COTOR Training Session II

GL Data: Long Tails, Volatility, Data Transforms

September 11, 2006

COTOR Session II Presenters

Doug Ryan

MBA Actuaries, Inc.

Phil Heckman

Heckman Actuarial Consulting

Assumptions and Verification

• Behavior of mean, variance, distribution (sometimes)

• Verify by examining– Descriptive statistics– Regression diagnostics– Scatter plots– Residual plots

GL Data: Chain Ladder

Mack GL Data

Cumulative LossesAY 1 2 3 4 5 6 7

1981 5,012 8,269 10,907 11,805 13,539 16,181 18,0091982 106 4,285 5,396 10,666 13,782 15,599 15,600

1983 3,410 8,992 13,873 16,141 18,735 22,214 22,8631984 5,655 11,555 15,766 21,266 23,425 26,083 27,0671985 1,092 9,565 15,836 22,169 25,955 26,1801986 1,513 6,445 11,702 12,935 15,8521987 557 4,020 10,946 12,3141988 1,351 6,947 13,112

1989 3,133 5,395

Incremental LossesAY 1 2 3 4 5 6 7

1981 5,012 3,257 2,638 898 1,734 2,642 1,8281982 106 4,179 1,111 5,270 3,116 1,817 11983 3,410 5,582 4,881 2,268 2,594 3,479 6491984 5,655 5,900 4,211 5,500 2,159 2,658 9841985 1,092 8,473 6,271 6,333 3,786 2251986 1,513 4,932 5,257 1,233 2,9171987 557 3,463 6,926 1,3681988 1,351 5,596 6,1651989 3,133 2,262

Cumulative Losses

0

5,000

10,000

15,000

20,000

25,000

30,000

0 1 2 3 4 5 6 7 8

Series1

Series2

Series3

Series4

Series5

Series6

Series7

Series8

Series9

Incremental Losses

0

1,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

9,000

0 1 2 3 4 5 6 7 8

Series1

Series2

Series3

Series4

Series5

Series6

Series7

Series8

Series9

Model: Variance of incremental is constant [ a(d, c(w,d)) = k(d) ]Implication 1: Linear Regression of 2 incrementals against Dev 1 cumulative (table format)

1Slope -0.109 5,113 InterceptSlope Std Error 0.349 1,066 Intercept Std ErrorR Square 0.014 1,948 SEy

0.098 7.000 Degrees of Freedom370,061 26,567,031

Implication 1: Linear2 3 4 5 6 7

Slope: -0.109 0.049 0.131 0.041 -0.100 0.005 Standard Error 0.349 0.309 0.283 0.071 0.114 0.107 T Statistic -0.312 0.160 0.463 0.586 -0.876 0.046 Degrees of Freedom 7.000 6.000 5.000 4.000 3.000 2.000 Student t probability 0.764 0.878 0.663 0.589 0.445 0.968

ConclusionNot Significant

From ZeroNot Significant

From ZeroNot Significant

From ZeroNot Significant

From ZeroNot Significant

From ZeroNot Significant

From ZeroIntercept: 5,113.372 4,311.471 1,687.179 2,061.069 4,064.460 767.753 Standard Error 1,066.162 2,440.121 3,543.141 1,164.742 2,241.921 2,189.568 T Statistic 4.796 1.767 0.476 1.770 1.813 0.351 Degrees of Freedom 7.000 6.000 5.000 4.000 3.000 2.000 Student t probability 0.002 0.128 0.654 0.152 0.167 0.759

ConclusionSignificant From Zero

Not Significant From Zero

Not Significant From Zero

Not Significant From Zero

Not Significant From Zero

Not Significant From Zero

R Squared: 0.014 0.004 0.041 0.079 0.204 0.001SE y 1,948.150 2,127.836 2,506.272 778.852 1,270.510 930.826

2 3 4 5 6 7Final Selections: m -0.109 0.049 0.131 0.041 -0.100 0.005

b 5,113.372 4,311.471 1,687.179 2,061.069 4,064.460 767.753

What are they?

