Risk & Return
Parameter Estimation
David Appel, Ph.D.
Milliman & Robertson
Richard A. Derrig, Ph.D.
Senior Vice President
Automobile Insurers Bureau of MA
Casualty Actuarial Society
Seminar on Ratemaking
March 12, 2001
Las Vegas, NV

 Risk and Return Models
 Time Value of Money
 Cash Flow Patterns & Levels
 Measuring Risk - CAPM
 IRR Models
 Allocating Capital
 Calendar Year Acct Models
 Risk Premium Project (COTOR)
 Summary


 Net Present Value Model:
Valuation of Policyholder Flows
 Internal Rate of Return Model:
Valuation of Shareholder Flows
 Calendar Year Accounting Model:
Valuation of Company Acct
Returns


 Net Present Value Model:
PV(P) = PV(L) + PV(E) + PV(T)
 Internal Rate of Return Model:
PV(Shareholder Inv. - Shareholder Dividends) = 0
 Calendar Year Accounting Model:
Return on Surplus = Return on Investment +
Return on Underwriting

Profit Models
Historical Prospective
CASH FLOW
NPV
(policyholder perspective)
IRR
(shareholder perspective)
ACCOUNTING
CYAM ISO
STATE-X


 Premiums and Capital In
 Expenses and Claims Out
 Risk-Free Interest Rates
 Risky Investments

2001 Massachusetts Private Passenger
Automobile Compilation of Treasury
Stirps - Wall Street Journal
Summary for 12 months Nov. 1999 - Oct. 2000
Maturity
(months)
3
6
9
12
15
18
Average
Yield
5.669
5.986
6.094
6.300
6.384
6.432
33
36
39
42
21
24
27
30
45
48
51
54
57
60
6.288
6.430
6.428
6.321
6.444
6.455
6.473
6.452
6.477
6.492
6.490
6.523
6.525
6.540


 Premium Payments
 Finance Charges
 Acquisition Costs
 General Expense
 Premium Taxes
 Capital Investment
 Investment Income
 Income Taxes


 Premium Payments
 Finance Charges
 Acquisition Costs
 General Expense
 Premium Taxes
 Capital Investment
 Investment Income
 Income Taxes


Income Taxes

Federal (35% Marginal Rate)

Change in Unearned Premium Reserve

Underwriting: Earned Premium - Formula
Discounted Loss Reserves

Investment: Deductions by Asset Class
• Deduct 70% Stock Dividends
• Deduct 85% Tax-Exempt Bond Income
• Alternate Minimum Tax
• Idea: 20% Minimum Rate on Net Income
• Actual: Consolidated Tax Returns

State (Specific Rate)

MA: Flat % of Investment Income


 Cost of Capital
 Capital Asset Pricing Model
(CAPM)
 Equity, Asset and Liability Betas
 Market Risk Premium
 Dividend Growth Models
 Risk-Adjusted Discount Rates

Capital Asset Pricing Model
(CAPM)
Investors are compensated for nondiversifiable (i.e., “systematic”) risk only r adj
= r free
+ {  x (r market
- r free
)} where, r adj r free r market

= risk adjusted return
= return on risk free investments
= return on the market
= systematic risk coefficient

Capital Asset Pricing Model
(Theoretical Relationship)
Expected
Return Expected
Return
20.1
R*= 15.4
R f
= 6.0

R* = r f
1.0
1.5
+

* (market risk premium)

= Risk

Capital Asset Pricing Model
(Observed Empirical Relationship)
Return

Beta, Firm Size, Market/Book ratio
Returns are higher than predicted for: Low Beta
Small Size
Low M/B
Expected
Return
Empirical
Observation

Returns higher for:
Insurers tend to be:
Low beta firms
Small firms
Low M/B firms
Average beta
Relatively small market cap
Relatively low market/book
Also, insurers subject to interest rate risk not priced by CAPM

•Sample Selection
•Estimation of beta :
Value Line, S&P,Merrill Lynch, Wilshire
•Market risk premium
Ibbotson Associates
Greenwich Associates

Capital Asset Pricing Model
Beta Coefficients
Company
21 st
Century
ACE Limited
Allmercan Finan
Allstate Corp.
Amer. Intl. Group
American Finan
Berkley (W.R.)
Chubb
Cincinnati Fin
Fremont General
GAINSCO
HCC Ins. Hldngs
HSB Group
Hartford Finan
Mercury General
Ohio Casualty
Old Republic
Progressive
RLI Corp.
Reliance
Selective
St Paul
Transatlantic
Unitrin
XL Capital Lmtd.
SAFECO
1.00
1.35
0.75
1.10
0.65
1.05
0.70
0.75
1.20
0.80
0.90
0.65
1.10
0.90
1.20
1.20
0.80
0.85
1.10
Beta Coefficients
0.75
1.20
1.15
1.25
1.35
0.95
0.85
Average 0.98
Sources: Value Line Investment Survey, Part 3, The Ratings & Reports, June 2, 2000 and June 30, 2000.

