Addressing Three Questions Regarding
an Insurance Company’s Operations
July 13 - 14, 1998
Boston, Massachusetts
Presented by:
Susan E. Witcraft
Milliman & Robertson, Inc.
0
BACKGROUND
During each of the past several years, an
insurance company’s actual experience has
been much worse than the plan provided to
its Board.
1
QUESTIONS
1. What is the probability that the insurance company will
meet or exceed the earnings estimates provided to its Board
for the following year?
2. Are the assumptions underlying the earnings estimates
overly optimistic, or has the company had a run of bad luck?
3. What elements of the company’s business are its sources of
greatest risk?
2
OVERVIEW OF COMPANY
Direct Written Premium (000s)
Line
Property
1997
1998
1999
$64,889
$65,668
$68,951
General Liability
31,000
31,000
31,000
Workers’ Compensation
21,586
21,586
21,586
Commercial Auto
38,638
38,638
38,638
Personal Auto
57,018
60,636
64,435
3
MODEL USED
Asset
Accounting
Financial
Statements
Economic
Scenarios
Liability
Scenarios
Report
Generator
Liability
Accounting
Loss/Cat
Expense
Other
4
RISKS MODELED

Economy

Investment Yields & Returns

Premium

Losses

Fixed Expenses

Statutory Assessments
5
RISKS NOT MODELED
Mass Torts
 Loss Payment Patterns
 Reserve Strengthening
 LAE Ratios
 Reinsurance Pricing
 Illiquid Assets
 Reduction in Best’s Rating
 Credit Risk

6
DATA

Management’s Three-Year Financial Plan

Five Years of Statutory Annual Statements

Company’s Analysis of Direct Ultimate Losses and LAE

Corresponding Payment Triangles and Earned Premium

Paid & Incurred Development Triangles of Individual Large
Losses

Probability Distributions of Catastrophe Losses

Policy Limits Profiles

List of Large Catastrophe Losses for the Past 10 Years
7
INPUTS

Economic

Assets

Premium

Losses

Expenses

Reinsurance
8
ECONOMIC ASSUMPTIONS
GDP is autoregressive model
Yi = a + b Yi -1 + c Yi -2 + d (Li -1 - Si-1 ) + f (Li -2 - Si-2 )
+ g (Si -1 ) + ei
where
Yi
Li
= GDP change in period i;
Si
= short-term interest rates; and
ei
= random error term.
= long-term interest rates;
9
ECONOMIC ASSUMPTIONS
Remaining system of equations is recursive
Ii
=  (Yi , Ii-1 . . . Ii-4 , Yi-1 . . . Yi- 4 , ei )
Si
=  (Ii , Si-1 , Si-2 , ei )
Li
=  (Si , Li-1 , Li-2 , Si-1 , Si- 2 , ei )
SPi =  (Li , Yi+1 , Yi , Ii , ei )
Di
=  (Di-1 , SPi-1 , SPi- 2 )
10
ASSET ASSUMPTIONS

Book, Acquisition & Par Values of
Bonds by Coupon and Maturity

Book, Acquisition & Market Values of
Other Asset Classes

Investment Strategy
11
PREMIUM ASSUMPTIONS
Growth
Personal Lines  N(0.05, 0.025)
Commercial Lines  N(0,
0.025)
Percent Earned in Year
Percent Collected in Year
12
LOSS ASSUMPTIONS

Small Losses

Large Claims (xs $500,000)

Catastrophe Losses (xs $2 Million)
13
SMALL LOSS RATIO:
PROPERTY
Accident
Year
(1)
Ultimate
Direct
Losses
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
$13,172
13,654
18,904
23,952
29,352
24,484
27,086
41,806
33,618
35,466
(2)
Losses
on Large
Claims
$
0
0
1,929
3,870
6,174
4,356
5,561
12,059
6,401
7,012
(3)
Catastrophe
Losses
$
0
0
0
0
2,460
0
0
9,750
0
0
(4)
Direct
Earned
Premium
(5)
Small
Loss Ratio
[(1)-(2)-(3)]/(4)
$31,893
37,408
38,580
43,002
47,038
46,459
49,427
53,597
60,247
62,330
41.3%
36.5%
44.0%
46.7%
46.7%
43.3%
43.6%
46.4%
45.2%
45.7%
Selected
43%
14
SMALL LOSS RATIOS
l / r j ,k  a  b( l / r j 1,k ) 
where
 c x ( l / r j , x )   d x ( l / r j 1 , x ) 
xk
l/r
j
x
k
I
a,b,c,d, and f
e
xk
=
=
=
=
=
=
=
f  ij  ej
loss ratio
year
line of business
specific line of business being modeled
interest rate
constants
Normal random variable
15
LARGE LOSS ASSUMPTIONS:
GENERAL LIABILITY
Projected
Average
Cost (000S)
Accident
Year
Projected
Frequency
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
0.00
0.00
0.11
0.10
0.00
0.10
0.11
0.33
0.23
0.37
--$ 700
1,000
-1,000
800
1,000
900
1,100
0.35
0.30
0.225
Selected $1,200
High
Middle
Low
16
SELECTED LARGE LOSS
ASSUMPTIONS
Line
Property
Workers’ Compensation
Commercial Auto
Personal Auto
Expected
Frequency
0.15
0.05
0.25
0.01
Expected
Severity
(000s)
$1,000
1,500
700
600
17
SELECTED CATASTROPHE
ASSUMPTIONS
Probability
Amount
Probability
Amount
0.5%
3.0%
1.5%
1.5%
1.5%
2.5%
2.5%
2.5%
2.5%
2.5%
2.5%
$200
130
110
90
70
60
50
44
38
32
26
2.5%
5.0%
5.0%
5.0%
5.0%
5.0%
9.5%
10.0%
10.0%
10.0%
10.0%
$20
18
16
14
12
10
9
8
7
6
5
Expected Number/Year 0.25
18
SUMMARY OF LOSS RATIO
ASSUMPTIONS
(1)
(2)
(3)
Line
Small
Loss
Ratio
(4)
Direct
Large Catastrophe Loss
Loss
Loss
Ratio
Ratio
Ratio
(1)+(2)+(3)
Property
43.0%
15.0%
8.7%
66.7%
General Liability
25.0%
36.0%
0.0%
61.0%
Workers’ Compensation
67.5%
7.5%
0.0%
75.0%
Commercial Auto
45.0%
17.5%
0.0%
62.5%
Personal Auto
68.0%
0.6%
0.0%
68.6%
19
SUMMARY OF LAE RATIO
ASSUMPTIONS
Line
Property
General Liability
Workers’ Compensation
Commercial Auto
Personal Auto
ALAE/Loss
Ratio
10.5%
15.0%
8.0%
8.5%
8.0%
ULAE/Loss
Ratio
6.0%
5.0%
4.5%
7.0%
7.0%
20
EXPENSES
Commissions
Premium Taxes
17.3% of GWP
2.7% of GWP
(GWPt  GWPt 1 )
 N (0,0.01))
FE t  FE t 1 * (1  CPI t  0.5
GWPt
21
STATUTORY ASSESSMENTS
Probability
Statutory
Assessments/
Direct Written
Premium
95%
0.5%
3%
1.0%
1%
2.0%
1%
5.0%
22
EXCESS OF LOSS
REINSURANCE