• Slope standard error• R square: Percentage

of variance explained by regression

• Intercept standard error

• Degrees of Freedom: # Observations - # Parameters

2

2

( )1

( )

i

i

Y Y

Y Y

22

2

( )

( )

ia

i

xs

n x x

2

2( )B

i

s

x x

Cumulative Losses + Projected Incremental LossesAY 1 2 3 4 5 6 7

1981 5,012 8,269 10,907 11,805 13,539 16,181 18,009 1982 106 4,285 5,396 10,666 13,782 15,599 15,600 1983 3,410 8,992 13,873 16,141 18,735 22,214 22,863 1984 5,655 11,555 15,766 21,266 23,425 26,083 27,067 1985 1,092 9,565 15,836 22,169 25,955 26,180 27,076 1986 1,513 6,445 11,702 12,935 15,852 18,338 19,195 1987 557 4,020 10,946 12,314 14,886 17,468 18,321 1988 1,351 6,947 13,112 16,517 19,263 21,410 22,282 1989 3,133 5,395 9,973 12,967 15,566 18,081 18,937

Fitted Incremental LossesAY 1 2 3 4 5 6 7

1981 5,012 4,568 4,720 3,116 2,551 2,717 847 1982 106 5,102 4,523 2,394 2,503 2,692 844 1983 3,410 4,742 4,756 3,505 2,731 2,199 876 1984 5,655 4,498 4,882 3,753 2,943 1,732 895 1985 1,092 4,994 4,784 3,762 2,981 1,480 896 1986 1,513 4,949 4,630 3,220 2,598 2,486 857 1987 557 5,053 4,510 3,121 2,572 2,582 853 1988 1,351 4,966 4,655 3,405 2,746 2,147 872 1989 3,133 4,772 4,578 2,994 2,599 2,515 856

Fitted Squared Error (in millions)AY 1 2 3 4 5 6 7

1981 0.00 1.72 4.33 4.92 0.67 0.01 0.961982 0.00 0.85 11.64 8.27 0.38 0.77 0.711983 0.00 0.71 0.02 1.53 0.02 1.64 0.051984 0.00 1.97 0.45 3.05 0.61 0.86 0.011985 0.00 12.10 2.21 6.61 0.65 1.581986 0.00 0.00 0.39 3.95 0.101987 0.00 2.53 5.84 3.071988 0.00 0.40 2.281989 0.00 6.30

Total 94.14n 48p 12

Fit Error 0.073

A Key Diagnostic: Standard Residual

• Standardize by subtracting mean (should be zero) and divide by standard deviation

• A z-score– Z = (x – mean)/sd

Standard Residuals

-2.000

-1.500

-1.000

-0.500

0.000

0.500

1.000

1.500

2.000

0 1 2 3 4 5 6 7 8

Series1

Series2

Series3

Series4

Series5

Series6

Series7

Series8

Series9

Two Factor Model

• One factor model: incremental loss =f(prior cumulative)– Compute separate function for each

development age– Can use Excel regression functions

• Two factor model: incremental loss = f(accident period, development age)– Bornhuetter-Ferguson is an example– Nonlinear function, Use solver

GL Data: Two-Factor ModelImplication 2: q(w, d) = h(w) * f(d)

f(d) 4.986 10.845 9.787 7.256 5.482 3.883 1.790

Fitted Incremental Lossesh(w) AY 1 2 3 4 5 6 7341 1981 1,700 3,698 3,337 2,474 1,869 1,324 610 351 1982 1,748 3,801 3,431 2,544 1,922 1,361 627 503 1983 2,508 5,455 4,923 3,650 2,758 1,953 900 581 1984 2,899 6,305 5,690 4,219 3,187 2,258 1,040 673 1985 3,358 7,303 6,590 4,886 3,692 2,615 1,205 428 1986 2,135 4,643 4,190 3,107 2,347 1,663 766 406 1987 2,023 4,401 3,971 2,944 2,224 1,576 726 536 1988 2,674 5,815 5,248 3,891 2,940 2,082 960 282 1989 1,405 3,056 2,758 2,045 1,545 1,094 504