 Definition: MRP = RM - RF
 RF depends on horizon length
 RF = T-Bill, Int. Govt, Long Govt.
 MRP = RM-Tbill,
RM-Int .Govt.
RM-Long Govt.

 Problem 1: How Do I Estimate MRP Value?
 Problem 2: Does RF + Beta * MRP Work?

2001 Massachusetts Private Passenger Auto
Ibbotson Market Data 1926-99*
Year Common Stocks
Total Annual
Returns
1926
1927
1928
1929
1930
11.62%
37.49%
43.61%
8.42%
-24.90%
U.S. Treasury
Bills
Total annual
Returns
3.27%
3.12%
3.56%
4.75%
2.41%
Arithmetic
Equity
Risk Premia
8.36%
34.36%
40.05%
-13.16%
-27.31%

1926
1927
1928
1929
1930
2001 Massachusetts Private Passenger Auto
Backward Moving Average Risk Premium
Year Risk
Premium
8.36%
34.36%
40.05%
-13.16%
-27.31%
# Years in
Avg.
Yr. X to 1999
Average Risk
Premium
74
73
72
71
70
9.44%
9.46%
9.11%
8.68%
8.99%

2001 Massachusetts Private Passenger Auto
Forward Moving Average Risk Premium
Year Risk
Premium
# Years in
Avg.
1926 8.36%
1927 34.36%
1928 40.05%
1929 -13.16%
1926-99
1
2
3
4
74
Average Risk
Premium
8.36%
21.36%
27.59%
17.40%
9.44%

M arke t Risk Pre mium
Long-term Mean =9.44%
60%
40%
20%
0%
-20%
-40%
-60%
1920
Mean 1960 to 1999 = 7.32%
Mean 1926 to 1959 = 11.94%
1930
Actual MRP
1940 1950 1960
Y e a r
1970
Long-term Mean
1980
Mean pre-1960
1990 2000
Mean 1960+
2010

Decile
Simple CAPM is Deficient
Add Small Stock Effect
Arithmetic
Mean
Return
Return on Decile
In Excess of
Risk-Free Rate Beta
Size
Beta * MRP Premium
7
8
5
6
9
3
4
1
2
10
7
8
5
6
9
10
3
4
1
2
12.13%
13.55%
13.92%
14.55%
15.28%
15.44%
15.75%
16.80%
17.59%
20.73%
6.93%
8.34%
8.71%
9.35%
10.08%
10.24%
10.54%
11.60%
12.38%
15.52%
0.90
1.04
1.08
1.12
1.15
1.18
1.23
1.27
1.33
1.43
*Note: Analysis done using 30-day Treasury bill as riskless rate.
Decile Largest Company Capitalization
General Electric Company
Ynisys Corp.
Readers Digest Association
Sterling Software Inc.
Steris Corp.
Unova Inc.
Trammell Crow Co.
Transaction Network Services Inc.
Donna Karen International Inc.
Delta Financial Corp.
$369,722,214
$10,498,796
$4,221,601
$2,203,671
$1,304,131
$872,220
$577,778
$381,830
$214,640
$97,914
7.28%
8.36%
8.76%
9.07%
9.31%
9.49%
9.90%
10.22%
10.77%
11.57%
-0.35%
-0.02%
-0.05%
0.28%
0.76%
0.74%
0.64%
1.38%
1.61%
3.95%

 Shareholders commit equity capital to support sale of insurance
 In return, receive rights to cash flows from underwriting and investment activities
Underwriting cash flow (net of tax)
Investment income on reserves and surplus
Flow of surplus
 IRR cash flows conditional on accounting conventions (usually SAP) and tax rules

CashFlow

CF i


UW i
 i

1
 t i

R i

S i


T i
 
S i

IRR = r = discount rate such that
(Cf i
) (1 + r) -i = 0
Set price such that IRR = target return (COK)

IRR - Key Inputs/Assumptions
 Cash Flow Patterns (especially premium + loss)
 Investment Yield Rate (usually current yield)
 Leverage (reserves/premiums/other)
 Surplus Runoff (flow/block)

