# How Valid Are Your Assumptions A Basic Introduction to

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```					    How Valid Are Your Assumptions?
A Basic Introduction to Testing the Assumptions of
Loss Reserve Variability Models

Casualty Loss Reserve Seminar
Renaissance Waverly Hotel
Atlanta, Georgia
presented by
F. Douglas Ryan, FCAS, MAAA
Philip E. Heckman, ACAS, MAAA, PhD
September 11, 2006
Class for Regional Affiliates
Schedule
8:30 am Optional Class A – The mathematics of regression
Optional Class B - The reserving problem in casualty insurance
10:00    Full session 1
11:45    Break for lunch and Regional Affiliate business
1:00     Full session 2
1:30     Break out into work groups of four to six persons
3:00     Full session 3
5:00     End of class
Participants who do not wish to take an optional early session are expected to
travel to the meeting in the morning and return home the same day.
Class for Regional Affiliates
Who Is It For?
The class is designed to be offered by Regional
Affiliates to all people interested in the technical
side of the estimation of liabilities using data in
“loss development triangles”. Regional Affiliates
are encouraged to invite students and faculty in
actuarial programs, actuaries at all levels of
experience with the setting of loss reserves, and
senior management.
Class for Regional Affiliates
The Scope from 10:00 am to 5:00 pm
•   Introduce triangles of cumulative loss costs and note their
obvious features
•   Algorithms: The “actuarial methods” and Excel
instructions viewed as algorithms
•   BLUE: “Best” Linear Unbiased Estimates
•   Multiple Models and “The Reserving Problem”
•   Testing the BLUE Assumptions
•   Testing a BLUE Model: Validation
•   Model Design
Class for Regional Affiliates
The Participant Should Be Able To:
1.1 Participate in a discussion of how two
triangles differ
1.2 Contrast the insights gained from two,
three or four methods of viewing triangles
1.3 Use Chart Wizard to create “XY Scatter”
charts of possible functional relations
Class for Regional Affiliates
The Participant Should Be Able To:
2.1 Contrast the insights gained from
normalizing using exposure rather than
2.2 Contrast the uses of price deflators with
the uses of time series
2.3 Appreciate the complications introduced
by introducing a triangle such as claim
counts
Class for Regional Affiliates
The Participant Should Be Able To:
3.1 Explain to a non-actuary what an
algorithm is, why the “actuarial methods”
are algorithms, and why Mack and Venter
method as an algorithm.
Class for Regional Affiliates
The Participant Should Be Able To:
algorithms and, if appropriate, draw an analogy with
Russian dolls or some other system of nesting
subroutines
4.2 Define in the Excel-user’s style (not as a statistician) the
meaning of each function: FORECAST, RSQD,
INTERCEPT, STEYX, TREND and LINEST
4.3 Give examples of what the Excel Function Wizard means
by “known_y’s” and “known_x’s”
4.4 Use each of these Excel functions for a specific, limited
purpose (although not be able yet to assemble them into
a more sophisticated algorithm)
Class for Regional Affiliates
The Participant Should Be Able To:
5.1 Using the acronym BLUE, recall the four aspects
of a regression algorithm.
5.2 Examine a triangle of data and note its qualities
as a set of “actual” observations.
5.3 Discuss the argument that regression is
inappropriate for extrapolating into the future.
5.4 Discuss the argument that big differences
between actual and expected imply a poor tool
for forecasting.
Class for Regional Affiliates
The Participant Should Be Able To:
6.1 Adjust a triangle of data (e.g., by taking logs or
using incremental data) and discuss how the
adjusted data is more or less appropriate as a set
of “actual” data.
6.2 Give examples of real-world problems for which
the linear assumption of regression is clearly
inappropriate. [This may be difficult, but if it is,
it simply shows the usefulness of linear models.]
6.3 Contrast BLUE models of paid data with BLUE
models of incurred data (paid+case reserves).
Class for Regional Affiliates
The Participant Should Be Able To:
7.1 Explain the differences between accounting
summaries of loss costs and the “sufficient
statistics” of a statistical approach such as GLM.
7.2 Explain how the “actuarial methods” rely on a
subset of the accounting data rather than the
entire triangles of paid and incurred.
7.3 Discuss the value of having a number of
estimators that reflect a diversity of possible
world-views and are independent of one another.
Class for Regional Affiliates
The Participant Should Be Able To:
8.1. List five ways that patterns in the residuals
might indicate that one or more assumptions is
not appropriate.
8.2. Create a chart in Excel of a set of residuals and
appropriateness of the BLUE assumptions.
9.1. Give reasons why the fitted values should look
like the observations
9.2. Give examples of using subsets of the larger
data set to test the reasonableness of a regression
estimate
Minimum Limits Auto Dataset
California Auto Liability Insurance
Minimum Statutory Limits Policies (15/30)
Claims Screen: Claimant Is Not Represented by an Attorney
Data Reflects Only Claims Closed as of the Evaluation
Data Used with the Permission of The Qestrel Companies