Fitted Weighted Squared Error (in millions)AY 1 2 3 4 5 6 7

1981 10.97 0.19 0.49 2.49 0.02 1.74 1.481982 2.70 0.14 5.38 7.43 1.43 0.21 0.391983 0.81 0.02 0.00 1.91 0.03 2.33 0.061984 7.60 0.16 2.19 1.64 1.06 0.16 0.001985 5.13 1.37 0.10 2.09 0.01 5.711986 0.39 0.08 1.14 3.51 0.321987 2.15 0.88 8.73 2.491988 1.75 0.05 0.841989 2.98 0.63

Total 93.38n 48p 16

Fit Error 0.091

Cumulative Losses + Projected Incremental LossesAY 1 2 3 4 5 6 7

1981 5,012 8,269 10,907 11,805 13,539 16,181 18,009 1982 106 4,285 5,396 10,666 13,782 15,599 15,600 1983 3,410 8,992 13,873 16,141 18,735 22,214 22,863 1984 5,655 11,555 15,766 21,266 23,425 26,083 27,067 1985 1,092 9,565 15,836 22,169 25,955 26,180 27,385 1986 1,513 6,445 11,702 12,935 15,852 17,515 18,281 1987 557 4,020 10,946 12,314 14,538 16,114 16,840 1988 1,351 6,947 13,112 17,003 19,943 22,025 22,985 1989 3,133 5,395 8,153 10,198 11,743 12,838 13,342

Standardized ResidualsAY 1 2 3 4 5 6 7

1982 -0.672 -0.426 -0.834 -0.180 0.842 1.5221983 0.575 -1.413 1.443 1.592 0.291 -0.7831984 0.193 -0.026 -0.731 -0.218 0.975 -0.3141985 -0.616 -0.901 0.678 -1.370 0.256 -0.0711986 1.782 -0.195 0.765 0.126 -1.5271987 0.440 0.650 -0.991 0.7591988 -1.428 1.800 -0.8341989 -0.334 0.559

Standard Residuals

-2.000

-1.500

-1.000

-0.500

0.000

0.500

1.000

1.500

2.000

0 1 2 3 4 5 6 7 8

Series1

Series2

Series3

Series4

Series5

Series6

Series7

Series8

GL Data: 3-Factor ModelImplication 3: q(w, d) = h(w) * f(d) * g(w+d)

f(d) 77.632 176.191 150.305 102.460 67.959 49.500 18.921

Fitted Incremental Lossesg(w+d) h(w) AY 1 2 3 4 5 6 73.2936 20 1981 5,012 2,664 3,245 2,030 1,516 1,193 521 0.7714 22 1982 1,311 4,247 3,326 2,552 1,828 1,522 646 1.1014 29 1983 2,509 5,226 5,019 3,695 2,801 2,265 984 1.0109 31 1984 2,394 6,116 5,635 4,389 3,233 2,676 826 1.1380 33 1985 2,903 7,115 6,936 5,250 3,958 2,326 1,009 1.2291 19 1986 1,768 4,584 4,342 3,365 1,801 1,488 645 1.4045 17 1987 1,855 4,676 4,534 2,494 1,877 1,551 672 1.5595 20 1988 2,476 6,387 4,397 3,400 2,559 2,114 917 1.7727 12 1989 1,671 3,061 2,962 2,291 1,724 1,424 618 1.4305

Fitted Squared Error (in millions)1 2 3 4 5 6 7

1981 0.00 0.35 0.37 1.28 0.05 2.10 1.711982 1.45 0.00 4.91 7.39 1.66 0.09 0.421983 0.81 0.13 0.02 2.04 0.04 1.47 0.111984 10.64 0.05 2.03 1.23 1.15 0.00 0.031985 3.28 1.84 0.44 1.17 0.03 4.421986 0.06 0.12 0.84 4.54 1.251987 1.69 1.47 5.72 1.271988 1.27 0.63 3.131989 2.14 0.64

