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CAS Ratemaking Seminar March 2005 INT-6 Introduction to Profit Provision Calculations
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Transcript of CAS Ratemaking Seminar March 2005 INT-6 Introduction to Profit Provision Calculations
CAS Ratemaking SeminarMarch 2005
INT-6Introduction to Profit Provision
Calculations
Ira Robbin, PhDPartner 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 anti-trust 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: Can They All Be Right?
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
UW Profit: Basic Equations U = P-L-X = UPM*P
X = Expense including premium tax
CR = (L+X)/P= 1- UPMUPM of –100% yields CR =200%
X = FX +VXR*PFX = Fixed expenseVXR = Variable expense ratio
P= (L+FX)/(1-VXR-UPM)
UW Profit Provision Chart
Profit Provision
Fixed Expense
Variable Expense
Loss Provision
Pre
miu
m
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!
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
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
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
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
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 ExpenseNet of Receivables
Loss+ LAE Reserves PLR·(LRES/INCL)CY Reserves-to-Incurred Ratio PLR =Permissible Loss RatioRatio of Loss Reserve to Incurred Loss
CY II Offset- Example
UEPR 400 Earned Prem 1,000
LRES 1,200 Inc’d Loss 800
RECV 260 PPACQR 10.0%
UPM0 5.0% PLR 60.0%
After-tax Yield 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 UPMPVDELLR = Present Value Differential
Present Value Differential PVDELLR = PLR·(PV(X0)- PV(X))
X0 = Loss Pattern for Reference LOBX = Loss Pattern for Review LOB Interest Rate: New money after-tax
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%
PLR 60.0%
Traditional UPM 5.0%
PVDELLR = (.98-.93)*.60 = 3.0%
UPM = .05-.03 = 2.0%
Method 3: Ratemaking CY ROE Start with ROE equation:
EQ
TINVU
EQ
INCROE
Assume S= EQ Simplify taxes Split INV into INV on PHSF vs INV on S
Ratemaking CY ROE
ATAT i)PHSFiUPM)t1((ROE
Premium to Surplus Ratio
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
)t1(
PHSFiiROEUPM ATATetargt
Ratemaking CY ROE - Example
UPM Calculation Validation
After-taxProfit
as % of P
PHSF 110.00% PHSF 4.40% 2.00 II on S 2.00%
After-tax yield 4.00% UPM -0.40%tax rate 35.00% Total 6.00%
target ROE 12.00%Surplus 50.00%
UPM -0.62% ROE 12.00%
Method 4: Discounted Cash Flow Prospective cash flow approach
founded in modern economic theory UPM = -krf +(E[rm] – rf)
k = funds generating coefficient rf = risk-free new money rate rm= market return = systematic covariance
What is Beta? Beta from CAPM
Capital Asset Pricing Model rA = rf +A(E[rm] – rf) Key CAPM Pricing Concept
Reward for taking systematic risk No reward for diversifiable risk
Beta: Covariance of Stock with Market
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
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:
)r PV(FIT, )r PV(X, )r PV(L,)r PV(P, ffAf rf = risk-free new money raterA = risk-adjusted rateFIT= income tax
Loss discounted at risk-adjusted rate
Risk-Adjusted Rate rA = rf + rm] – rf ) = Cov of liabilities with market While >0 for assets, the here is
for liabilities. Thus: <0 and rA < rf
How to get by LOB?
Simplified Taxes in RA DCF
)S on II(tPV
)r t)PV(X,-(1 )r PV(L,)t1(
)r t)PV(P,-(1
fA
f
)r1)(t1(
/Prt)r PV(X, )r PV(L,
)r PV(P,
f
ffA
f
Risk-Adjusted DCF Example
FV PV Factor DiscountedLoss (PV at RA rate) 50.00 0.95 47.50Fixed Expense 30.00 1.00 30.00Variable Expense 13.95 1.00 13.95
Subtotal 93.95 91.45
Capital 50.00Inv Inc on Capital 3.00 0.96 2.88Tax on II on Capital 1.05 0.96 1.01Pre-tax Provision 1.55
Premium 93.00 1.00 93.00
UPM -1.0% CR 101.0%
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
Equity Flow DiagramInvestment Income
UW Cash Flows Income Tax
Assets Liabilities and Surplus UW GainInvestables UEPR Investment Income EarnedReceivables Loss Reserves Realized Capital GainsRecoverables Expense Reserves TaxesOther Surplus Net
Equity Flows
Pool of Equity
Balance Sheet Income Statement
Single Policy Company
- +
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
Single Policy Company: UW Income and Cash Flow
timeEarned Prem
Paid Premium
Inc'd Loss
Paid Loss
Inc'd Expense
Paid Expense
UW Income
0 0 50 0 0 30 16 -301 100 50 62 20 5 10 332 0 0 0 30 0 5 03 0 0 0 12 0 4 0
total 100 100 62 62 35 35 3
time UEPRLoss
ReservesExpense
Reserves Surplus
Total Liab and
Surplus Recv'bleInv'stble Assets
Inv Income
0 100 0 14 40 154 50 1041 0 42 9 10 61 0 61 5.22 0 12 4 4 20 0 20 3.13 0 0 0 0 0 0 0 1.0
Single Policy Company: Assets and Investment Income
Single Policy Company: Equity Flow and IRR
Pre-tax IRR 14.2%
timeUW
IncomeInv
IncomeTotal
IncomeChange in
SurplusEquity
Flow
0 -30 0.0 -30.0 40 -70.01 33 5.2 38.2 -30 68.22 0 3.1 3.1 -6 9.13 0 1.0 1.0 -4 5.0
total 3 9.3 12.3 0 12.3
IRR
IRR is comparable to the rate of interest on a loan
0t
tt xv0
Given flows xt , IRR is the interest rate, y, (if it exists) which solves:
1)y1(v
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 of INC at t=0 or t=1? PV of Balance Sheet account?
)r PV(EQB,
)r PV(INC,PVI/PVE
f
f
Equity Balance
Single Policy Company: PVI/PVE
Pre-tax PVI/PVE = 18.1%
time Income yearEquity
balance
0 -30.001 37.20 1 40.002 3.10 2 10.003 1.05 3 4.00
total 11.35 total 54.00
PV0 53.15
PV1 9.60
Chart of Methods
CY Invesment OffsetPV Loss Differential Offset
CY ROEIRR on Equity FlowsPVI/PVE
Find UPM DCFFind P Risk-adjusted DCF
Investment Income Offset
Target Return Methods Single Policy Company Model
Economic Component Methods
Conclusion No single “right” answer Use appropriate method for
situation Select parameters consistent with
method used Questions