The Over-Dispersed Skellam (ODS) Reserving Model

Abstract

We introduce a new stochastic reserving model tailored to incurred claims triangles, particularly suited for lines of business with long settlement delays. In addition, we propose a parametric bootstrap procedure that delivers substantial computational efficiency gains compared to standard approaches.

Interest in uncertainty quantification has grown steadily in recent years, driven both by compliance requirements and by the need for effective risk management. This development has also encouraged a wider adoption of stochastic reserving methods to quantify the uncertainty of future claims in insurance portfolios.

Let (Xij)(i,j)∈{1,…,I}×{0,…,J} denote the incremental claim amounts and

$$D_I = {X_{ij} : 1 \le i \le I, \ 0 \le j \le J, \ i+j \le I}$$

denote the observed incremental claim amounts at the end of calendar period I. Note that the Xij’s can be e.g. claim payments or incurred claims.

It is of interest to find a modeling framework for the predictive distribution of the remaining incremental claim amounts Xij after calendar period I:

$$\sum_{i+j>I} X_{ij}.$$

Such a framework is often desired for the standard case where Xij ≥ 0 for all (i, j). Here, it can be suitable to use the over-dispersed Poisson (ODP) reserving model from England and Verrall (2002) and apply the non-parametric bootstrap approach implemented by e.g. Gesmann et al. (2025) in the ChainLadder::BootChainLadder-function.

In practice, however, the reserving actuary may often experience Xij < 0 for some lines of business. This can be the case for accident insurance when considering the incremental incurred claims triangle even though chain-ladder might provide a completely reasonable actuarial best estimate. This has been our motivation for introducing a new reserve uncertainty modeling framework consistent with the underlying triangle of such a best estimate.

The ODS Reserving Model

Model 1. Given θ ∈ R I+2J+2 ≥0 defined by

$$\theta := (\alpha_1, \dots, \alpha_I, \beta_0^+, \dots, \beta_J^+, \beta_0^-, \dots, \beta_J^-)$$

and φ ∈ (1, ∞), assume that (Xij)(i,j)∈{1,…,I}×{0,…,J} are mutually independent with

$$\frac{X_{ij}}{\varphi} \sim \text{Skellam}\left(\frac{\alpha_i \beta_j^+}{\varphi}, \frac{\alpha_i \beta_j^-}{\varphi}\right).$$

Remark 1. Defining β^j := β + ^j − β − j for j = 0, . . . , J, the ODS model satisfies

$$\mathbb{E}[X_{ij} \mid \theta] = \alpha_i , \beta_j,$$

$$\operatorname{Var}(X_{ij} \mid \theta, \varphi) = \varphi , \alpha_i \left(\beta_j + 2\beta_j^-\right) \geq \varphi , \alpha_i \beta_j$$ for $(i, j) \in {1, \dots, I} \times {0, \dots, J}.$

Remark 2. The ODP model appears as a special case of the ODS model obtained by setting β − ^j = 0 for j = 0, . . . , J. Equivalently, an ODS random variable can be constructed as the difference of two independent ODP random variables with the same dispersion parameter.

Remark 3. For simplicity in this report, the dispersion parameter φ is estimated using Pearson residuals applied to the absolute values of both the observed Xij and the corresponding predictions.

Results

We apply the parametric bootstrap implementation of the ODS model to the synthetic triangle in Table 3. The ODS parameters were calibrated to match the chain-ladder estimates for comparability.

For reference, we compare the numerical results to those obtained from the BootChainLadder function based on the ODP reserving model, a widely applied method in practice.

Table 1: Summary statistics of simulated reserves

The two approaches yield very similar reserve estimates and percentiles. Importantly, however, is that the ODS model remains consistent with the negative increments Xij for j ∈ {1, 2} observed in Table 3, while the reference ODP reserving model cannot accommodate such values.

Table 2: Benchmark results

It appears evident that one can improve the computational performance remarkably by applying this new parametric bootstrap method.