Total 77.45n 48p 25

Fit Error 0.15

Standard Residuals1 2 3 4 5 6 7

1981 0.735 -0.385 -0.640 0.245 1.029 1.5251982 -0.085 -1.403 1.535 1.449 0.210 -0.7531983 0.441 -0.087 -0.806 -0.233 0.862 -0.3911984 -0.267 -0.902 0.627 -1.208 -0.013 0.1851985 1.683 -0.421 0.612 -0.194 -1.4921986 0.431 0.579 -1.204 1.2561987 -1.503 1.515 -0.6361988 -0.981 1.1201989 -0.990

Standard Residuals by Duration

-2.000

-1.500

-1.000

-0.500

0.000

0.500

1.000

1.500

2.000

0 1 2 3 4 5 6 7 8

Series1

Series2

Series3

Series4

Series5

Series6

Series7

Series8

Series9

GL Data: Log Chain LadderModel: log(L(d)) = log(L(d-1)) + N(m(d),s(d))Log Transform of Chain-Ladder ModelLOG Cumulative Losses

AY 1 2 3 4 5 6 7 Current Projected @ 71981 8.5196 9.0203 9.2972 9.3763 9.5133 9.6916 9.7986 18,009 18,009 1982 4.6634 8.3629 8.5934 9.2748 9.5311 9.6550 9.6550 15,600 15,600 1983 8.1345 9.1041 9.5377 9.6891 9.8381 10.0085 10.0373 22,863 22,863 1984 8.6403 9.3549 9.6656 9.9649 10.0616 10.1690 10.2061 27,067 27,067 1985 6.9958 9.1659 9.6700 10.0065 10.1641 10.1728 10.2170 26,180 27,365 1986 7.3218 8.7711 9.3675 9.4677 9.6711 9.7911 9.8353 15,852 18,682 1987 6.3226 8.2990 9.3007 9.4185 9.5867 9.7068 9.7510 12,314 17,172 1988 7.2086 8.8461 9.4813 9.7564 9.9246 10.0447 10.0889 13,112 24,075 1989 8.0497 8.5932 9.1235 9.3986 9.5668 9.6868 9.7311 5,395 16,833

Why use logarithms?

• Descriptive statistics indicate data not normal

• A-priori belief that model is mutiplicative

• Residuals increase with value of dependent variable

Log Link RatiosAY 1 2 3 4 5 6 7

1981 8.5196 0.5007 0.2769 0.0791 0.1371 0.1783 0.10701982 4.6634 3.6994 0.2305 0.6814 0.2563 0.1238 0.00011983 8.1345 0.9696 0.4336 0.1514 0.1490 0.1703 0.02881984 8.6403 0.7146 0.3107 0.2993 0.0967 0.1075 0.03701985 6.9958 2.1701 0.5042 0.3364 0.1577 0.00861986 7.3218 1.4492 0.5965 0.1002 0.20341987 6.3226 1.9765 1.0017 0.11781988 7.2086 1.6375 0.63521989 8.0497 0.5435

1 2 3 4 5 6 7m(d) = 7.3174 1.5179 0.4987 0.2522 0.1667 0.1177 0.0432s(d) = 1.2524 1.0213 0.2513 0.2140 0.0558 0.0680 0.0454

m(d)+0.5*s(d)2 = 8.1016 2.0394 0.5302 0.2751 0.1682 0.1200 0.0443

Standard ResidualsAY 1 2 3 4 5 6 7

1981 -0.9960 -0.8827 -0.8088 -0.5314 0.8910 1.40561982 2.1361 -1.0672 2.0054 1.6072 0.0903 -0.95101983 -0.5369 -0.2589 -0.4710 -0.3166 0.7742 -0.31801984 -0.7866 -0.7480 0.2198 -1.2552 -0.1505 -0.13661985 0.6386 0.0219 0.3934 -0.1617 -1.60501986 -0.0673 0.3892 -0.7104 0.65771987 0.4490 2.0021 -0.62831988 0.1171 0.54351989 -0.9541

