CAS Ratemaking Seminar March 2005 INT-6 Introduction to Profit Provision Calculations

1. CAS Ratemaking SeminarMarch 2005INT-6Introduction 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: Can They All Be Right?

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% • P= (50 + 30)/(1-.15-.05) = 100 • L=50 FX=30 • VXR=15% UPM = -1% • 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. Regulatory Context • Philosophy of Regulation • State controlled vs free market approaches • Affordability and availability • Solvency focus of statutory accounting • Rate Regulation Environments • Prior approval/File and use/Use and file • Differences by LOB and size of risk • ROE vs Return on Sales

13. Corporate Context • Pricing Analysis • UPM needed by LOB • Individual large account pricing • Economic Return Net of Risk • Consistency across business units • Reconciliation vs actual return

14. 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

15. 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

16. 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

17. CY II Offset- Example

18. 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

19. PV Differential Offset- Example

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

21. Ratemaking CY ROE Premium to Surplus Ratio

22. 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

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

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

25. Ratemaking CY ROE - Example

26. 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

27. What is Beta? • Beta from CAPM • Capital Asset Pricing Model • rA = rf +bA(E[rm] – rf) • Key CAPM Pricing Concept • Reward for taking systematic risk • No reward for diversifiable risk • Beta: Covariance of Stock with Market

28. Applying CAPM to Insurance • Insurance Betas by LOB? • Few single LOB insurance companies • These don’t represent much of the market • Beta based on covariance of LOB UPM with stock market return? • Not right theoretically, but has been used • CY UPMs vs PY Ratemaking – prior year • CATs uncorrelated with Stock Market

29. DCF - Example

30. 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

31. 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?

32. Simplified Taxes in RA DCF

33. Risk-Adjusted DCF Example

34. 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

35. Equity Flow Diagram

36. 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

37. Single Policy Company: UW Income and Cash Flow

38. Single Policy Company: Assets and Investment Income

39. Single Policy Company: Equity Flow and IRR

40. 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:

41. 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

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

43. Single Policy Company: PVI/PVE

44. Chart of Methods

45. Conclusion • No single “right” answer • Use appropriate method for situation • Select parameters consistent with method used • Questions