Cumulative Losses
AY                1              2                     3          4           5
1998          34,691        1,402,761           1,544,751   1,547,344   1,549,844
1999        1,018,092       4,532,796           5,063,937   5,093,082
2000        4,161,397      16,336,148          18,067,843
2001        4,077,332      20,892,524
2002        3,415,880
Data Limitations
• To fit on overhead – only 5 years
• More stable than “real” data
– Only one state
– Only closed claims
– Only minimum limits
AGE TO AGE FACTORS
1 to 2       2 to 3         3 to 4         4 to 5      5 to Ult
Wgtd Avg         4.6456      1.1080         1.0048         1.0016
f(d)          3.6456      0.1080         0.0048         0.0016
% Reported       0.1930      0.7037         0.0968         0.0048        0.0016

AY                    1         1 to 2       2 to 3         3 to 4          4 to 5
1998            34,691      1,402,761    1,544,751      1,547,344       1,549,844
1999         1,018,092      4,532,796    5,063,937      5,093,082      5,101,311
2000         4,161,397     16,336,148   18,067,843    18,154,613      18,183,945
2001         4,077,332     20,892,524  23,148,432     23,259,601      23,297,180
2002         3,415,880    15,868,658   17,582,104     17,666,541      17,695,084
Incremental Losses
AY            1         1 to 2             2 to 3    3 to 4   4 to 5
1998      34,691      1,368,070           141,990     2,593    2,500
1999   1,018,092      3,514,703           531,142    29,145
2000   4,161,397     12,174,751         1,731,695
2001   4,077,332     16,815,192
2002   3,415,880

AY              1       1 to 2        2 to 3        3 to 4        4 to 5
1998      34,691      126,468       151,466         7,419         2,500
1999   1,018,092    3,711,512       489,437        24,319        8,229
2000   4,161,397   15,170,600     1,763,925       86,769        29,332
2001   4,077,332   14,864,137    2,255,908       111,169        37,580
2002   3,415,880  12,452,778     1,713,447        84,437        28,543
Exploratory Graphical Analysis
Severity

Year
Exploratory Graphical Analysis:
Residual
• A residual = Actual value for Yi – Fitted
value for Yi
• Used to assess fit of model
• There should be no pattern, just random
points
AY                  1 to 2      2 to 3        3 to 4        4 to 5
1998            1,241,602      (9,475)       (4,825)           (0)
1999             (196,809)     41,705         4,825
2000           (2,995,848)    (32,230)
2001            1,951,055

STD DEV          2,188,000      37,867         6,824             -

CHAIN LADDER METHOD STANDARDIZED RESIDUALS (INCREMENTAL)
AY                  1 to 2       2 to 3       3 to 4         4 to 5
1998              0.5675      (0.2502)     (0.7071)       (0.0000)
1999             (0.0899)      1.1014       0.7071
2000             (1.3692)     (0.8511)
2001              0.8917

AGE 1 T O 2 CHAI N LADDER M ET HOD RESI DUAL VERSUS AGE 1 CUM ULAT I VE
( I N 0 0 0 ' s)

3,000

2,000

1,000

0
RESIDUAL

0     500       1,000     1,500          2,000            2,500   3,000   3,500   4,000   4,500

- 1,000

- 2,000

- 3,000

- 4,000
CUMULATIVE LOSS

AGE 2 T O 3 CHAI N LADDER M ET HOD RESI DUAL VERSUS AGE 2 CUM ULAT I VE
( I N 0 0 0 ' s)

50

40

30

20

10
RESIDUAL

0
0   2,000     4,000     6,000         8,000            10,000   12,000   14,000   16,000   18,000