INTERNAL RATE OF RETURN ANALYSIS
ALL LINES COMBINED (Policy Year)
TABLE III: PORTFOLIO YIELD AND TAX RATE - CURRENT YIELD
(1)
Investable
Asset
Bonds
U.S. Govt
States & territories
Special revenue
Public
Utilities
Industrial
(2)
Percent of
Assets
14.6
21.0
13.8
1.2
19.0
(3)
Expected Pre-
Tax Return
6.37%
5.49%
4.98%
7.45%
7.25%
(4) (5)
Tax Rate Expected Post-
Tax Return
35.00%
5.25%
5.25%
35.00%
35.00%
4.14%
5.20%
4.72%
4.84%
4.71%
Preferred stock
Common stock
Mortgage
Loans
Real estate
Cash & short-term invs.
4.5
18.2
1.0
1.0
5.7
6.89%
14.77%
8.30%
9.65%
5.92%
14.18%
28.89%
35.00%
35.00%
35.00%
5.91%
10.50%
5.40%
6.27%
3.85%
Rate of Return Pre-Inv Exp and Prior to MA Inv Tax
MA Inv Tax
100.0
7.75%
Investment Expenses
Portfolio Rate of Return
0.47%
7.28%
Sources:
Value Line Investment Survey, Part II, Selection and Opinion.
Federal Reserve Statistical Release, H.15(519).
Moody's Credit Perspectives.
24.98%
0.260%
35.00%
24.61%
5.82%
5.80%
0.31%
5.49%
Ibbotson Associates, Stocks, Bonds, and Inflation: 1999 Yearbook.
Ibbotson and Siegel, AREUEA Journal, 1984.
OneSource 7/31/99 CD, Asset Page, Part 1, Part 1A and Schedule D, Top 30 MA Groups, 12/98.
Return on other bonds from Major Lehman Indices, Asset Class U.S. Aggregate, @3/31/99.
Return on other assets from TOP302.WK4.

IRR CASH FLOWS IN
NOMINAL AND PRESENT VALUE
Time
2
3
0
1
Final NPV of Final
Cash Flow Cash Flow
-75.0
68.0
15.6
8.1
-75.0
58.4
11.5
5.1
SUM =
IRR = 16.5%
0.0

Setting Target Returns
(a.k.a. Estimating Cost of Capital)
Two Important Methods
 Dividend valuation (DCF) model
 CAPM
Also, comparable earnings

DCF Model
Price of stock equals present value of future cash flows
P o
= i



1
Di /( 1
 r ) i
If D grows at constant annual rate, g, then and
P o r


D o

1

D o
1
1

 r
 g

P o g


 g
D o

1
 g

2
( 1
 r )
2

......

 Sample selection
 Estimation of growth rate (g)
- Historical data or analysts forecasts
- Earnings, dividends or book value
 Single growth rate or multi-stage model

Discounted Cash Flow Analysis
Estimated Dividend Yield
21th Century
ACE Limited
Allmerican Finan
Allstate Corp.
Amer Intl Group
American Finan
Berkley (W.R.)
Chubb
Cincinnati Fin
Fremont General
GAINSCO
HCC Ins Holdings
HSB Group
Hartford Finan
Mercury General
Ohio Casualty
Old Republic
Progressive
RLI Corp
Reliance
Selective
St. Paul
Transatlantic
Unitrin
XL Capital Limited
SAFECO
Average
Estimated Dividend Yield
3.3
0.4
1.8
nil
3.2
1.1
5.7
1.6
3.8
4.4
3.1
0.6
7.0.
3.6
3.5
2.84
3.0
1.9
0.5
2.8
0.2
3.7
2.6
2.1
2.2
7.6
1.4
Sources: Value Line Investment Survey, Part 3, The Ratings & Reports, June 2, 2000 and June 30,
2000

