Chain Ladder Method and CredibilityNM

0 1 2 3 4

1994 580,222 499,679 240,001 133,601 54,912

1995 494,534 499,293 192,227 159,007

1996 551,136 509,075 238,245

1997 648,031 664,188

1998 746,003

STEP 1: DEVELOP A TABLE OF CUMULATIVE PAYMENTS

0 1 2 3 4

1994 580,222 1,079,901 1,319,902 1,453,503 1,508,415

1995 494,534 993,827 1,186,054 1,345,061

1996 551,136 1,060,211 1,298,456

1997 648,031 1,312,219

1998 746,003

m1/0 = 1.955

m2/1 = 1.214

m3/2 = 1.117

m4/3 = 1.038

minf/4 = 1.112

0 1 2 3 4

1994 580,222 1,079,901 1,319,902 1,453,503 1,508,415

1995 494,534 993,827 1,186,054 1,345,061 1,395,876

1996 551,136 1,060,211 1,298,456 1,450,070 1,504,853

1997 648,031 1,312,219 1,592,954 1,778,956 1,846,163

1998 746,003 1,458,645 1,770,707 1,977,464 2,052,170

1395876 = 1345061 * m4/3 1450070 = 1298456 * m3/2

PAYMENTS ($) MADE IN DEVELOPMENT YEARYEAR OF

ORIGIN

CHAIN-LADDER METHOD WITHOUT INFLATION ADJUSTMENT

STEP 3: FILL IN THE BLANKS (yellow)

-->(1,453,503 + 1,345,061) / (1,319,902 + 1,186,054 )

--> (1,508,415/1,453,503)

--> (1,079,901 + 993,827 + 1060211+1312219) / ( 580,222 +494,534 +551,136+648,031)

--> (1319,902 + 1,186,054 + 1,298,456)/ (1,079,901+993,827 +1,060,211)

1504853 = 1450070 * m 4/3

1592954 = 1312219* m2/1

--> (1,677,402/1508415)

CUMULATIVE PAYMENTS ($) UP TO AND INCLUDING DEVELOPMENT YEAR

STEP 2: RATIO OF THE TOTAL CLAIMS SETTLED

YEAR OF

ORIGIN

YEAR OF

ORIGIN

CUMULATIVE PAYMENTS ($) UP TO AND INCLUDING DEVELOPMENT YEAR

*First fill in

the yellow

blanks

1846163 = 1778956 * m4/3

1778956 = 1592954*m3/2

STEP 4: OUTSTANDING CLAIM PROVISION

1994

1995

1996

1997

1998

Total 3,028,006

YEAR OF

ORIGIN

OUTSANDING CLAIM

PROVISION ($)

168,987

207,194

374,984

740,769

1,536,071

*next compute the estimated total liability (blue)

*Calculate Outstanding Claim estimate (green)

2052988 = 1846163 * m inf/4

2282074 = 2052170 * m inf/4

374984 = 11673440 - 1298456

740769 = 2052988 - 1312219

1536071 = 2282074 - 746003

1552255 = 1395876 * m inf/4

1673440 = 1504853 * m inf/4

207,194 = 1552255 - 1345061

168,987

168,987 1,677,402 *to get value for estimated total liability

= 168,987 + 1,508,415

168,987 1,677,402

207,194 1,552,255

374,984 1,673,440

740,769 2,052,988

1,536,071 2,282,074

OUTSTANDING

CLAIM ESTIMATE

AT 31/12/98 ($)

ESTIMATED TOTAL

LIBILITY ($)

-->(1,453,503 + 1,345,061) / (1,319,902 + 1,186,054 )

--> (1,508,415/1,453,503)

--> (1,079,901 + 993,827 + 1060211+1312219) / ( 580,222 +494,534 +551,136+648,031)

--> (1319,902 + 1,186,054 + 1,298,456)/ (1,079,901+993,827 +1,060,211)

1592954 = 1312219* m2/1

--> (1,677,402/1508415)

ESTIMATED TOTAL

LIBILITY ($)

OUTSTANDING

CLAIM ESTIMATE

AT 31/12/98 ($)

1458645 = 746003 * m1/0

1770707 = 1458645* m2/1

OUTSTANDING

CLAIM ESTIMATE

AT 31/12/98 ($)

ESTIMATED TOTAL

LIBILITY ($)