- 10

- 20

- 30

- 40
CUMULATIVE LOSS

AGE 3 T O 4 CHAI N LADDER M ET HOD RESI DUAL VERSUS AGE 3 CUM ULAT I VE
( I N 0 0 0 ' s)

6

4

2
RESIDUAL

0
0       1,000           2,000                  3,000              4,000   5,000   6,000

-2

-4

-6
CUMULATIVE LOSS

CHAI N LADDER M ET HOD RESI DUALS OVER T I M E
( ACROSS ACCI DENT YEARS)
2.0

1.5

1.0
STD RESIDUALS

0.5

0.0
1998            1999                                     2000            2001

- 0.5

- 1.0

- 1.5                                                                  Age 1 t o 2
Age 2 t o 3
Age 3 t o 4
- 2.0
ACCIDENT YEAR
CHAI N LADDER M ET HOD RESI DUALS OVER T I M E
( ACROSS DEVELOPM ENT PERI ODS)

2.0

1.5

1.0
STD RESIDUALS

0.5

0.0
1     2                               3              4             5

- 0.5

- 1.0

- 1.5
AY 1998
AY 1999
AY 2000
- 2.0
AGE
Exploratory Graphical Analysis:
Scatterplots
• Plot of points of dependent variable vs
independent variable
• Should suggest the shape of the relationship
between the variables
– Does it look like a linear relationship?
– Does it look non-linear?
– Is any relationship suggested: If not – mean of
Y is its best estimate
Incremental vs. Cumulative - 1

AGE 1 TO 2 I NCREMENTAL VERSUS AGE 1 CUMULATI VE
( I N 0 0 0 's)

18,000

16,000

14,000

12,000
INCREMENTAL LOSS

10,000

8,000

6,000

4,000

2,000

0
0   500      1,000    1,500             2,000                  2,500   3,000   3,500   4,000   4,500
CUMULATIVE LOSS
Incremental vs. Cumulative - 2

AGE 2 TO 3 I NCREMENTAL VERSUS AGE 2 CUMULATI VE
( I N 0 0 0 's)
2,000

1,800

1,600

1,400
INCREMENTAL LOSS

1,200

1,000

800

600

400

200

0
0   2,000    4,000     6,000         8,000               10,000   12,000   14,000   16,000   18,000
CUMULAT I VE LOSS
Incremental vs. Cumulative - 3

AGE 3 T O 4 I NCREM ENT AL VERSUS AGE 3 CUM ULAT I VE
( I N 0 0 0 ' s)

35

30

25
INCREMENTAL LOSS

20

15

10

5

0
0   1,000             2,000                 3,000              4,000   5,000   6,000
CU MU LATI VE LO SS
Assumes a Linear Function
Linear Function                 Non-linear Function

Y-VAriable
Y-Variable

X-Variable                    X-Variable
Linear Regression
• A to B incremental against age A cumulative
• Use Excel function to model y = mx + b
• LINEST(known y’s, known x’s,const,stat)
– Result is a 5 x 2 array
– Arguments
•   known y’s – column of incremental
•   known x’s – column of cumulative
•   const – “false” sets b = 0
•   stat – “true” provides full array
Input Data for 1 to 2 Incremental
vs. 1 Cumulative
AY            X            Y

1998      34,691    1,368,070
1999   1,018,092    3,514,703
2000   4,161,397   12,174,751
2001   4,077,332   16,815,192
Create Array of Regression
Statistics
•   Select 5 x 2 matrix of cells
•   Type LINEST function in upper left cell
•   Keystroke <CTRL><SHIFT><ENTER>

LINEST ARRAY TABLE (1 TO 2)
m               3.297           809,596 b
SEm               0.689         2,038,734 SEb
r2              0.920          2,523,361 SEy
F Statistic          22.871              2.000 Degrees of Freedom
Reg SS 145,625,078,580,873 12,734,698,887,975 Res SS
Statistics Based on Residual
• Compute variance of regression as sum of squared
residuals divided by the degrees of freedom (the
mean square error, MSE)
– Its square root, s, is standard error of regression
• The R2 or percent of explained variance: 1-R2 =
divide MSE by total variance
• Standard deviation of constant
– Use to test significance of constant
• Standard deviation of coefficient
– Use to test significance of coefficient
Selecting Array Components
• INDEX Function in Excel will return array elements

