CAS Ratemaking Seminar March 2004 INT-7 Introduction to Profit Provision Calculations

1. CAS Ratemaking SeminarMarch 2004INT-7Introduction to Profit Provision Calculations Ira Robbin, PhD Partner Re

2. The purpose of this session is to educate actuaries in various methods used to compute the underwriting profit provision. There will be no discussion of the adequacy of the premium charge for any particular consumer or particular class of consumers. All attendees should scrupulously follow anti-trust guidelines. Ground Rules

3. No statements of Partner Re’s corporate position will be made or should be inferred. While some methods may be similar to methods promulgated by regulatory authorities, practitioners should follow actual regulatory instructions. While some methods to be discussed are similar to methods in the Part 9 Study Note, students should consult the Study Note for exact details. Disclaimers

4. Examples are for illustrative purposes only. Do not use the results from any example in real-world applications. The profit load indicated from a model often depends critically on the assumptions and parameters. For ease of presentation, assumptions have been greatly simplified and hypothetical parameters have been selected. Cautions

5. Overview • UW Profit Basics • Overview of Different Methods • Corporate and Regulatory Contexts • Offset Formulas • DCF and Risk-Adjusted DCF • Single Policy Company Models • Conclusion

6. Different Types of UW Profit • Actual Achieved • Booked to Date vs Ultimate • PY, AY, CY • Direct, Gross, Ceded, Net • Stat vs GAAP • Provision in Manual Rate • Indicated, Filed, Approved • Provision in Charged Premium

7. UW Profit: Basic Equations • U = P-L-X = UPM*P X = Expense including premium tax • CR = (L+X)/P= 1- UPM UPM of –100% yields CR =200% • X = FX +VXR*P FX = Fixed expense VXR = Variable expense ratio • P= (L+FX)/(1-VXR-UPM)

8. UW Profit Provision Chart

9. Examples • L=50 FX=30 • VXR=15% UPM = 5% • Result: • P= (50 + 30)/(1-.15-.05) = 100 • L=50 FX=30 • VXR=15% UPM = -1% • Result: P= (50 + 30)/(.86) = 93 • Note UPM can be negative!

10. UPM Calculation Approaches • Investment Income Adjustment • CY Inv Offset and PV Differential • Adequate Total Return • Ratemaking CY ROE • Economic return via Single Policy model • IRR on Equity Flow and PVI/PVE • Economic Components • DCF and Risk-adjusted DCF

11. Different UPM Calculation Methods • 1. CY Inv Offset • 2. PV Differential • 3. Ratemaking CY ROE • 4. DCF • 5. Risk-Adjusted DCF • 6. IRR on Equity Flow • 7. PVI/PVE

12. Corporate vs Regulatory Contexts • Corporate: UPM targets by LOB • Maximize economic return net of risk • Regulatory: Allowed UPMs • Manual rates by LOB • Philosophy of regulation • State controlled vs free market approaches • Affordability and availability • Legislated rate environments • File and use/Use and file • Market pricing for large risks

13. Recap of UW Profit Regulation • 1920’s – 1970’s: Low interest era • No consideration of investment income • 5.0% UPM for most lines • 2.5% for WC • 1970’s – 90’s: High rate era • Investment income offsets • CAPM, DCF and Risk-Adjusted DCF • IRR on Equity Flows and PVI/PVE

14. Method 1: CY Investment Income Offset (State X) • UPM = UPM0 – IIOffset • UPM0 = Traditional UPM • IIOffset = Investment Income Offset • IIOffset = iAT ·PHSF • Based on After-tax realized CY returns • Actual portfolio mix of invested assets • Base of PH-Supplied Funds

15. Policyholder Supplied Funds • Unearned Premium Balances • UEPR(1-PPACQR) - RECV Net of Pre-paid Acquisition Expense Net of Receivables • Loss+ LAE Reserves • PLR·(LRES/INCL) CY Reserves-to-Incurred Ratio PLR =Permissible Loss Ratio Ratio of Loss reserve to incurred loss