Standard Residuals

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

0 1 2 3 4 5 6 7 8

Series1

Series2

Series3

Series4

Series5

Series6

Series7

Series8

Series9

GL Data: Log 2-Factor ModelLog transform 2: ln(q(w, d)) = h(w) + f(d) Solved by iterative reweighting

Cumulative LossesAY 1 2 3 4 5 6 7

1981 5,012 8,269 10,907 11,805 13,539 16,181 18,0091982 106 4,285 5,396 10,666 13,782 15,599 15,6001983 3,410 8,992 13,873 16,141 18,735 22,214 22,8631984 5,655 11,555 15,766 21,266 23,425 26,083 27,0671985 1,092 9,565 15,836 22,169 25,955 26,1801986 1,513 6,445 11,702 12,935 15,8521987 557 4,020 10,946 12,3141988 1,351 6,947 13,1121989 3,133 5,395

Iterative Least Squares

• Start with all weights = 1

• Estimate by minimizing weighted sum of squares

• Calculate new weights =

1/(1+ Old Weight*Squared Error)

• Reëstimate. Stop when weights stop changing.

Incremental LossesAY 1 2 3 4 5 6 7

1981 5,012 3,257 2,638 898 1,734 2,642 1,8281982 106 4,179 1,111 5,270 3,116 1,817 11983 3,410 5,582 4,881 2,268 2,594 3,479 6491984 5,655 5,900 4,211 5,500 2,159 2,658 9841985 1,092 8,473 6,271 6,333 3,786 2251986 1,513 4,932 5,257 1,233 2,9171987 557 3,463 6,926 1,3681988 1,351 5,596 6,1651989 3,133 2,262

f(d) 1.231 2.334 2.184 1.707 1.735 1.210 0.000

Fitted Log Incremental Lossesh(w) AY 1 2 3 4 5 6 7

5.832 1981 7.063 8.166 8.016 7.538 7.567 7.042 5.832 5.859 1982 7.090 8.194 8.043 7.566 7.594 7.070 5.859 6.221 1983 7.451 8.555 8.405 7.927 7.955 7.431 6.221 6.365 1984 7.596 8.700 8.549 8.072 8.100 7.575 6.365 6.512 1985 7.743 8.846 8.696 8.219 8.247 7.722 6.512 6.059 1986 7.290 8.394 8.243 7.766 7.794 7.270 6.059 6.006 1987 7.236 8.340 8.190 7.712 7.741 7.216 6.006 6.285 1988 7.515 8.619 8.468 7.991 8.019 7.495 6.285 5.641 1989 6.872 7.975 7.825 7.348 7.376 6.851 5.641

Initial WeightsAY 1 2 3 4 5 6 7

1981 1.000 1.000 1.000 1.000 1.000 1.000 1.0001982 1.000 1.000 1.000 1.000 1.000 1.000 1.0001983 1.000 1.000 1.000 1.000 1.000 1.000 1.0001984 1.000 1.000 1.000 1.000 1.000 1.000 1.0001985 1.000 1.000 1.000 1.000 1.000 1.0001986 1.000 1.000 1.000 1.000 1.0001987 1.000 1.000 1.000 1.0001988 1.000 1.000 1.0001989 1.000 1.000

Iterated Weights Copy values left and reestimate.AY 1 2 3 4 5 6 7

1981 0.320 0.994 0.981 0.647 0.988 0.588 0.2621982 0.145 0.980 0.485 0.498 0.832 0.841 0.0281983 0.682 0.995 0.992 0.961 0.991 0.656 0.9391984 0.478 1.000 0.960 0.774 0.848 0.912 0.7831985 0.642 0.962 0.998 0.778 1.000 0.1581986 0.999 0.988 0.905 0.704 0.9671987 0.545 0.965 0.701 0.8061988 0.914 1.000 0.9371989 0.419 0.941