• Form - INDEX(ARRAY, ROW #, COLUMN #)

–   INDEX(LINEST, 1, 1) = M
–   INDEX(LINEST, 1, 2) = B
–   INDEX(LINEST, 2, 1) =SEm
–   INDEX(LINEST, 2, 2) =SEb
–   INDEX(LINEST, 4, 2) =Degrees of Freedom
Test Significance Of Intercept
• Calculate Student t statistic, B / SEb
• Select Significance Level –Judgment
(selected 0.05 for this example)

• Excel Function for probability value from Student t
distribution
- TDIST(ABS(B/ SEb), Degrees Freedom, Tails)
- Set Tails = 2 For 2-tail Distribution

• Compare result to selected significance level
Results Significance of Intercept

1 to 2             2 to 3        3 to 4
Intercept:          809,596.433          20,993.037    -9,061.620
Standard Error           2,038,734             38,195              0
T Statistic             0.397              0.550           N/A
Degrees of Freedom                 2.000              1.000         0.000
Student t probability             0.730              0.680           N/A
Not Significant    Not Significant
Conclusion                                                      N/A
From Zero          From Zero
The “One Factor Model”
• Age-toAged Factor=
incrementald loss/cumulatived-1
• Incremental lossd = f(d) * cumulatived-1
• This is equivalent to weighted regression
where
–   Y is incremental at d
–   X is cumulative at d-1
–   B, the coefficient, is the (LDF – 1.0)
–   The regression has no constant term
–   Weights are xi/S xj instead of xi2/S xj2
Linear Regression Through the
Origin-Statistic Array
LINEST ARRAY TABLE (1 TO 2)
3.512073632                0.000
0.361831528                #N/A
0.91324316        2,139,999.524
31.57940603                     3
144,620,983,575,556   13,738,793,893,292
Results Linear Regression
Through the Origin
1 to 2         2 to 3             3 to 4
Slope:                           3.512          0.107              0.005
Standard Error                  0.362          0.002              0.001
T Statistic                     9.706        51.969               4.754
Degrees of Freedom              3.000          2.000              1.000
Student t probability           0.002          0.000              0.132
Significant    Significant    Not Significant
Conclusion
From Zero     From Zero           From Zero
R Squared:                       0.913          0.998              0.897
SEy                      2,139,999.524    34,947.730           6,023.359

1 to 2         2 to 3             3 to 4
Selected f(d):                   3.512          0.107              0.005
Linear Regression Projections
CUMULATIVE LOSSES + MODELED INCREMENTAL LOSSES
AY                      1       1 to 2         2 to 3        3 to 4          4 to 5
1998              34,691    1,402,761      1,544,751     1,547,344       1,549,844
1999           1,018,092    4,532,796      5,063,937     5,093,082      5,093,082
2000           4,161,397   16,336,148     18,067,843   18,165,559      18,165,559
2001           4,077,332   20,892,524    23,123,108    23,248,164      23,248,164
2002           3,415,880  15,412,702     17,058,235    17,150,490      17,150,490

MODELED INCREMENTAL LOSSES
AY                   1        1 to 2       2 to 3           3 to 4
1998           34,691       121,837      149,765            8,354
1999        1,018,092     3,575,616      483,943           27,387
2000        4,161,397    14,615,132    1,744,124          97,716
2001        4,077,332    14,319,891   2,230,584          125,056
2002        3,415,880   11,996,822    1,645,533           92,255
Residual Results
RESIDUALS
AY                        1 to 2           2 to 3          3 to 4
1998            1,246,232.8090      (7,775.2208)    (5,761.2629)
1999              (60,912.4134)     47,199.1589      1,757.4699
2000           (2,440,380.5837)    (12,428.7173)
2001            2,495,301.5777

STD DEV          2,109,837.6096     33,164.5420      5,316.5470

STANDARDIZED RESIDUALS
AY                     1 to 2        2 to 3                3 to 4
1998                 0.5907       (0.2344)              (1.0836)
1999                (0.0289)       1.4232                0.3306
2000                (1.1567)      (0.3748)
2001                 1.1827
Graphs Based on Residual
•   Plot residual versus Y
•   Plot residual versus predicted Y
•   Plot residual versus X
•   Plot residual versus variables not in
regression (i.e., age, calendar year)

