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Dal Moro, E. 2021. “The Skewness of Bornhuetter-Ferguson.” Variance 14 (2).

Abstract

The Bornhuetter-Ferguson method is among the more popular methods of projecting non-life paid or incurred triangles. For this method, Thomas Mack developed a stochastic model allowing the estimation of the prediction error resulting from such projections. Mack’s stochastic model involves a parametrization of the Bornhuetter-Ferguson method based on incremental triangles of incurred or paid. Hence, that parametrized method differs from how Bornhuetter-Ferguson is usually applied on cumulative triangles of incurred or paid. Based on that proposed stochastic model, this paper provides a first approach for the estimation of the third moment, i.e., the skewness, of the resulting reserving distribution. An estimate of the third moment is useful in the context of IFRS 17, which directs that the quantile corresponding to the addition of a risk margin on top of the best estimate must be disclosed. To illustrate the proposed method, a few numerical examples are provided.

Accepted: February 15, 2020 EDT

Appendices

Appendix A. Proof of Equation (4)

\(\begin{align}\text{SKEW}\left( {\widehat{R}}_{i}^{\text{BF}} \right) &= E\left\lbrack \left( {\widehat{R}}_{i}^{\text{BF}} - R_{i} \right)^{3} \right\rbrack \\& = E\left( {\widehat{R}}_{i}^{BF\ \ 3} \right) - 3E\left( {\widehat{R}}_{i}^{BF\ \ 2}R_{i} \right) \\&\ \ \ \ + 3E\left( {\widehat{R}}_{i}^{\text{BF}}\ R_{i}^{2} \right) - E\left( R_{i}^{3} \right)\end{align}.\)

As \({\widehat{R}}_{i}^{\text{BF}}\) and \(R_{i}\) are independent, we have

\(\begin{align}\text{SKEW}\left( {\widehat{R}}_{i}^{\text{BF}} \right) &= E\left( {\widehat{R}}_{i}^{BF\ \ 3} \right) \\&\ \ \ \ - 3E\left( {\widehat{R}}_{i}^{BF\ \ 2} \right)E\left( R_{i} \right) \\&\ \ \ \ + 3E\left( {\widehat{R}}_{i}^{\text{BF}} \right)E\left( \ R_{i}^{2} \right) - E\left( R_{i}^{3} \right).\end{align}\)

As we have defined

\(\begin{align}K\left( R_{i} \right) &= E\left\lbrack \left( R_{i} - E\left( R_{i} \right) \right)^{3} \right\rbrack \\&= E\left( R_{i}^{3} \right) - 3E\left( R_{i}^{2} \right)E\left( R_{i} \right) + 2E\left( R_{i} \right)^{3},\end{align}\)

we have

\(\begin{align}\text{SKEW}\left( {\widehat{R}}_{i}^{\text{BF}} \right) &= K\left( {\widehat{R}}_{i}^{\text{BF}} \right) - K\left( R_{i} \right) +3E ( \widehat{R}_{i}^{BF\text{ }2})E(R_i^{BF})\\&\ \ \ \ -2E(R_i)^3 -3E(\widehat{R}_i^2)E(R_i)+2E(R_i)^3 \\&\ \ \ \ -3E\left( {\widehat{R}}_{i}^{BF\text{ }2} \right)E\left( R_{i} \right) + 3E\left( {\widehat{R}}_{i}^{\text{BF}} \right)E\left( \ R_{i}^{2} \right).\end{align}\)

After rearranging, we find

\[\begin{align}\text{SKEW}\left( {\widehat{R}}_{i}^{\text{BF}} \right) =\ & K\left( {\widehat{R}}_{i}^{\text{BF}} \right) - K\left( R_{i} \right) + E\left( {\widehat{R}}_{i}^{\text{BF}} - R_{i} \right)\\ &\times \Bigl\lbrack 3E\left( {\widehat{R}}_{i}^{BF\ \ 2} \right) + 3E\left( \ R_{i}^{2} \right) - 2{E\left( {\widehat{R}}_{i}^{{BF}} \right)}^{2}\\ &- 2{E\left( R_{i} \right)}^{2} - 2E\left( {\widehat{R}}_{i}^{\text{BF}} \right)E\left( R_{i} \right) \Bigr\rbrack.\end{align}\]

