2. CAS Ratemaking Seminar March 2005 INT-6 Introduction to Profit Provision Calculations Ira Robbin, PhD Partner Re

3. Ground Rules 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.

4. Disclaimers 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.

5. Cautions 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.

6. 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?

7. 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

8. 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)

9. UW Profit Provision Chart

10. 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!

11. 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

12. 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

13. 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

14. 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

15. 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

16. Method 1: CY Investment Income Offset (State X) UPM = UPM 0 – IIOffset UPM 0 = Traditional UPM IIOffset = Investment Income Offset IIOffset = i AT · PHSF Based on After-tax realized CY returns Actual portfolio mix of invested assets Base of PH-Supplied Funds

17. 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

18. CY II Offset- Example UEPR400Earned Prem1,000 LRES1,200Inc’d Loss800 RECV260PPACQR10.0% UPM 0 5.0%PLR60.0% After-tax Yield4.0% PHSF = (.4·(1-.1)-.26) +.6·1.5 =1.00 UPM =.05 -.04·1.00 = 1.0%

19. Method 2: Offset for PV Differential UPM = UPM 0 - PVDELLR UPM 0 = Traditional UPM PVDELLR = Present Value Differential Present Value Differential PVDELLR = PLR·(PV(X 0 )- PV(X)) X 0 = Loss Pattern for Reference LOB X = Loss Pattern for Review LOB Interest Rate: New money after-tax

20. PV Differential Offset- Example PV(REF Loss Pattern)98.0% PV(REV Loss Pattern)93.0% Risk-free New Money Rate after tax 2.8% PLR60.0% Traditional UPM 5.0% PVDELLR = (.98-.93)*.60 = 3.0% UPM =.05-.03 = 2.0%

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

22. Ratemaking CY ROE Premium to Surplus Ratio

23. 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

24. Surplus in ROE Equation S = Target Statutory Surplus S = P/  Premium-to-Surplus leverage ratio varies by LOB Equity vs Surplus

25. Solve for UPM Find UPM to hit CY ROE target

26. Ratemaking CY ROE - Example

27. Method 4: Discounted Cash Flow Prospective cash flow approach founded in modern economic theory UPM = -kr f +  (E[r m ] – r f ) k = funds generating coefficient r f = risk-free new money rate r m = market return  = systematic covariance

28. What is Beta? Beta from CAPM Capital Asset Pricing Model r A = r f +  A (E[r m ] – r f ) Key CAPM Pricing Concept Reward for taking systematic risk No reward for diversifiable risk Beta: Covariance of Stock with Market

29. 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

30. DCF - Example Risk-free rate5.0% Funds Generating Coefficient1.50 Beta for LOB1.20 E[Market yield]10.0% UPM = -1.50*.05+ 1.20(.10-.05) =-1.5%

31. Method 5:Risk-Adjusted DCF Solve for UPM so that:  r f = risk-free new money rate  r A = risk-adjusted rate  FIT= income tax Loss discounted at risk-adjusted rate

32. Risk-Adjusted Rate r A = r f +  r m ] – r f )  = Cov of liabilities with market While  >0 for assets, the  here is for liabilities. Thus:  <0 and r A < r f How to get  by LOB?

33. Simplified Taxes in RA DCF

34. Risk-Adjusted DCF Example

35. 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 -  S

36. Equity Flow Diagram

37. 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

38. Single Policy Company: UW Income and Cash Flow

39. Single Policy Company: Assets and Investment Income

40. Single Policy Company: Equity Flow and IRR

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

42. 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

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

44. Single Policy Company: PVI/PVE

45. Chart of Methods

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