Discounted Cash Flow Analysis
Earnings Per Share Experience
Company
21 st
Century
ACE Limited
Allmercan Finan
Allstate Corp.
Amer. Intl. Group
Berkley (W.R.)
Chubb
Cincinnati Fin
Fremont General
GAINSCO
HCC Ins. Hldngs
HSB Group
Hartford Finan
Mercury General
Ohio Casualty
Old Republic
Progressive
RLI Corp.
Reliance
Selective
St Paul
Transatlantic
Unitrin
XL Capital Lmtd.
SAFECO
7.0
N.A.
22.5
7.0
11.5
7.5
8.5
2.5
4.0
7.0
7.5
19.0
N.A.
10.0
1990-1999
9.5
N.A.
N.A.
N.A.
14.5
N,.A.
12.0
9.0
13.0
N.A.
N.A.
Average
Average Dividend Growth Rate
10.08
Annual Rate of Change
1995-1999
N.A.
27.5
N.A>
24.5
13.5
3.5
12.0
7.5
12.0
17.5
N.A.
3.0
N.A.
18.5
4.5
16.0
4.5
8.0
N.A.
0.5
7.0
17.0
20.0
26.0
9.5
12.38
3.0
10.0
9.0
6.0
11.0
3.0
7.0
Nil
4.0
4.0
12.0
19.5
9.0
4.5
2000-2004
5.5
13.5
5.5
11.0
13.5
Nil
7.5
6.0
10.5
5.5
10.0
8.15
10.20
Sources: Value Line Investment Survey, Part 3, The Ratings & Reports, June 2, 2000 and June 30, 2000.

Discounted Cash Flow Analysis
Earnings Per Share Experience
Company
21 st
Century
ACE Limited
Allmercan Finan
Allstate Corp.
Amer. Intl. Group
American Finan
Berkley (W.R.)
Chubb
Cincinnati Fin
Fremont General
GAINSCO
HCC Ins. Hldngs
HSB Group
Hartford Finan
Mercury General
Ohio Casualty
Old Republic
Progressive
RLI Corp.
Reliance
Selective
St Paul
Transatlantic
Unitrin
XL Capital Lmtd.
SAFECO
Average
Average Dividend Growth Rate
19.0
4.0
12.0
17.5
11.0
N.A.
4.0
7.0
14.5
27.5
N.A.
4.0
1990-1999
0.50
N.A.
N.A.
N.A.
13.5
N,.A.
-5.5
7.0
8.5
10.0
6.0
N.A.
1.0
N.A.
8.318
Annual Rate of Change
1995-1999
N.A.
27.00
N.A>
49.50
15.0
9.5
-5.5
8.0
6.5
13.0
27.5
N.A.
12.0
11.0
19.0
20.0
20.0
-1.0
-1.0
-30.0
24.0
10.0
37.0
13.5
3.0
10.5
12.98
6.0
5.0
6.0
17.5
8.0
6.0
7.0
5.0
9.0
19.5
-1.5
4.0
2000-2004
Nil
7.00
11.50
1.00
13.0
14.0
15.5
10.0
11.5
20.0
40.0
10.5
4.5
3.5
10.14
10.47
Sources: Value Line Investment Survey, Part 3, The Ratings & Reports, June 2, 2000 and June 30, 2000.


 Standard Allocations
 Premium
 Liabilities
 Discounted Liabilities
 Myers-Cohn Allocation
 Myers-Read

Calendar Year Accounting Model - CYAM
Total Return = (UW Profit + IY Reserves) + IY Surplus
= Return on Operations +
Return from Investment of Surplus
Return on
Operations = Return attributable to undertaking the risk of the insurance transaction

ROE

UW



1
 t u


P
S
 i

1
 t i
 reserve
Inc .
LLAE


Inc .
LLAE
 premium  
P
S
 i

1
 t i

Set Total Return = Target Return (COK) and Solve for UW


CYAM - Key
Inputs/Assumptions
•Investment Yield Rate = i
•Investible Balance =
•Leverage =


P
S




Re se rve
Inc.LLAE




Inc.LLAE
Pre mium

 i


Reserve
Inc.LLAE

 is usually embedded yield is usually estimated using recent historical data


P 

S is usually normative value; rarely varies by line

CYAM - Likely Problems
 Embedded yield not necessarily good proxy for expected earnings rate
 Investible balance may be distorted due to variations in historical growth or loss experience
 Leverage is typically insensitive to risk and,
TIMING OF CASH FLOWS IS IGNORED

CALENDAR/ACCIDENT YEAR ANALYSIS
STEADY STATE / GROWTH RATE = 0%
4
5
6
0
1
2
3
AY1 AY2 AY3 AY4
-75.0
68.0
-75.0
15.6
68.0
-75.0
8.1
15.6
68.0
-75.0
8.1
15.6
68.0
8.1
15.6
8.1
Calendar Year ROE = (68+15.6+8.1)/75 = 22.3%

