...Figure 2.7 Hu/SCS Case Study, bivariate densities...

3 Chapter 3Figure 3.1 Taxonomy of insurance-related risk.Figure 3.2 The original Lee diagram. Source...Figure 3.3 Different ways of computing...Figure 3.4 The random variables, distribution...Figure 3.5 Random variables, functions of an explicit state.Figure 3.6 Survival functions of the outcome.Figure 3.7 Lee diagrams, function of a dual...Figure 3.8 Different ways of computing the...Figure 3.9 Sample layering functions for Exercise 26.Figure 3.10 Insurance variables in a Lee diagram.Figure 3.11 Two ways of computing expected loss from a discrete sample.Figure 3.12 Accounting for the effect of adding 1 to each outcome.Figure 3.13 Lee diagram showing relationship between...Figure 3.14 Potential functional forms for risk...

4 Chapter 4Figure 4.1 The two ways a distribution...Figure 4.2 Three types of discontinuous behavior...Figure 4.3 Five independent Bernoulli random...Figure 4.4 Quantile function for the outcomes...Figure 4.5 Left: the distribution function F corresponding...Figure 4.6 Left: the interpolated distribution function...Figure 4.7 Crossed (darker) and uncrossed or...Figure 4.8 Left: TVaR (solid) and quantile...Figure 4.9 Discrete example, TVaR, and VaR...Figure 4.10 Tame Case Study, TVaR, and VaR...Figure 4.11 Cat/Non-Cat Case Study, TVaR,...Figure 4.12 Hu/SCS Case Study, TVaR, and...Figure 4.13 Left: the graph of...Figure 4.14 TVaR, CTE, and quantile functions for X .Figure 4.15 VaR and TVaR as solutions to an...

5 Chapter 5Figure 5.1 The basic structure of Ins...Figure 5.2 Relationships between properties...Figure 5.3 The graph of...Figure 5.4 Acceptance sets and corresponding...Figure 5.5 The distribution (left) and...Figure 5.6 The impact of...Figure 5.7 A typical real valued convex...Figure 5.8 Cherny’s generator set for...Figure 5.9 Different modes of convergence for random variables.

6 Chapter 8Figure 8.1 Services provided by an insurance...Figure 8.2 Interactions between the product and...Figure 8.3 A simple (left) and realistic capital structure (right).Figure 8.4 Capital, surplus, and equity under market value...Figure 8.5 Market value and accounting value balance...Figure 8.6 Sample geometric Brownian motion paths,...Figure 8.7 Sample compound Poisson paths,...

7 Chapter 9Figure 9.1 Left column: Pollaczeck-Khinchine formula

8 Chapter 1Figure 10.1 Cash flows corresponding to bid...Figure 10.2 The function...Figure 10.3 Left: graph of a distortion function...Figure 10.4 Left: Graph of the survival function...Figure 10.5 Continuum of risk sharing varying by layer...Figure 10.6 Reconciliation of Figure 10.3 and Figure 10.5....Figure 10.7 Relationships between the variables that...Figure 10.8 Plot of p against q(p) (dashed black)...

9 Chapter 11Figure 11.1 The Lee diagram illustrating objective and distorted...Figure 11.2 Distortion envelope for Discrete Example...Figure 11.3 Distortion envelope for Tame Case Study...Figure 11.4 Distortion envelope for Cat/Non-Cat Case...Figure 11.5 Distortion envelope for Hu/SCS...Figure 11.6 Tame Case: Variation in premium, loss ratioFigure 11.7 Cat/Non-Cat Case: Variation in premium...Figure 11.8 Hu/SCS Case: Variation in premium, loss...Figure 11.9 Tame Case: Variation in SRM properties as the asset...Figure 11.10 Cat/Non-Cat Case: Variation in SRM...Figure 11.11 Hu/SCS Case: Variation in SRM properties...Figure 11.12 Spread (ROL) vs. EL on US wind (hurricane...Figure 11.13 Number of issues, amount of issuance...Figure 11.14 LEP parameters i and t *by year implied by...Figure 11.15 A convex envelope distortion defined by...Figure 11.16 A bagged (bootstrap aggregated) convex...Figure 11.17 As Figure 11.16, on log scale.

10 Chapter 12Figure 12.1 Development of insurance pricing financial model...Figure 12.2 The order of decision making in an...Figure 12.3 The order of decision making in an insurance market...Figure 12.4 Assumptions from various important...

11 Chapter 14Figure 14.1 Computation of absolute semideviation with...Figure 14.2 Contact functions for TVaR, illustrating the...Figure 14.3 Conditions and properties for allocations of coherent...Figure 14.4 Relationship between game theory, allocations...

12 Chapter 15Figure 15.2 Gross Tame Case Study, CCoC distortion.Figure 15.3 Net Tame Case Study, TVaR distortion.Figure 15.4 Gross Cat/Non-Cat Case Study, dual moment distortion.Figure 15.5 Net Cat/Non-Cat Case Study, proportional hazard distortionFigure 15.6 Gross Hu/SCS Case Study, blended distortion.Figure 15.7 Net Hu/SCS Case Study, Wang transform.Figure 15.8 Capital density by layer for the Tame Case.Figure 15.9 Capital density by layer for the Cat/Non-Cat Case.Figure 15.10 Capital density by layer for the Hu/SCS Case.Figure 15.11 Loss spectrums by Case (top row)....Figure 15.12 Tame Case Study percentile layer of capital...Figure 15.13 Cat/Non-Cat Case Study percentile layer of capital...Figure 15.14 Hu/SCS Case Study percentile layer of capital...

13 Chapter 18Figure 18.1 Sample path for surplus process...Figure 18.2 Left: sample path for surplus process with...

14 Chapter 19Figure 19.1 Technical premium for net...

1 Chapter 2 Table 2.1 Investor and insured... Table 2.2 Simple Discrete Example... Table 2.3 Discrete Example estimated... Table 2.4 Loss distribution assumptions... Table 2.5 Tame Case Study... Table 2.6 Cat/Non-Cat Case Study... Table 2.7 Hu/SCS Case Study...

2 Chapter 3 Table 3.1. Representing risks: event... Table 3.2. Simple Discrete Example...1 Table 3.3 Solution to Exercise 30. Table 3.4. Computing the limited expected...

3 Chapter 4 Table 4.1. Standard return periods... Table 4.2. VaRp(X) estimated as E[X0]... Table 4.3. Estimated occurrence and... Table 4.4. The sum of two independent... Table 4.5. Starting X is shown in the first...Table 4.6. Tame Case Study estimated...Table 4.7. Cat/Non-Cat Case StudyTable 4.8. Hu/SCS Case Study estimated...

4 Chapter 6Table 6.1. A compendium of risk measures

5 Chapter 7Table 7.1. Risk measures applied to...Table 7.2. Pricing summary by Case...Table 7.3. Details of reinsurance for...

6 Chapter 8Table 8.1. Total expenses by function...Table 8.2. Comparison of types of insurer capitalTable 8.4. Comparison of solvency frameworks...Table 8.5. Actuarial vs. Black-Scholes pricing