AGE 1 T O 2 CHAI N LADDER M ODEL RESI DUAL VERSUS AGE 1 CUM ULAT I VE
( I N 0 0 0 ' s)

3,000

2,000

1,000
RESIDUAL

0
0     500      1,000     1,500           2,000             2,500   3,000   3,500   4,000   4,500

- 1,000

- 2,000

- 3,000
CUMULATIVE LOSS

AGE 2 T O 3 CHAI N LADDER M ODEL RESI DUAL VERSUS AGE 2 CUM ULAT I VE
( I N 0 0 0 ' s)

60

50

40

30
RESIDUAL

20

10

0
0   2,000    4,000     6,000          8,000           10,000   12,000   14,000   16,000   18,000

- 10

- 20
CUMULATIVE LOSS

AGE 3 T O 4 CHAI N LADDER M ODEL RESI DUAL VERSUS AGE 3 CUM ULAT I VE
( I N 0 0 0 ' s)

3

2

1

0
0      1,000           2,000                 3,000              4,000   5,000   6,000

-1
RESIDUAL

-2

-3

-4

-5

-6

-7
CUMULATIVE LOSS

CHAI N LADDER M ODEL RESI DUALS OVER T I M E
( ACROSS ACCI DENT YEARS)
5.0

4.0

3.0
STD RESIDUALS

2.0

1.0

0.0
1998           1999                                    2000            2001

- 1.0
Age 1 t o 2
Age 2 t o 3
Age 3 t o 4
- 2.0
ACCIDENT YEAR

CHAI N LADDER M ODEL RESI DUALS OVER T I M E
( ACROSS DEVELOPM ENT PERI ODS)

5.0

4.0

3.0
STD RESIDUALS

2.0

1.0

0.0
2            3                                    4          5

- 1.0
1998
1999
2000
- 2.0
AGE
• What to do if tests of assumptions
• Alternative Models
–   Linear with Constant
–   Bornhuetter-Ferguson (BF)
–   Cape Cod-Special Case of BF (CC)
Approach to CC and BF
• Estimate Ultimate Claim Counts
• Parameterize incremental severity by functional
form f(d)*h(w), where f(d) varies by development
age and h(w) varies by accident year
• Fitting accomplished by an iterative approach with
• Cape Cod assumes h(w) constant over all accident
years (reduces parameters)
Claim Count
CUMULATIVE COUNT
AY      1          2              3       4        5
1998    11          215          226     227      228
1999   133          598          684     688
2000   632        2,731        3,004
2001   877        3,577
2002   665

INCREMENTAL COUNT
AY      1       1 to 2      2 to 3    3 to 4   4 to 5
1998    11          204          11         1        1
1999   133          465          86         4
2000   632        2,099        273
2001   877        2,700
2002   665
Count
AGE TO AGE FACTORS
1 to 2        2 to 3       3 to 4         4 to 5   5 to Ult
Wgtd Avg      4.3079       1.1044       1.0055         1.0044
f(d)       3.3079       0.1044       0.0055         0.0044
% Reported    0.2081       0.6884       0.0936         0.0054     0.0044

AY            1          1 to 2          2 to 3     3 to 4    4 to 5
1998         11             215             226       227       228
1999        133             598             684       688       691
2000        632           2,731           3,004     3,021     3,034
2001        877           3,577          3,950      3,972     3,990
2002        665          2,865           3,164      3,181     3,195
Claim Count Incremental vs Cumulative-1

AGE 1 T O 2 I NCREM ENT AL VERSUS AGE 1 CUM ULAT I VE

3,000

2,500

2,000
INCREMENTAL COUNT

1,500

1,000

500

0
0   100   200       300       400              500             600   700   800   900   1,000
CUMULATIVE COUNT
Claim Count Incremental vs Cumulative-2

AGE 2 T O 3 I NCREM ENT AL VERSUS AGE 2 CUM ULAT I VE

300

250

200
INCREMENTAL COUNT

150

100

50

0
0   500              1,000                    1,500                2,000   2,500   3,000
CUMULATIVE COUNT
Claim Count Incremental vs Cumulative-3

AGE 3 TO 4 I NCREMENTAL VERSUS AGE 3 CUMULATI VE

5

4
INCREMENTAL COUNT

3

2

1

0
0   100      200        300                 400                    500   600   700   800
CUMULATIVE COUNT
Claim Count Residual - 1