CALENDAR/ACCIDENT YEAR ANALYSIS
STEADY STATE / GROWTH RATE = 16.5%
4
5
6
2
3
0
1
AY1 AY2 AY3 AY4
-75.0
68.0
-87.4
15.6
79.2
-101.8
8.1
18.2
9.4
92.3 -118.5
21.2
107.5
11.0 24.7
12.8
NPV OF
AY4
-118.5
Calendar Year ROE =(92.3+18.2+8.1)/101.8 = 16.5%
92.3
18.2
8.1

CALENDAR/ACCIDENT YEAR ANALYSIS
STEADY STATE / GROWTH RATE = 25.0%
4
5
6
2
3
0
1
AY1 AY2 AY3
-75.0
68.0
-93.8
15.6
85.0
-117.2
AY4
8.1
19.5
106.3
-146.5
10.1
24.4
12.7
132.8
30.5
15.8
NPV OF
AY4
-146.5
114.0
22.5
10.0
Calendar Year ROE = (106.3+19.5+8.1)/117.2 = 14.2%


 NPV: Underwriting Profit Provision
 IRR: Expected Return to Capital
 CYA: Return to Capital:
Actual or Expected
 Model Outputs Consistent if Inputs are Consistent


 Guaranty Funds
 Residual Markets
 Reinsurance
 Default Risk
 Risk Premium Project
 Excess Capital

 Committee on Theory of Risk
 Discount Rate for Liabilities
 Literature Review
 Actuarial: Process and Parameter Risk
 Financial: Systematic Risk
 Academic: Dave Cummins, Rich Phillips
 Industry: Bob Butsic, Rich Derrig http://casact.org/cotor/rpp.htm

 Small Stock Effect: Smaller Decile (MKT CAP)
Returns Exceed CAPM Expected
 Theory: Non-Systematic Risk Based on
Information Flow and Liquidity
 Practice: Deciles 5 to 10, 1926-1998 0.87% (5) to
3.75% (10) Excess of CAPM
 Example: MA Companies 1.3%
 Ibbotson, Kaplan & Peterson (1997): Cross-
Autocorrelations in Returns; “Sum Beta” adds
One Lag; Sum  =  + 
-1
 Sum Beta “Explains” Some of Small Stock Effect

 Problem: Public Firms not all “Pure Play”
 Solution: Industry Equity Beta via
Sales Weighted Full
Market Regression
 P & C: Equity Beta 12/31/1998 of 0.92;
3/31/2000 of 1.15

 Surplus by Company stands behind all lines
 Surplus by Line needed to allocate taxes and other by line Costs.
 Myers-Read (1999): Theory Allows Unique
Additive Allocation of Capital by “Fairness” to
Guaranty Fund Criteria and Options Pricing
Methods
 Properties: Higher Line Covariance with Liab
(Asset) Portfolio Implies Higher (Lower) Surplus
 Key Equation: Default Option = F (Liabilities,
Assets, A/L)

 CAPM Loss Beta (Fairley, 1979) has
 = F(A,L,T,S, More (?)), No Default
 Problem: All Liability Dollars Have Same Risk
 Butsic (1999): Unique Surplus Allocation if Price
Homogeneity (Same Marginal Default Option).
 Surplus Allocation Across Coverage Layers (Loss
Distribution)
 Layer Beta and Surplus Increasing by Limit
 Risk Loads by Layer
 Example: Catastrophe Risk, Layer Betas 0.18 to 8.29
 Stay Tuned for More Developments

 Almagro, Manuel and Thomas L. Ghezzi, (1988), Federal Taxes Provisions
Affecting Property-Casualty Insurers, Proceedings of the Casualty Actuarial
Society , LXXV.
 Brealey, Richard A. and Stewart C. Myers, (2000), Principles of Corporate
Finance, Sixth Edition, McGraw-Hill Higher Education.
 Butsic, Robert P., (1991), Loss Reserve Valuation Using A Risk-Adjusted
Discounting Interest Rate, Managing the Insolvency Risk of Insurance
Companies, J. David Cummins and R. A. Derrig (Eds), Kluwer Academic
Publishers
 Butsic, Robert P., (1999), Capital Allocation for Property-Liability Insurers:
A Catastrophe Reinsurance Application, Casualty Actuarial Society Forum ,
Spring.
 Cummins, J. David, (1990), Asset Pricing Models and Insurance
Ratemaking, ASTIN Bulletin , 20:2.

 Cummins, J. David, (1990), Multi-Period Discounted Cash Flow
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
 Models follow policyholder or shareholder perspectives
 Cash flows are modelled according to perspective
 Pricing models are prospective and by line of business
 Capital must be allocated
 Model outputs are consistent with consistent parameters