1846163 = 1778956 * m4/3

1778956 = 1592954*m3/2

1977464 = 1770707*m3/2

2052170 = 1977464 * m4/3

2052988 = 1846163 * m inf/4

2282074 = 2052170 * m inf/4

374984 = 11673440 - 1298456

740769 = 2052988 - 1312219

1536071 = 2282074 - 746003

1552255 = 1395876 * m inf/4

1673440 = 1504853 * m inf/4

207,194 = 1552255 - 1345061

INFLATION RATES OVER THE PERIOD 1999-2003

1994 11.00% 1.11

1995 12.00% 1.12

1996 10% 1.1

1997 15.00% 1.15

1998 10% 1.1

FUTURE INFLATION 20% 1.2

0 1 2 3

1994 580,222 1,079,901 1,319,902 1,453,503

1995 494,534 993,827 1,186,054 1,345,061

1996 551,136 1,060,211 1,298,456

1997 648,031 1,312,219

1998 746,003

0 1 2 3

1994 580,222 499,679 240,001 133,601

1995 494,534 499,293 192,227 159,007

1996 551,136 509,075 238,245

1997 648,031 664,188

1998 746,003

STEP 1: DEVELOP AN INFLATION-ADJUSTED PAYMENTS TABLE (ASSUME PAYMENT MADE MID YR)

0 1 2 3

1994 1,003,733 778,740 333,961 169,005

1995 770,721 694,766 243,167 174,908

1996 766,906 643,980 262,070

1997 819,759 730,607

1998 820,603

YEAR OF ORIGINPAYMENTS (DECEMBER 1998 PRICES) IN DEVELOPMENT YEAR

CHAIN-LADDER METHOD WITH INFLATION ADJUSTMENT

YEAR OF ORIGIN

CUMULATIVE PAYMENTS (RM) MADE IN DEVELOPMENT YEAR

YEAR OF ORIGIN

PAYMENTS ($) MADE IN DEVELOPMENT YEAR


11.00% 1.11 4

12.00% 1.12 5

10% 1.1 6

15.00% 1.15 7

10% 1.1 8

FUTURE INFLATION 20% 1.2

*Calculation :

770721 = 494534*1.12*1.1*1.15*1.1 694766 = 499293 *1.1*1.15*1.1

766906 = 551136 * 1.1*1.15*1.1 643980 = 509075*1.15*1.1

819759= 648031*1.15*1.1 730607 = 664188*1.1

820603 = 746003* 1.1

STEP 2: DEVELOP A TABLE OF CUMULATIVE INFLATION-ADJUSTED PAYMENTS

0 1 2 3

1994 1,003,733 1,782,473 2,116,435 2,285,440

1995 770,721 1,465,488 1,708,655 1,883,562

1996 766,906 1,410,886 1,672,955

1997 819,759 1,550,366

1998 820,603

Example : column 1 Column 2

1782473 = 1003733 + 778740

1465488 = 770721 + 694766

*To get value for estimated total liability =

m1/0 = 1.847

m2/1 = 1.180

m3/2 = 1.090

m4/3 = 1.026

minf/4 = 1.072

0 1 2 3

1994 1,003,733 1,782,473 2,116,435 2,285,440

1995 770,721 1,465,488 1,708,655 1,883,562

1996 766,906 1,410,886 1,672,955 1,823,370

1997 819,759 1,550,366 1,829,633 1,994,135

1998 820,603 1,515,953 1,789,022 1,949,872

STEP 3: RATIO OF THE TOTAL CLAIMS SETTLED

YEAR OF ORIGIN

CUMULATIVE PAYMENTS (DECEMBER 1998 PRICES) TO DEVELOPMENT YEAR

STEP 4: FILL IN THE BLANKS

778740 = 499679 *1.12*1.1*1.15*1.1

YEAR OF ORIGIN

1999

1003733 = 580222 * 1.11*1.12*1.1*1.15*1.1


2000

2001

2002

2003

2116435 = 1782473+333961

1708655 = 1465488 + 243167

168987+2345843 = 2514830

STEP 5: SEPARATE THE CONSTANT-DOLLAR CUMULATIVE PAYMENTS INTO PAYMENTS BY DEVELOPMENT YEAR

0 1 2 3

1994

1995

1996 150,415

1997 279,267 164,502

1998 695,350 273,069 160,851

0 1 2 3

1994

1995

1996 164,771

1997 305,922 216,243

1998 761,718 358,958 253,732

54533 = 49782 * 1.2^0.5

63349= 48191 * 1.2^1.5

216243 = 164502 * 1.2^1.5 83138 = 52704 * 1.2^2.5

97551 = 51534* 1.2^3.5

11.00% 1.11 4

12.00% 1.12 5

10% 1.1 6

15.00% 1.15 7

10% 1.1 8

20% 1.2 9

YEAR OF ORIGIN

1994

1995 Calculation

1996

1997

OUTSANDING CLAIM PROVISION ($)

