CAS Ratemaking Seminar March 2004 INT-7 Introduction to Profit Provision Calculations Ira Robbin, PhD Partner Re 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 antitrust guidelines. 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. 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. Overview UW Profit Basics Overview of Different Methods Corporate and Regulatory Contexts Offset Formulas DCF and Risk-Adjusted DCF Single Policy Company Models Conclusion Different Types of UW Profit Actual Achieved Provision in Manual Rate Booked to Date vs Ultimate PY, AY, CY Direct, Gross, Ceded, Net Stat vs GAAP Indicated, Filed, Approved Provision in Charged Premium 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) UW Profit Provision Chart Fixed Expense Variable Expense Loss Provision Premium Profit Provision 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! UPM Calculation Approaches Investment Income Adjustment Adequate Total Return CY Inv Offset and PV Differential Ratemaking CY ROE Economic return via Single Policy model IRR on Equity Flow and PVI/PVE Economic Components DCF and Risk-adjusted DCF 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 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 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 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 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 CY II Offset- Example UEPR LRES 400 1,200 Earned Prem Inc’d Loss 1,000 800 RECV 260 PPACQR 10.0% UPM0 5.0% PLR After-tax Yield 60.0% 4.0% PHSF = (.4·(1-.1)-.26) + .6·1.5 =1.00 UPM = .05 - .04·1.00 = 1.0% 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 PV Differential Offset- Example PV(REF Loss Pattern) PV(REV Loss Pattern) 98.0% 93.0% Risk-free New Money Rate after tax PLR Traditional UPM 3.0% 60.0% 5.0% PVDELLR = (.98-.93)*.60 = 3.0% UPM = .05-.03 = 2.0% Method 3: Ratemaking CY ROE Start with ROE equation: INC U INV T ROE EQ EQ Assume S= EQ Simplify taxes Split INV into INV on PHSF vs INV on S Ratemaking CY ROE Premium to Surplus Ratio ROE ((1 t ) UPM iATPHSF) iAT 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 Surplus in ROE Equation S = Target Statutory Surplus S = P/ Premium-to-Surplus leverage ratio varies by LOB Equity vs Surplus Solve for UPM Find UPM to hit CY ROE target UPM ROE t arg et iAT iAT PHSF (1 t ) Ratemaking CY ROE - Example UPM Calculation PHSF After-tax yield tax rate target ROE UPM Validation 110.00% 2.00 4.00% 35.00% 12.00% -0.62% PHSF II on S UPM Total Surplus ROE After-tax Profit as % of P 4.40% 2.00% -0.40% 6.00% 50.00% 12.00% 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 Applying CAPM to Insurance CAPM risk–reward concept Insurance Betas by LOB? Reward for taking systematic risk No reward for diversifiable risk Beta =Cov of Company Stock with Market 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 DCF - Example Risk-free rate 5.0% Funds Generating Coefficient 1.50 Beta for LOB 1.20 E[Market yield] 10.0% UPM = -1.50*.05+ 1.20(.10-.05) =-1.5% Method 5:Risk-Adjusted DCF Solve for UPM so that: PV(P, rf ) PV(L, rA ) PV(X, rf ) PV(FIT, rf ) rf = risk-free new money rate rA = risk-adjusted rate FIT = income tax Loss discounted at risk-adjusted rate 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? Risk-Adjusted DCF Example Risk-free rate 0.90 Risk-Adjusted Rate 0.95 FV PV Factor Discounted Loss Fixed Expense Variable Expense Total 50.00 30.00 13.68 93.68 0.95 1.00 1.00 47.50 30.00 13.68 91.18 Premium 91.18 1.00 91.18 PV Factor for Loss Combined Ratio UPM 102.7% -2.7% 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 Equity Flow Diagram Investment Income UW Cash Flows Income Tax Single Policy Company Balance Sheet Assets Liabilities and Surplus Investables UEPR Receivables Loss Reserves Recoverables Expense Reserves Other Surplus - + Pool of Equity Income Statement UW Gain Investment Income Earned Realized Capital Gains Taxes Net Equity Flows Income and Cash Flow UW Gain = EP –IncLoss –IncExpense Inv Inc = II on Invested Assets Invested Assets Defined by accounting rules Does not depend on UW cash flows Assets- Recvbl’s -Recovs Assets = Reserves + Surplus Balance sheet must balance UW Cash flows impact Invested Assets Single Policy Company: UW Income and Cash Flow time 0 1 2 3 total Earned Paid Prem Premium 0 50 100 50 0 0 0 0 100 100 Inc'd Loss 0 62 0 0 62 Paid Inc'd Paid Loss Expense Expense 0 30 16 20 5 10 30 0 5 12 0 4 62 35 35 UW Income -30 33 0 0 3 Single Policy Company: Assets and Investment Income time 0 1 2 3 UEPR 100 0 0 0 Loss Expense Reserves Reserves 0 14 42 9 12 4 0 0 Surplus 40 10 4 0 Total Liab and Surplus Recv'ble 154 50 61 0 20 0 0 0 Inv'stble Assets 104 61 20 0 Inv Income 5.2 3.1 1.0 Single Policy Company: Equity Flow and IRR Pre-tax IRR time 0 1 2 3 total UW Income -30 33 0 0 3 Inv Income 0.0 5.2 3.1 1.0 9.3 Total Income -30.0 38.2 3.1 1.0 12.3 Change in Surplus 40 -30 -6 -4 0 14.2% Equity Flow -70.0 68.2 9.1 5.0 12.3 IRR Given flows xt , IRR is the interest rate, y, (if it exists) which solves: 0 v xt t t 0 v (1 y )1 IRR is comparable to the rate of interest on a loan 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 Method 7: PVI/PVE Generalize ROE: PV(INC, rf ) PVI/PVE PV(EQB, rf ) Equity Balance PV of INC at t=0 or t=1? PV of Balance Sheet account? Single Policy Company: PVI/PVE Pre-tax PVI/PVE = time 0 1 2 3 total PV1 Income -30.00 37.20 3.10 1.05 11.35 9.60 18.1% year 1 2 3 total PV0 Equity balance 40.00 10.00 4.00 54.00 53.15 Chart of Methods Investment Income Offset Target Return Methods Economic Component Methods CY Invesment Offset PV Loss Differential Offset CY ROE Single Policy IRR on Equity Flows Company Model PVI/PVE Find UPM Find P DCF Risk-adjusted DCF Conclusion No one right answer Use appropriate method for situation Select parameters consistent with method used Questions