CLAI M COUNT RESI DUAL PLOT OVER T I M E
( ACROSS ACCI DENT YEARS)
2.0

1.0
STD RESIDUALS

0.0
1998         1999                               2000           2001

- 1.0

Age 1 t o 2
Age 2 t o 3
Age 3 t o 4
- 2.0
ACCIDENT YEAR
Claim Count Residual - 2

CLAI M COUNT RESI DUAL PLOT OVER T I M E
( ACROSS DEVELOPM ENT PERI ODS)

2.0

1.0
STD RESIDUALS

0.0
1    2                            3            4          5

- 1.0

1998
1999
2000
- 2.0
AGE
Significance of Intercept-Claim
Count
1 to 2             2 to 3             3 to 4
Intercept:                       136.181              6.901             -0.480
Standard Error                        66                 22                  0
T Statistic                       2.079              0.307                N/A
Degrees of Freedom                2.000              1.000              0.000
Student t probability             0.173              0.810                N/A
Not Significant    Not Significant    Not Significant
Conclusion
From Zero          From Zero          From Zero
R Squared:                         0.997              0.981              1.000
Linear Regression Results-Claim
Count
1 to 2         2 to 3         3 to 4
Slope:                         3.168         0.102          0.006
Standard Error                0.114         0.007          0.000
T Statistic                 27.858        14.484         13.464
Degrees of Freedom            3.000         2.000          1.000
Student t probability         0.000         0.005          0.047
Significant   Significant    Significant
Conclusion
From Zero     From Zero      From Zero
R Squared:                     0.990         0.979          0.979
SEy                         123.868        19.682           0.305

1 to 2         2 to 3         3 to 4
Selected f(d):                3.168          0.102          0.006
Linear Regression Projections
Cumulative Count + Modeled Incremental Count
AY                1          1 to 2           2 to 3           3 to 4             4 to 5
1998             11             215              226             227                228
1999            133             598              684             688                688
2000            632           2,731            3,004           3,021              3,021
2001            877           3,577           3,941            3,963              3,963
2002            665          2,772            3,054            3,071              3,071

Modeled Incremental Count
AY                1           1 to 2           2 to 3             3 to 4
1998            11               35               22                  1
1999           133             421                61                  4
2000           632           2,002              278                  17
2001           877           2,778              364                  22
2002           665           2,107              282                  17
Observed Severity
Severity Per File
AY         1            2                3       4        5
1998      152       6,152             6,775   6,787    6,798
1999    1,480       6,588             7,360   7,403
2000    1,377       5,407             5,980
2001    1,029       5,272
2002    1,112

Incremental Severity
AY         1       1 to 2           2 to 3   3 to 4   4 to 5
1998      152        6,000             623        11       11
1999    1,480        5,109             772        42
2000    1,377        4,030             573
2001    1,029        4,243
2002    1,112
Initial Seed for f(d) and h(w)
Cape Cod
H(w) BASED ON THE ULTIMATES FROM THE CHAIN
LADDER MODELS (LOSS DIVIDED BY CLAIM COUNTS)

5,943 FOR ALL YEARS

F(d) BASED ON THE PAYMENT PATTERN FROM THE

CL Model                 1      1 to 2    2 to 3    3 to 4    4 to 5
Final Selections:               3.512     0.107     0.005     0.000
Cumulative            1.000     4.512     4.994     5.021     5.021
Incremental           1.000     3.512     0.482     0.027     0.000
Starting Values
% Reported          19.92%    69.95%     9.59%     0.54%     0.00%
Iterative Process Results-CC

F(d)          15.86%          74.50%       9.65%      0.00%     0.00%
Fitted                          FITTED INCREMENTAL SEVERITY
h(w)           AY            1           1 to 2      2 to 3    3 to 4    4 to 5
6,507          1998     1,032           4,848         628        -         -
6,507          1999     1,032           4,848         628        -         -
6,507          2000     1,032           4,848         628        -         -
6,507          2001     1,032           4,848         628        -         -
6,507          2002     1,032           4,848         628        -         -
Fitted Ultimates and Projected Development-CC