5 7 12 13 14 15 17 20 22 25 $1 \quad 8951595 \quad 5046153 \quad 3382754 \quad 3835268 \quad 4624117 \quad 5276622 \quad 5846706 \quad 6392039 \quad 6772110 \quad 6904717 \quad 7014634 \quad 7035127 \quad 7095887 \quad 7109155 \quad 7114284 \quad 7159873 \quad 7196603 \quad 7196603 \quad 7196632 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 \quad 7196638 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\ 8066113\ \ 8079153\ \ \ 8166280\ \ 8214387\ \ 8215755\ \ 8226544\ \ 8226544\ \ 8226544\ \ 8226523\ \ 8325023\ \ 8325023\ \ 8325023\ \ 8325023\ \ 8325023\ \ 8325023\ \ 8325023\ \ 8325023\ \ 8325023\ \ 8325023\ \ 8325023\ \ 8325023\ \ 8325023\ \ 8325023\ \ 8325023\ \ 8325023\ \ 8325023\ \ 8325023\ \ 8325023\ \ 8325023\ \ 8325023\ \ 8325023\ \ 8325023\ \ 8325023\ \ 8325023\ \ 8325023\ \ 8325023\ \ 8325023\ \ 8325023\ \ 8325023\ \ 8325023\ \ 8325023\ \ \ 8325023\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $ 3 8131 447 5326485 2936379 3169631 4778378 6147337 6527736 7116285 7286537 7387158 7528281 7554990 7573144 7573867 7758681 7758681 7748189 7777367 7777370 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 7777384 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7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 7908608\ 790860$ $9\ \ 10452507\ \ 6662352\ \ 4920695\ \ 5234901\ \ 6905260\ \ 7442374\ \ 8129971\ \ 8378650\ \ 8796485\ \ 8875578\ \ 8880601\ \ 10109086\ \ 9164924\ \ 9189591\ \ 9191593\ \ 9191850\ \ 9191850\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 9191928\ \ 91$ 11 9.43.833 5.831.717 2.949.786 3.228.792 4.053.898 4.637.755 5.053.268 5.415.717 5.609.892 5.777.706 5.822.275 5.966.045 6.063.864 6.067.119 6.077.880 6.077.880 6.102.345 6.102.345 6.102.345 6.102.345 12 9801326 6798285 4602901 5065046 6908109 7337725 8195468 8593298 8847721 9050069 9076027 9209927 9213217 9213223 9290449 9294870 9294870 9294870 9295557 $13 \quad 9102, 293 \quad 6054, 974 \quad 3657, 698 \quad 3, 932, 428 \quad 5, 366, 592 \quad 6, 135, 938 \quad 6, 910, 782 \quad 7, 347, 205 \quad 7, 504, 525 \quad 7, 666, 527 \quad 7, 708, 575 \quad 7, 778, 699 \quad 7, 783, 118 \quad 7, 788, 002 \quad 7, 791, 888 \quad 7, 831, 506 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7, 972, 262 \quad 7$ $14\ \ 10436802\ \ 7244662\ \ 4600151\ \ 4718759\ \ 5466844\ \ 6253152\ \ 6562116\ \ 7138782\ \ 7622062\ \ 7800825\ \ 8137385\ \ \ 8220837\ \ 8235188\ \ 8237108\ \ 8238047\ \ 8238713\ \ 8238713$ $15\ 12,093,876\ 9614626\ 7016,128\ 7234,831\ 7974,871\ 8,901,774\ 9,074,726\ 9,208,839\ 9,460,336\ 9,507,553\ 9,610,719\ 9,679,981\ 9,692,852\ 9,702,277\ 9,703,682\ 9,703,682$ 16 9676641 7461761 5535711 5906418 6785688 7771332 8510709 8916409 9205741 9226167 9261543 9312317 9318948 9319143 9319143 17 9899082 4929471 2183571 2609419 3865564 4639632 5220476 5555842 5891915 5966643 5986429 6072795 6075814 6095585 18 9628 067 6 581 142 4 019 544 4 299 758 6 108 151 7 045 541 7 499 017 8 223 831 8 650 831 8 982 650 9 059 501 9 070 280 9 213 123 19 9047 334 6186435 3 361 217 4 024 488 5 472 144 6 002 957 6 427 372 6 609 792 6 664 129 6 757 370 6 953 849 7 227 718 $20\ \ 11458177\ \ 7484858\ \ 5071566\ \ 5542022\ \ 6307883\ \ 7129400\ \ \ 7786078\ \ 8174210\ \ 8366392\ \ 8606511\ \ 8694726$ $21\ 10707397\ 6131950\ 3536434\ 3900676\ 4587413\ 5992276\ 7070149\ 7502303\ 7900511\ 8011567$ 22 10 369 476 6 674 262 4 623 655 4 925 803 6 512 534 7 860 643 8 712 454 9 262 707 9 489 201 23 11354803 8019112 6100185 6726389 7879139 8802965 9261664 9744075 $24\ 11,889,879\ 7,877,538\ 4,370,958\ 4,988,026\ 6,250,857\ 7,019,025\ 7,559,530$ 25 11665035 8538842 5007965 5397504 6593600 7868410 26 9937940 6129223 2548694 2750784 3616102 27 12 008 085 7 630 815 5 073 961 5 410 128 28 11 686 453 8 335 617 5 503 376 29 11 850 154 7 252 701 30 11845473

Table 3: Synthetic quarterly cumulative incurred triangle – Accident insurance

References

England, P. and R.J. Verrall (2002). ‘Stochastic claims reserving in general insurance’. British Actuarial Journal 8(3), 443–518.

Gesmann, M., Daniel Murphy, Yanwei (Wayne) Zhang, Alessandro Carrato, Mario Wuthrich, Fabio Concina and Eric Dal Moro (2025). ChainLadder: Statistical Methods and Models for Claims Reserving in General Insurance. R package version 0.2.20.