A Day’s Work for New Dimensions an International Consulting Firm

1. A Day’s Work for New Dimensionsan International Consulting Firm Glenn Meyers Insurance Services Office, Inc. CAS/ARIA Financial Risk Management Seminar

2. DFA - Dynamic Financial Analysis • Coined by the CAS in 1994. • Best defined in terms of the problems it seeks to solve. • How much capital does an insurer need? • For how much time is the capital needed? • What decisions does an insurer make to provide the greatest return on its capital? • Underwriting • Asset management (Include hedges)

3. Outline of Talk • Multi-dimensional aspects of insurer capital management • Provide simple (perhaps artificial) examples focusing on particular dimensions. • Short and long tailed lines • Catastrophe options and reinsurance • Describe (but not solve) a multi-dimensional insurer problem in capital management. • Compare approach with efficient frontier methods.

4. Assignment #1 Lineland Life Insurance Company • Writes one life insurance policy • Face value $1 • t is the term of the policy • Mortality assumptions • Probability of death in [0,t] = q • Uniform distribution of deaths within [0,t]

5. Assignment #1 Lineland Life Insurance Company • Investors provide $1 of capital. • Capital is invested at rate I compounded continuously. • In return for exposing the capital to loss they demand a return of R compounded continuously. R > I • Find minimum premium, P, it must get.

6. Assignment #1 Lineland Life Insurance Company Case 1 - Claim occurs at time T The return is a continuous annuity of I

7. Assignment #1 Lineland Life Insurance Company Case 2 - Claim does not occur Return = PV[Annuity] + PV of Capital

8. Assignment #1 Lineland Life Insurance Company • Receives P immediately. • Receives annuity until claim occurs or the term ends. 1 = P + E[PV with Claim] + E[PV without Claim]

9. Assignment #1 Lineland Life Insurance Company P increases when capital must be held longer.

10. Background - Capital RequirementsDefine Terms

11. Background - Capital RequirementsThree Formulas #1 Probabililty of Ruin  is determined by judgment of insurer management. Insurer management always knows what the rating agencies - NAIC, Best, S&P think they should have. Value at Risk -- VaR = C+E[X]

12. Background - Capital RequirementsThree Formulas #2 Expected Policyholder Deficit (EPD)  is determined by judgment of insurer management. Sensitive to amount of insolvency

13. Background - Capital RequirementsThree Formulas #3 Standard Deviation Formula C  T T is determined by judgment of insurer management. Normal approximation to ruin formula, but you can use this formula as is. Easiest to work with

14. Assignment #2 Lineland Property Insurance Company • Losses have a Gamma(100,100) distribution. • Claims settle quickly • Time value of money is not an issue. • Investors expect 10% ROE. • Find the Cost of Capital.

15. Gamma Distribution Mathematics Cumulative Distribution Function Excel Formula Expected Value

16. Gamma Distribution Mathematics Limited Expected Value (LEV) Function Excel Formula Variance

17. Assignment #2 Lineland Property Insurance Company Probability of Ruin • E[X] = 10,000 • F(12,472) = 0.99 Capital = 2,472 @ 1.0% Level • Cost of capital = 247

18. Assignment #2 Lineland Property Insurance Company Expected Policyholder Deficit • E[X] = 10,000 • LEV[12,091] = 9,990 Capital = 2,091 @ 0.10% Level • Cost of Capital = 209

19. Assignment #2 Lineland Property Insurance Company Standard Deviation • E[X] = 10,000 • Std[X] = 1000 • Select T = 2.33 Capital = 2,330 • Cost of Capital = 233

20. Cost of Capital Depends Upon: Economic Environment e.g. interest rates How long Capital is held Volatility of Net Worth

21. Parameter Uncertaintyfor Gamma(,) • Let  be a random variable • E[] = 1 • Var[] = b • Select  at random • Conditional distribution given  Gamma(,)

22. Parameter Uncertainty for Gamma(,)A simple, but nontrivial example E[] = 1 and Var[] = b

23. Assignment # 2´Capital Requirements with Parameter Uncertainty

24. Assignment # 2´Capital Requirements with Parameter Uncertainty Probability of Ruin • E[X] = 10,000 • FU(14,443) = 0.99 Capital = 14,443 @ 1.0% Level • Cost of capital = 444

25. Assignment # 2´Capital Requirements with Parameter Uncertainty

26. Assignment #3Lineland Property Insurance CompanyConsiders Renewing a Policy • The renewal business has a Gamma(100,1) loss distribution. • Lineland has a Gamma(100,99) loss distribution without the renewal. Property of the Gamma Distribution • Lineland has a Gamma(100,100) loss distribution with the renewal. This Property Assumes Independence

27. Assignment #3Lineland Property Insurance CompanyConsiders Renewing a Policy • What is the marginal capital needed for the renewal business? • Calculate capital needed without the business. • Calculate capital needed with the business. • Marginal capital is the difference.