266,567

318,165

534,368

1,007,219

164771 = 150415 * 1.2^0.5

FUTURE INFLATION

534368 = 164771+63349+306248

* Example ofCalculation

YEAR OF ORIGINPAYMENTS (DECEMBER 1998 PRICES) MADE IN DEVELOPMENT YEAR

2003

STEP 6: ALLOW FOR INFLATION AFTER 31/12/98 (PAYMENTS ARE ASSUMED TO BE MADE ON AVERAGE IN THE MIDDLE OF THE DEVELOPMENT YEAR, AND

THOSE IN THE RANGE 5-INF IN THE MIDDLE OF DEVELOPMENT YEAR 7)

318165 = 54533 + 263632


1999

2000

2001

2002

YEAR OF ORIGINPAYMENTS (DECEMBER 1998 PRICES) MADE IN DEVELOPMENT YEAR

1998

Total

ASSUME OUTSTANDING CLAIM PROVISIONS ARE EXPECTED TO EARN INTEREST AT 12% PER ANNUM

INTEREST 12% 1.12

0 1 2 3

1994

1995

1996 155,694

1997 289,069 182,438

1998 719,755 302,842 191,131

PAYMENTS (DECEMBER 1998 PRICES) MADE IN DEVELOPMENT YEAR

0 1 2 3

1994

1995

1996 150415

1997 279267 164502

1998 695350 273069 160851

YEAR OF ORIGIN

1994

1995 Calculation

1996

1997

1998

total 3,077,411

228,840

393,044

YEAR OF ORIGIN

1,505,100

OUTSANDING CLAIM PROVISION ($)

200,799

PROVISION REQUIRED AT 31/12/98 TO MEET PAYMENTS DUE IN DEVELOPMENT YEAR

1007219 = 305922 + 216243+83138+401916

1943552 = 761718 + 358958+253732+97551+471593

YEAR OF ORIGIN

1,943,552

4,069,872

150415 X (1.2/1.12)^0.5 = 155694

164502 X (1.2/1.12)^1.5 = 182438

160851X (1.2/1.12)^2.5 = 191131

749,628

41,508,415 168,987 1677402

4

54,912 168,987

STEP 1: DEVELOP AN INFLATION-ADJUSTED PAYMENTS TABLE (ASSUME PAYMENT MADE MID YR)

4

60,403 168,987

PAYMENTS (DECEMBER 1998 PRICES) IN DEVELOPMENT YEAR

CHAIN-LADDER METHOD WITH INFLATION ADJUSTMENT

OUTSTANDING

CLAIM ESTIMATE

AT 31/12/98 ($)

CUMULATIVE PAYMENTS (RM) MADE IN DEVELOPMENT YEAR

OUTSTANDING

CLAIM ESTIMATE

AT 31/12/98 ($)

PAYMENTS ($) MADE IN DEVELOPMENT YEAR

ESTIMATED

TOTAL LIBILITY

($)

ESTIMATED

TOTAL LIBILITY

($)

OUTSTANDING

CLAIM ESTIMATE

AT 31/12/98 ($)

ESTIMATED

TOTAL LIBILITY

($)

169005= 133601*1.15*1.1

243167 = 192227*1.15*1.1 174908= 159007*1.1

262070 = 238245*1.1

4

2,345,843 168,987 2,514,830

4

2,345,843 168,987 2,514,830

1,933,344 2,072,616

1,871,561 2,006,382

2,046,840 2,194,287

2,001,407 2,145,582

CUMULATIVE PAYMENTS (DECEMBER 1998 PRICES) TO DEVELOPMENT YEAROUTSTANDING

CLAIM ESTIMATE

AT 31/12/98 ($)

ESTIMATED

TOTAL LIBILITY

($)