FITTED CUMULATIVE SEVERITY                     ULT CLAIMS    ULT LOSS
AY            1             2            3          4        5
1998      1,032         5,880        6,507      6,507    6,507            228    1,483,663
1999      1,032         5,880        6,507      6,507    6,507            688    4,477,018
2000      1,032         5,880        6,507      6,507    6,507          3,021   19,659,486
2001      1,032         5,880        6,507      6,507    6,507          3,963   25,789,508
2002      1,032         5,880        6,507      6,507    6,507          3,071   19,984,231

PROJECTED LOSS DEVELOPMENT
AY                  1              2            3            4           5
1998          34,691      1,402,761    1,544,751    1,547,344   1,549,844
1999       1,018,092      4,532,796    5,063,937    5,093,082  5,093,082
2000       4,161,397     16,336,148   18,067,843  18,067,843 18,067,843
2001       4,077,332     20,892,524  23,380,324   23,380,324 23,380,324
2002       3,415,880    18,303,297   20,231,087   20,231,087 20,231,087
Initial Seed for f(d) and h(w)
Bornhuetter-Ferguson
H(w) BASED ON THE ULTIMATES FROM THE CHAIN
LADDER MODELS (LOSS DIVIDED BY CLAIM COUNTS)
1998         1999         2000     2001    2002
6,798       7,403        6,013     5,866   5,585

F(d) BASED ON FINAL PATTERN FROM CAPE COD MODEL

1           1 to 2        2 to 3    3 to 4    4 to 5
Starting Values
% Reported         15.86%         74.50%           9.65%     0.00%     0.00%
Iterative Process Results-BF

F(d)     14.69%          75.60%      9.71%        0.00%     0.00%
Fitted                       FITTED INCREMENTAL SEVERITY
h(w)       AY           1           1 to 2     2 to 3      3 to 4    4 to 5
7,666     1998     1,126           5,795        744          -         -
6,895     1999     1,013           5,212        669          -         -
5,484     2000       806           4,146        532          -         -
5,663     2001       832           4,281        550          -         -
7,570     2002     1,112           5,723        735          -         -
Fitted Ultimates and Projected Development-BF

FITTED CUMULATIVE SEVERITY                     ULT CLAIMS      ULT LOSS
AY                 1             2           3            4        5
1998          1,126         6,922       7,666        7,666    7,666            228       1,747,831
1999          1,013         6,225       6,895        6,895    6,895            688       4,743,672
2000            806         4,952       5,484        5,484    5,484          3,021      16,569,195
2001            832         5,113       5,663        5,663    5,663          3,963      22,443,025
2002          1,112         6,835       7,570        7,570    7,570          3,071      23,249,242

PROJECTED LOSS DEVELOPMENT
AY                  1               2            3            4                          5
1998          34,691       1,402,761    1,544,751    1,547,344                  1,549,844
1999       1,018,092       4,532,796    5,063,937    5,093,082                 5,093,082
2000       4,161,397      16,336,148   18,067,843  18,067,843                 18,067,843
2001       4,077,332      20,892,524  23,071,364   23,071,364                 23,071,364
2002       3,415,880     20,992,140   23,249,250   23,249,250                 23,249,250
Summary of Projected Ultimates
Ultimate Losses (Age 5)
AY          CL Method               CL Model         Cape Cod            FH
1998          1,549,844             1,549,844         1,549,844    1,549,844
1999          5,101,311             5,093,082         5,093,082    5,093,082
2000         18,183,945            18,165,559        18,067,843   18,067,843
2001         23,297,180            23,248,164        23,380,324   23,071,364
2002         17,695,084            17,150,490        20,231,087   23,249,250
Total        65,827,364            65,207,139        68,322,181   71,031,383

% Reported (incremental)
Dev       CL Method              CL Model         Cape Cod             FH
1            19.30%               19.92%          15.86%          14.69%
2            70.37%               69.95%          74.50%          75.60%
3              9.68%               9.59%            9.65%          9.71%
4              0.48%               0.54%            0.00%          0.00%
5              0.00%               0.00%            0.00%          0.00%

Cape Cod and FH Model are driven by AY 1998, Dev 1 loss amount
Comparison of Prediction of
Incremental Loss
CL Model     Cape Cod           BF
SSE (000,000's)     13,741,273   13,213,917   2,818,248
DF                          11           11           7
MSE (000,000's)      1,249,207    1,201,265     402,607

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