28. Assignment #3´Find Marginal Capital Assuming Parameter Uncertainty • The random variable  affects all business (including renewal) simultaneously. • The renewal’s  parameter changes at the same time as the  for the remaining business. • The renewal’s losses are correlated with the rest of the losses. In case you are interested --  = 0.195

29. Assignment #3 and #3´Results With Parameter Uncertainty Total Capital  Double Marginal Capital  Triple +

30. How do you use the marginal cost of capital? • Allocate the total cost of capital in proportion to the marginal cost of capital. • No consensus among actuaries yet. • Add the allocated cost of capital to the expected loss and expense to see if you can make money at the “going market premium.” • Can be done at individual insured level, or the line of business level.

31. Assignment #4Flatland Casualty Insurance Company • Claim count distribution is negative binomial - by settlement lag. • Claim severity distribution is mixed exponential - by settlement lag.

32. Assignment #4Flatland Casualty Insurance CompanyOutstanding Aggregate Loss Statistics The aggregate loss model included parameter uncertainty affecting all claim count distributions simultaneously. (g =.02 - analogous to b =.02 above.)

33. Assignment #4Flatland Casualty Insurance CompanyCapital is released over time as losses are paid.

34. Assignment #4Flatland Casualty Insurance Company What is the cost of providing the capital? i = Interest rate on invested capital r = Rate of return needed to attract capital. C0 = Capital needed at beginning of year 0. The cost of capital, R, satisfies:

35. Assignment #4Given i = 6% and r = 10%What is the cost of providing the capital?

36. Asset Management Reinsurance and Catastrophe Options • “Value will be determined not by the ability of an [insurance] enterprise to accumulate capital and sit on it. • Rather it will be determined by a company’s franchise with its customers and its ability to originate risk. • In this scenario the capital markets become the more efficient warehouse of [insurance] risk.”

37. Asset Management Reinsurance and Catastrophe Options • Reduce the cost of financing insurance • Expected insurer costs • Cost of Capital • Cost of Capital Substitutes • Reinsurance • Contracts on a catastrophe index • Find the right mix of capital and capital substitutes

38. Quantifying the Cost of Capital • We use the “easy” formula • Cost of Capital = K  T  • Where: •  = Standard deviation of total loss • T = Factor reflecting risk aversion • K = Rate of return needed to attract capital

39. Quantifying Basis Risk Ran RMS cat model through insurers and index. + about 9000 more • Compare variability before and after • Is the risk reduction worth the cost?

40. Minimize Sum of Cost Elements • Insurer Capital • Cost of Capital = K  T (Net Losses) • Reinsurance • Transaction Cost + Expected Cost • Cat index contracts • Transaction Cost + Expected Cost • Use cat model results to back out transaction costs.

41. References Missing transaction costs are in the first paper. • “The Cost of Financing Catastrophe Insurance” by Glenn Meyers and John Kollar - 1998 DFA Call Paper Program • Catastrophe Risk Securitization: Insurer and Investor Perspectives” by Glenn Meyers and John Kollar - 1999 CAS Spring Meeting Call Paper Program

42. Assignment #5 Analyze Three Insurers • Insurer #1 - A medium national insurer • Highly correlated with the index • Insurer #2 - A large national insurer • Moderately correlated with the index • Insurer #3 - A small regional insurer • Slightly correlated with the index

43. Search for Best Strategy to Minimize Cost of Financing Insurance • Search for the combination of index and reinsurance purchases that minimizes total cost of providing insurance. • Questions • How many index contracts at each strike price? • What layer of reinsurance?

44. Results of Search

45. Financing With Reinsurance and Catastrophe Options

46. Financing Without Reinsurance and Catastrophe Options

47. Differences in Costs

48. Assignment #6Spaceland Property and Casualty • Short tailed property exposure • Include catastrophe exposure • Long tailed casualty exposure • Include unsettled claims from prior years • Capital Management Questions • Catastrophe options/reinsurance? • Casualty reinsurance?

49. Assignment #6Spaceland Property and Casualty Underwriting Management Decisions • Allocate the cost of capital to the lines of insurance - in proportion to the marginal cost of capital. • Allocate the cost of reinsurance and/or catastrophe options to the lines of insurance - in proportion to the marginal costs.

50. Assignment #6Information and Technology Requirements • An Aggregate Loss Model • Size of loss distributions by settlement lag • Correlation structure between lines of insurance • A catastrophe model • Exposure underlying catastrophe index