As \(E\left( {\widehat{R}}_{i}^{\text{BF}} - R_{i} \right) = 0\), we finally have

\(\text{SKEW}\left( {\widehat{R}}_{i}^{\text{BF}} \right) = K\left( {\widehat{R}}_{i}^{\text{BF}} \right) - K\left( R_{i} \right).\)

Appendix B. Estimators of \(E\left( {\widehat{U}}_{i} \right)\), \(Var\left({\widehat{U}}_{i} \right)\), \(E\left(1 -\widehat{z}_{n + 1 - i} \right)\) and \(Var\left( {\widehat{z}}_{n + 1 - i} \right)\)

\(E\left( {\widehat{U}}_{i} \right)\) is estimated by \(\nu_{\text{i}}{\widehat{q}}_{i},\) where \(\nu_{i}\) is the premium volume of accident year i and \({\widehat{q}}_{i}\) is the ultimate claims ratio.

As mentioned in Mack (2008), \(\text{Var}\left( {\widehat{U}}_{i} \right)\) is best obtained from the pricing distribution (when available).

As for \(\text{E}\left( 1 - {\widehat{z}}_{n + 1 - i} \right)\), we use: \({\widehat{z}}_{k} = {\widehat{y}}_{1} + ... + {\widehat{y}}_{k}\).

Finally, \(\text{Var}\left( {\widehat{z}}_{k} \right) \approx min\left\lbrack \text{Var}\left( {\widehat{y}}_{1} \right) + ... + Var\left( {\widehat{y}}_{k} \right);Var\left( {\widehat{y}}_{k + 1} \right) + ... + Var\left( {\widehat{y}}_{n + 1} \right) \right\rbrack,\) where

\(\text{Var}\left( {\widehat{y}}_{k} \right) = \frac{{\widehat{s}}_{k}^{2}}{\sum_{i = 1}^{n + 1 - k}{\widehat{U}}_{i}}\).

Appendix C.

 
Triangle 1 Development Year
Accident Year Premium Initial LR Ui 1 2 3 4 5 6 7 8 9 10 11 12 13
2005 110 940 316 69,6% 77 176 365 4 626 711 7 833 956 9 237 575 4 075 137 10 828 853 2 000 000 1 916 455 1 000 000 2 136 161 873 469 -696 077 1 970 249 2 060 435
2006 136 881 755 54,1% 74 081 078 2 497 127 11 277 229 7 424 316 5 612 201 3 410 710 930 785 952 588 3 311 966 2 527 888 2 177 772 655 961 2 129 708
2007 148 066 614 74,1% 109 785 557 6 733 949 16 001 488 8 581 545 16 094 884 4 740 896 2 946 401 5 320 553 8 191 596 858 217 1 921 349 2 674 149
2008 142 419 083 64,8% 92 216 629 5 495 872 10 634 243 9 357 874 3 687 937 5 459 783 7 244 685 3 514 985 1 278 885 1 130 235 2 201 251
2009 141 000 201 85,5% 120 589 443 5 513 563 11 892 818 13 879 064 5 892 620 6 906 455 19 110 413 4 127 163 4 465 916 4 691 799
2010 148 748 619 79,4% 118 144 218 4 642 178 9 611 735 16 674 760 7 170 953 13 301 042 7 576 809 1 561 553 5 247 638
2011 181 023 013 76,1% 137 843 628 5 088 492 12 712 715 12 492 986 16 125 708 17 424 813 9 530 326 6 511 744
2012 195 545 471 95,0% 185 764 666 5 911 892 14 485 648 15 199 655 20 501 488 15 352 506 10 000 000
2013 212 536 401 73,2% 155 613 255 7 261 343 13 285 662 16 307 615 12 557 638 11 727 223
2014 210 377 519 83,5% 175 714 686 5 210 360 23 258 993 10 144 201 23 137 200
2015 219 950 578 79,3% 174 402 671 6 709 466 14 819 711 20 470 286
2016 230 357 704 87,3% 201 091 730 13 556 673 13 519 627
2017 238 706 732 84,9% 202 706 418 8 255 514
 