Final WeightsAY 1 2 3 4 5 6 7

1981 0.511 0.994 0.972 0.734 0.989 0.828 0.5191982 0.327 0.980 0.597 0.605 0.851 0.990 0.1601983 0.783 0.995 0.997 0.974 0.992 0.883 0.9991984 0.634 1.000 0.949 0.792 0.867 0.999 0.9511985 0.678 0.964 1.000 0.795 1.000 0.3141986 0.997 0.988 0.928 0.775 0.9671987 0.617 0.966 0.772 0.8501988 0.879 1.000 0.9541989 0.589 0.943

Weighted Squared ErrorAY 1 2 3 4 5 6 7

1981 0.95744 0.00614 0.02880 0.36227 0.01110 0.20797 1.043521982 2.07004 0.02008 0.67602 0.65355 0.17443 0.00975 6.010141983 0.27682 0.00507 0.00295 0.02646 0.00839 0.13264 0.000041984 0.57871 0.00032 0.05372 0.26301 0.15313 0.00069 0.066951985 0.47388 0.03743 0.00018 0.25854 0.00003 2.191261986 0.00324 0.01170 0.07794 0.29108 0.033691987 0.62008 0.03529 0.29591 0.176351988 0.13737 0.00010 0.047921989 0.69870 0.06016

Total 19.25099n 48p 16

Fit Error 0.60159

Cumulative Losses + Projected Incremental LossesAY 1 2 3 4 5 6 7 Current Projected @ 7

1981 5,012 8,269 10,907 11,805 13,539 16,181 18,009 18,009 18,009 1982 106 4,285 5,396 10,666 13,782 15,599 15,600 15,600 15,600 1983 3,410 8,992 13,873 16,141 18,735 22,214 22,863 22,863 22,863 1984 5,655 11,555 15,766 21,266 23,425 26,083 27,067 27,067 27,067 1985 1,092 9,565 15,836 22,169 25,955 26,180 27,361 26,180 27,361 1986 1,513 6,445 11,702 12,935 15,852 18,567 19,318 15,852 19,318 1987 557 4,020 10,946 12,314 15,413 17,986 18,698 12,314 18,698 1988 1,351 6,947 13,112 16,963 21,059 24,459 25,399 13,112 25,399 1989 3,133 5,395 8,894 10,918 13,070 14,857 15,351 5,395 15,351

Standard ResidualsAY 1 2 3 4 5 6 7

1981 1.163 -0.130 -0.278 -0.857 -0.174 0.690 1.2231982 -1.368 0.233 -1.056 1.045 0.641 0.163 -1.6321983 0.774 0.118 0.090 -0.267 -0.152 0.569 -0.0101984 1.007 -0.030 -0.375 0.759 -0.606 -0.044 0.4191985 -0.942 0.316 0.022 0.753 -0.009 -1.3781986 -0.094 0.179 0.447 -0.789 0.3001987 -1.028 -0.307 0.794 -0.6441988 -0.578 0.017 0.3551989 1.066 -0.396

Standard Residuals by Duration

-2.000

-1.500

-1.000

-0.500

0.000

0.500

1.000

1.500

0 1 2 3 4 5 6 7 8

Series1

Series2

Series3

Series4

Series5

Series6

Series7

Series8

Series9

GL Data: Summary

AY f(d) f(d) plus constant h(w)f(d) h(w)f(d)g(w+d) log f(d) (LN) log h(w)f(d) (LN)1981 18,009 18,009 18,009 18,009 18,009 18,0091982 15,600 15,600 15,600 15,600 15,600 15,6001983 22,863 22,863 22,863 22,863 22,863 22,8631984 27,067 27,067 27,067 27,067 27,067 27,0671985 27,266 27,076 27,385 27,189 27,365 27,3611986 18,156 19,195 18,281 17,985 18,682 19,318

1987 16,389 18,321 16,840 16,414 17,172 18,6981988 22,003 22,282 22,985 22,102 24,075 25,3991989 14,203 18,937 13,342 14,414 16,833 15,351

Log Normal Analogs