778740 = 499679 *1.12*1.1*1.15*1.1

60403= 54912*1.1


2116435 = 1782473+333961

1708655 = 1465488 + 243167

OUTSTANDING

CLAIM ESTIMATE

ESTIMATED

TOTAL LIBILITY

333961= 24001*1.1*1.15*1.1

STEP 5: SEPARATE THE CONSTANT-DOLLAR CUMULATIVE PAYMENTS INTO PAYMENTS BY DEVELOPMENT YEAR

4

168,987 168,987

49,782 139,272

48,191 134,821

52,704 147,448

51,534 144,175

*139272=2072616-1933344

4

266,567

54,533 263,632

63,349 306,248

83,138 401,916

97,551 471,593

266567 = 168987 *1.2^ 2.5

54533 = 49782 * 1.2^0.5

63349= 48191 * 1.2^1.5

83138 = 52704 * 1.2^2.5 401916 = 147448*1.2^5.5

97551 = 51534* 1.2^3.5 471593= 144175*1.2^6.5

534368 = 164771+63349+306248

PAYMENTS (DECEMBER 1998 PRICES) MADE IN DEVELOPMENT YEAR

5-INF

STEP 6: ALLOW FOR INFLATION AFTER 31/12/98 (PAYMENTS ARE ASSUMED TO BE MADE ON AVERAGE IN THE MIDDLE OF THE DEVELOPMENT YEAR, AND

THOSE IN THE RANGE 5-INF IN THE MIDDLE OF DEVELOPMENT YEAR 7)

OUTSTANDING

CLAIM ESTIMATE 5-INF

306248 = 134821*1.2^4.5

263632 = 139272 *1.2^ 3.5

318165 = 54533 + 263632

*134821=2006382-1871561

PAYMENTS (DECEMBER 1998 PRICES) MADE IN DEVELOPMENT YEAR OUTSTANDING

CLAIM ESTIMATE

ASSUME OUTSTANDING CLAIM PROVISIONS ARE EXPECTED TO EARN INTEREST AT 12% PER ANNUM

4

200,799

51,529 177,311

53,445 183,905

62,626 215,494

65,610 225,762

4

49782

48191

52704

51534

719755+302842+191131+65610+225762= 1505100

PROVISION REQUIRED AT 31/12/98 TO MEET PAYMENTS DUE IN DEVELOPMENT YEAR OUTSTANDING

CLAIM ESTIMATE

1007219 = 305922 + 216243+83138+401916

1943552 = 761718 + 358958+253732+97551+471593

5-INF

155694+53445+183905 = 393044

289069+182438+62626+215494=749628

49782 X (1.2 /1.12)^0.5 = 51529

48191 X (1.2/1.12)^1.5 = 53445

52704 X (1.2/1.12)^2.5 = 62626

51534X (1.2/1.12)^3.5 = 65610

51529+177311=228840

CREDIBILITY

EXAMPLE:

Claim size (RM) Number of claims

0-400 2

400-800 24

800-1200 32

1200-1600 21

1600-2000 10

2000-2400 6

2400-2800 3

2800-3200 1

3200-3600 1

Over 3600 0

Total 100

$(200 X 2/100 + 600 X 24/100 + ..+3400 X 1/100)

$1,216

STEP 2: VARIANCE OF THE OBSERVED CLAIM SIZE DISTRIBUTION

(2002 X 2/100 +6002 X 24/100 + . +34002 X 1/100) - 12162

362,944

STEP 3: MINIMUM NUMBER OF EXPECTED CLAIMS FOR FULL CREDIBILITY

nF = n0 [1 + (std / mean)2]

n0 = y2 / k2

y from normal distribution (eg. 90%, y=1.645)

n0 = y2 / k2 (k= 0.05, P = 0.9)

n0 = y2 / k2 prob 90%, y=1.645

n0 = 1.6452 / 0.052

A SAMPLE OF 100 CLAIMS ON A GENERAL INSURER IN RESPECT OF A CERTAIN CLASS OF BUSINESS IS GIVEN IN TABLE BELOW.

THE CLAIM FREQUENCY RATE IS ABOUT 0.015. CALCULATE THE MINIMUM SIZE OF THE PORTFOLIO IF FULL CREDIBILITY

(k=0.05,P=0.9) IS TO BE ASSIGNED TO THE EXPERIENCE.

STEP 1: MEAN CLAIM SIZE

n0 = 1082

nF = n0 [1 + (std / mean)2] STD / MEAN = 0.5

nF = 1082 [1 + 0.52]

nF = 1353

STEP 4: FIND MINIMUM PORTFOLIO SIZE

IF 0.015, THEN Nf = 1353

THEREFORE, 100% = 1/0.015 * 13453= 90,200 POLOCIES

A SAMPLE OF 100 CLAIMS ON A GENERAL INSURER IN RESPECT OF A CERTAIN CLASS OF BUSINESS IS GIVEN IN TABLE BELOW.

THE CLAIM FREQUENCY RATE IS ABOUT 0.015. CALCULATE THE MINIMUM SIZE OF THE PORTFOLIO IF FULL CREDIBILITY

(k=0.05,P=0.9) IS TO BE ASSIGNED TO THE EXPERIENCE.

Chain Ladder Method and CredibilityNM

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