16. CY II Offset- Example

17. Method 2: Offset for PV Differential • UPM = UPM0 - PVDELLR UPM0 = Traditional UPM PVDELLR = Present Value Differential • Present Value Differential • PVDELLR = PLR·(PV(X0)- PV(X)) X0 = Loss Pattern for Reference LOB X = Loss Pattern for Review LOB Interest Rate: New money after-tax

18. PV Differential Offset- Example

19. Method 3: Ratemaking CY ROE • Start with ROE equation: • Assume S= EQ • Simplify taxes • Split INV into INV on PHSF vs INV on S

20. Ratemaking CY ROE Premium to Surplus Ratio

21. ROE in Ratemaking? • GAAP vs Statutory • Going-concern vs Solvency • Stat defined by state regulation • Calendar Yr vs Policy Yr • ROE is CY • Past decisions impact this CY • Ratemaking is PY and prospective

22. Surplus in ROE Equation • S = Target Statutory Surplus S = P/l l = Premium-to-Surplus leverage ratio l varies by LOB • Equity vs Surplus

23. Solve for UPM • Find UPM to hit CY ROE target

24. Ratemaking CY ROE - Example

25. Method 4: Discounted Cash Flow • Prospective cash flow approach founded in modern economic theory • UPM = -krf +b(E[rm] – rf) • k = funds generating coefficient • rf = risk-free new money rate • rm= market return • b = systematic covariance

26. Applying CAPM to Insurance • CAPM risk–reward concept • Reward for taking systematic risk • No reward for diversifiable risk • Beta =Cov of Company Stock with Market • Insurance Betas by LOB? • Few single LOB insurance companies • These don’t represent much of the market • Beta=Cov of LOB UPM with stock market? • Not right theoretically • CY achieved UPMs influenced by other factors

27. DCF - Example

28. Solve for UPM so that: Method 5:Risk-Adjusted DCF • rf = risk-free new money rate • rA = risk-adjusted rate • FIT= income tax • Loss discounted at risk-adjusted rate

29. Risk-Adjusted Rate • rA = rf + b (E[rm] – rf ) • b = Cov of liabilities with market • While b>0 for assets, the b here is for liabilities. Thus: • b<0 and rA < rf • How to get b by LOB?

30. Risk-Adjusted DCF Example

31. Method 6: IRR on Equity Flow • Equity flow: flow of $ between an equity investor and the insurance company • Model prospective equity flows for hypothetical insurance company writing one policy • Use accounting rules, surplus requirements, and other assumptions to derive income and surplus each time period. • EQF = INC - DS

32. Equity Flow Diagram

33. Income and Cash Flow • UW Gain = EP –IncLoss –IncExpense • Defined by accounting rules • Does not depend on UW cash flows • Inv Inc = II on Invested Assets • Invested Assets • Assets- Recvbl’s -Recovs • Assets = Reserves + Surplus • Balance sheet must balance • UW Cash flows impact Invested Assets

34. Single Policy Company: UW Income and Cash Flow

35. Single Policy Company: Assets and Investment Income

36. Single Policy Company: Equity Flow and IRR

37. IRR • IRR is comparable to the rate of interest on a loan • Given flows xt , IRR is the interest rate, y, (if it exists) which solves:

38. IRR on Equity Flows • Typical EQ Flows in P/C insurance • First flow is negative • Later flows are positive • One sign change • IRR on EQ Flow well-defined • Solve for premium to hit IRR target

39. Generalize ROE: Equity Balance Method 7: PVI/PVE • PV of INC at t=0 or t=1? • PV of Balance Sheet account?

40. Single Policy Company: PVI/PVE

41. Chart of Methods

42. Conclusion • No one right answer • Use appropriate method for situation • Select parameters consistent with method used • Questions