CAS Ratemaking Seminar
March 2004
INT-7
Introduction to Profit Provision
Calculations
Ira Robbin, PhD
Partner Re
Ground Rules
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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
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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
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
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
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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
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Actual Achieved
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Provision in Manual Rate
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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
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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
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Investment Income Adjustment
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Adequate Total Return
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CY Inv Offset and PV Differential
Ratemaking CY ROE
Economic return via Single Policy model
IRR on Equity Flow and PVI/PVE
Economic Components
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DCF and Risk-adjusted DCF
Different UPM Calculation
Methods
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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
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Corporate: UPM targets by LOB
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Maximize economic return net of risk
Regulatory: Allowed UPMs
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Manual rates by LOB
Philosophy of regulation
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State controlled vs free market approaches
Affordability and availability
Legislated rate environments
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File and use/Use and file
Market pricing for large risks
Recap of UW Profit Regulation
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1920’s – 1970’s: Low interest era
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No consideration of investment income
5.0% UPM for most lines
2.5% for WC
1970’s – 90’s: High rate era
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Investment income offsets
CAPM, DCF and Risk-Adjusted DCF
IRR on Equity Flows and PVI/PVE
Method 1: CY Investment
Income Offset (State X)
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UPM = UPM0 – IIOffset
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UPM0 = Traditional UPM
IIOffset = Investment Income Offset
IIOffset = iAT · PHSF
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Based on After-tax realized CY returns
Actual portfolio mix of invested assets
Base of PH-Supplied Funds
Policyholder Supplied Funds
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Unearned Premium Balances
 UEPR(1-PPACQR) - RECV
Net of Pre-paid Acquisition Expense
Net of Receivables
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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
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UPM = UPM0 - PVDELLR
UPM0 = Traditional UPM
PVDELLR = Present Value Differential
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Present Value Differential
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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
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Start with ROE equation:
INC U  INV  T
ROE 

EQ
EQ
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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?
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GAAP vs Statutory
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Going-concern vs Solvency
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Stat defined by state regulation
Calendar Yr vs Policy Yr
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ROE is CY
Past decisions impact this CY
Ratemaking is PY and prospective
Surplus in ROE Equation
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S = Target Statutory Surplus
S = P/
  Premium-to-Surplus leverage ratio
 varies by LOB
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Equity vs Surplus
Solve for UPM
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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
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Prospective cash flow approach founded
in modern economic theory
UPM = -krf +b(E[rm] – rf )
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k = funds generating coefficient
rf = risk-free new money rate
rm= market return
b = systematic covariance
Applying CAPM to Insurance
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CAPM risk–reward concept
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Insurance Betas by LOB?
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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?
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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
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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
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Loss discounted at risk-adjusted rate
Risk-Adjusted Rate
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rA = rf + b (E[rm] – rf )
b = Cov of liabilities with market
While b>0 for assets, the b here is for
liabilities. Thus:
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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
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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
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UW Gain = EP –IncLoss –IncExpense
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Inv Inc = II on Invested Assets
Invested Assets
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Defined by accounting rules
Does not depend on UW cash flows
Assets- Recvbl’s -Recovs
Assets = Reserves + Surplus
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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
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IRR is comparable to the rate of interest
on a loan
IRR on Equity Flows
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Typical EQ Flows in P/C insurance
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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
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Generalize ROE:
PV(INC, rf )
PVI/PVE 
PV(EQB, rf )
Equity Balance
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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
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No one right answer
Use appropriate method for situation
Select parameters consistent with
method used
Questions