Attachment Point
 $5 Million - General Liability
 $1 Million - All Other Lines

No Deductible, Maximum, or Ceding
Commission
23
EXCESS OF LOSS
REINSURANCE PREMIUM
Line
Property
General Liability
Workers’ Compensation
Commercial Auto
Personal Auto
1997 Ceded
Premium
(000s)
$ 360
1,440
600
360
2
24
QUOTA SHARE
REINSURANCE

General Liability Only

Cede 75% of Premium + Losses

Commission = Min(0.4, Max(0.18,0.25+0.8(0.55-l/r)))
25
CATASTROPHE
REINSURANCE

$50 Million Excess of $10 Million

2 Reinstatements at 5% Rate On Line

$4.5 Million Initial Premium
26
OBJECTIVE #1
Evaluate the likelihood that actual results will
equal or exceed those in the company’s plan.
27
STATUTORY RESULTS
Net After-Tax Income
1997
1998
1999
Surplus
Mean
$2,020
$1,740
$ 855
$120,852
Probability
(Min) 0%
1%
5%
10%
25%
50%
75%
90%
95%
99%
(Max)100%
$-40,231
-21,026
-10,998
-8,020
-2,754
2,213
6,992
10,720
12,952
16,341
22,616
$-40,456
21,320
-11,201
-8,213
-3,012
2,070
6,612
10,529
12,689
16,028
22,327
$-41,342
22,116
-12,089
-9,118
-3,887
1,137
5,698
9,628
11,754
15,117
21,472
$ 64,729
86,912
101,731
106,444
114,765
122,115
128,275
136,349
136,981
142,560
147,783
Plan
P{x>Plan}
4,000
38%
4,500
35%
5,000
28%
131,500
15%
28
OBJECTIVE #2
Identify differences in assumptions
between us and the company.
29
GENERAL LIABILITY
LARGE CLAIMS
Company:
M&R Low:
M&R Middle:
M&R High:
Expected Number
of Large Claims
Probability of Last
Three Year’s Results
6.0
7.2
9.2
10.7
0.1%
1.3%
17.4%
46.0%
30
FIXED EXPENSES

Inflationary Impact on Salaries
31
OBJECTIVE #3
Identify key drivers of results.
32
APPROACH

List Independent Variables

Use t-Test to Determine Statistical Significance

Calculate Impact on Income at 90th Percentile
33
LIST OF VARIABLES TESTED

Gross Written Premium

 Commercial Lines
 Personal Lines





Underwriting Expense Deviation
Statutory Assessments
Number of Catastrophes
Size of Each Catastrophe
Small Loss Ratio





Property
Commercial Auto
General Liability
Workers’ Compensation
Personal Auto
Number of Large Claims






Average Cost of Large Claims







Property
Commercial Auto
General Liability
Workers’ Compensation
Personal Auto
Property
Commercial Auto
General Liability
Workers’ Compensation
Personal Auto
Inflation
Short and Long Term Rates
34
KEY DRIVERS
Variable
Small Loss Ratio General Liability
Commercial Auto
Personal Auto
Workers’ Compensation
Property
Average
1997
Value
Net Income
Impact
if 10%
Worse Than
Expected
(thousands)
Probability
of 10%
Worse Than
Expected
Net Income
Impact of
90th Percentile
Adverse
Deviation
(millions)
25.0%
45.0%
$ 775
1,739
16%
19%
$-1.0
-2.6
68.0%
67.5%
43.0%
3,877
1,457
2,790
3%
22%
15%
-2.8
-2.6
-4.0
35
KEY DRIVERS
Variable
Number of Large Claims Property
General Liability
Commercial Auto
Workers’ Compensation
Number of Catastrophes
Underwriting Expenses
(Deviation from Expected)
Net Income
Impact
if 10%
Worse Than
Expected
(thousands)
Probability
of 10%
Worse Than
Expected
Net Income
Impact of
90th Percentile
Adverse
Deviation
(millions)
9.7
9.3
9.7
1.1
0.25
$ 970
1,116
679
165
141
36%
34%
31%
30%
25%
$-3.3
-4.4
-2.3
-2.9
-2.5
0%
N/A
N/A
-2.8
Average
1997
Value
36