Triangle 2 Development Year
Accident Year Premium Initial LR Ui 1 2 3 4 5 6 7 8 9 10 11 12 13
2005 227 650 968 57,2% 130 246 487 89 281 5 826 729 10 642 655 13 443 290 12 861 714 11 016 529 5 941 853 15 777 922 -2 016 362 3 897 878 2 017 884 3 991 344 4 599 524
2006 248 800 723 73,3% 182 459 114 253 461 10 141 425 15 502 109 14 653 424 19 038 614 13 526 518 11 920 567 11 637 234 3 152 799 2 832 256 1 596 518 1 957 742
2007 241 098 844 71,6% 172 508 136 285 602 11 280 177 13 467 685 17 570 170 16 533 966 11 438 498 10 882 011 11 879 412 12 716 240 3 081 753 2 959 545
2008 183 864 522 81,2% 149 329 997  056 11 224 085 14 034 547 17 843 506 16 022 003 11 874 150 11 319 798 4 775 666 1 954 680 1 837 360
2009 188 240 854 80,6% 151 761 168 964 209  600 107 16 442 016 14 689 707 15 510 264 11 733 827 17 713 234 3 013 169 3 668 265
2010 196  095 84,3% 165 366 633 671 876 10 913 835 16 157 793 14 372 512 17 147 487 14 286 587 11 287 031 9 345 582
2011 194 276 106 87,3% 169  372 335 616 8 956 636 12 122 076  394 437 17 350 796 16 863 903 10 246 783
2012 245 584 590 88,3% 216 964 899 762 265 11 836 973 17 712 645 23 477 706 18 519 857 18 934 294
2013 174 627 992 63,6% 111 092 658 859 883 9 284 060 13 453 513 11 872 282 11 664 474
2014 201 255 277 60,9% 122 553 304 1 002 904 8 720 759 13 041 075 12 514 625
2015 273 145 637 63,8% 174 317 096 285 729 7 943 122 15 639 699
2016 406 620 096 63,5% 258 284 240 536 767 9 569 288
2017 413 560 514 62,7% 259 455 540 513 235
 
Triangle 3 Development Year
Accident Year Premium Initial LR Ui 1 2 3 4 5 6 7 8 9 10 11 12 13
2005 306 442 90,0% 275 798 13 487 50 621 30 204 33 570 24 290 12 173 5 826 10 832 21 869 10 000 1 048 4 023 4 801
2006 360 077 90,0% 324 069 20 508 56 047 17 758 33 570 24 290 12 173 5 826 10 832 21 869 10 000 17 621 6 244
2007 526 754 90,0% 474 078 14 156 50 362 53 811 43 285 28 710 31 867 31 430 18 001 12 657 6 330 5 052
2008 465 852 90,0% 419 267 30 721 41 275 19 637 30 725 28 823 21 106 4 683 8 735 5 000 2 938
2009 429 724 90,0% 386 751 7 305 30 942 29 580 25 095 17 541 9 402 8 953 4 637 7 104
2010 381 545 90,0% 343 390 13 601 40 569 22 200 20 380 21 140 10 123 -572 5 787
2011 367 539 90,0% 330 785 9 657 38 809 47 837 31 987 22 077 6 375 7 262
2012 347 770 90,0% 312 993 7 085 41 073 55 745 46 024 17 225 19 820
2013 333 314 90,0% 299 982 18 237 38 891 51 624 39 890 29 183
2014 382 678 90,0% 344 410 15 008 41 664 40 047 25 834
2015 375 905 90,0% 338 314 21 592 40 788 26 595
2016 359 604 90,0% 323 644 17 061 33 068
2017 388 474 90,0% 349 627 17 291
 
Correlation matrix for Skewness - Triangle 1
ρ ij 1 2 3 4 5 6 7 8 9 10 11 12 13
1 100% 94% 89% 87% 84% 80% 74% 69% 60% 51% 40% 29% 15%
2 94% 100% 94% 92% 89% 84% 78% 73% 64% 54% 43% 31% 16%
3 89% 94% 100% 98% 94% 89% 83% 77% 68% 57% 45% 33% 17%
4 87% 92% 98% 100% 96% 91% 84% 79% 69% 58% 46% 33% 18%
5 84% 89% 94% 96% 100% 95% 88% 82% 72% 60% 48% 35% 18%
6 80% 84% 89% 91% 95% 100% 92% 86% 76% 63% 51% 37% 19%
7 74% 78% 83% 84% 88% 92% 100% 94% 82% 69% 55% 40% 21%
8 69% 73% 77% 79% 82% 86% 94% 100% 88% 73% 59% 42% 22%
9 60% 64% 68% 69% 72% 76% 82% 88% 100% 84% 67% 48% 26%
10 51% 54% 57% 58% 60% 63% 69% 73% 84% 100% 80% 58% 31%
11 40% 43% 45% 46% 48% 51% 55% 59% 67% 80% 100% 72% 38%
12 29% 31% 33% 33% 35% 37% 40% 42% 48% 58% 72% 100% 53%
13 15% 16% 17% 18% 18% 19% 21% 22% 26% 31% 38% 53% 100%
 
Correlation matrix for Skewness - Triangle 2
ρ ij 1 2 3 4 5 6 7 8 9 10 11 12 13
1 100% 92% 89% 86% 83% 79% 70% 61% 51% 40% 29% 18% 4%
2 92% 100% 96% 93% 90% 85% 76% 66% 55% 44% 32% 19% 4%
3 89% 96% 100% 97% 94% 89% 79% 68% 58% 45% 33% 20% 4%
4 86% 93% 97% 100% 96% 91% 81% 70% 59% 47% 34% 21% 5%
5 83% 90% 94% 96% 100% 95% 84% 73% 62% 49% 35% 21% 5%
6 79% 85% 89% 91% 95% 100% 89% 77% 65% 51% 37% 23% 5%
7 70% 76% 79% 81% 84% 89% 100% 87% 73% 58% 42% 25% 6%
8 61% 66% 68% 70% 73% 77% 87% 100% 84% 66% 48% 29% 6%
9 51% 55% 58% 59% 62% 65% 73% 84% 100% 79% 57% 35% 8%
10 40% 44% 45% 47% 49% 51% 58% 66% 79% 100% 73% 44% 10%
11 29% 32% 33% 34% 35% 37% 42% 48% 57% 73% 100% 60% 13%
12 18% 19% 20% 21% 21% 23% 25% 29% 35% 44% 60% 100% 22%
13 4% 4% 4% 5% 5% 5% 6% 6% 8% 10% 13% 22% 100%
 
Correlation matrix for Skewness - Triangle 3
ρ ij 1 2 3 4 5 6 7 8 9 10 11 12 13
1 100% 96% 93% 89% 85% 79% 75% 72% 66% 57% 46% 34% 16%
2 96% 100% 96% 92% 89% 82% 78% 74% 68% 59% 48% 35% 17%
3 93% 96% 100% 96% 92% 85% 81% 77% 71% 61% 49% 36% 18%
4 89% 92% 96% 100% 96% 89% 85% 81% 74% 64% 52% 38% 19%
5 85% 89% 92% 96% 100% 93% 88% 84% 77% 67% 54% 40% 19%
6 79% 82% 85% 89% 93% 100% 95% 90% 83% 72% 58% 43% 21%
7 75% 78% 81% 85% 88% 95% 100% 95% 87% 76% 61% 45% 22%
8 72% 74% 77% 81% 84% 90% 95% 100% 92% 80% 64% 47% 23%
9 66% 68% 71% 74% 77% 83% 87% 92% 100% 87% 70% 51% 25%
10 57% 59% 61% 64% 67% 72% 76% 80% 87% 100% 80% 59% 29%
11 46% 48% 49% 52% 54% 58% 61% 64% 70% 80% 100% 74% 36%
12 34% 35% 36% 38% 40% 43% 45% 47% 51% 59% 74% 100% 49%
13 16% 17% 18% 19% 19% 21% 22% 23% 25% 29% 36% 49% 100%