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Python | 3.14.7 on Windows-11-10.0.26200-SP0
aggregate | 1.7.1, source at git commit 0036744
greater_tables | 8.1.0Preface
This book reproduces published figures, tables and results using aggregate. Every number in it is computed by the library, never transcribed. A chapter is recomputed when its text changes, and every chapter is recomputed whenever the library or any other installed package changes, so all the numbers in a given copy come from the one environment described below. A match therefore demonstrates the correctness of the library version named here, down to the git commit of its source.
A computational library must be correct. aggregate has an extensive set of internal correctness checks embodied in its validation system: these check for a match between the first three theoretical and empirical moments. However, a published number computed by somebody independent using a different method is a good external check. We work through results from twenty eight papers. Most of them confirm the library. Several of them corrected it, or located a limit in it that was not previously known. Nine of them turn out to contain an error, an internal inconsistency or a misprint, which is not a criticism of the authors so much as evidence that the check runs in both directions.
Solving external problems tests aggregate in a second way: it shows the library is flexible enough to solve problems posed by other people! Obviously, it can solve problems I think are important. The library’s success on external problems bodes well when you come to use it.
The first chapter is the pith: five papers, each with the DecL program that reproduces the published exhibit and the answer set beside the paper’s own numbers. It highlights how a simple DecL statement can answer an externally set question.
WarningAI Generated Work!
The mechanical work in this book was conducted by Claude Code Opus 5 model with high-level human review. In a sense, the work checks itself, but the reader should be aware it was essentially auto generated by AI.
0.1 The structure of each chapter
Each chapter follows the same four-part structure: the context and the problem the paper is solving, the paper’s own solution and the gold standard it publishes, the aggregate reproduction, and comments. A simple reproduction can be very short. Something unusual, or unfamiliar to actuaries, needs more context.
When reviewing the code mentally discount the Python scaffolding and focus on the DecL nugget that does the work. Since the reproductions have to recreate the paper’s work, they include a lot of code that is conceptually irrelevant other than helping to display correctness.
Jin et al. (2016) turned out not to be reproducible because of aggregate limitations, as the chapter on the paper records. It models a common shock count mode for bivariate frequencies.
0.2 Three categories of paper
The papers fall into three categories.
A model paper poses an insurance structure with explicit parameters. Reproduction means building it in DecL and matching the published numbers, and the paper is the authority. Venter (1983), Mack and Fackler (2003), Mata and Verheyen (2005), Bodoff (2017), Homer and Clark (2003) and Bear and Nemlick (1990) are of this kind, and they make up the first part.
A method paper proposes a competing numerical technique for the aggregate distribution. Here aggregate is the exact ground truth against which the paper’s own approximation is measured, so the roles reverse: the paper is the thing being checked. Whether reproduction is possible at all depends on whether the paper published concrete input parameters and tabulated output. Several rare-event papers publish neither, reporting only sample counts and CPU seconds, and that is a large part of why the later parts read the way they do.
A theory paper is a third kind, and the two Denuit, Dhaene, et al. (2025) and Denuit, Ortega-Jiménez, et al. (2025) chapters are the instances. These prove results about the conditional expectation given the sum, \(\mathsf E[X_i \mid S = s]\), and illustrate them with figures rather than tables. There is no insurance contract in either, and no compound distribution: each agent carries one claim with certainty, so what is under test is the allocation kernel and the grid it runs on. Success here means reproducing a shape, and it is checkable only because both papers state something numerical about the shape, a turning point in one case and a monotonicity failure in the other. Reproducing a figure is weaker evidence than reproducing a table, so these chapters lean on constraints the model must satisfy whatever the picture looks like: that shares add up, and that identically distributed agents get identical shares.
0.3 Conclusions: correctness, flexibility, errors, and exactness
The reproductions exercise confirms aggregate‘s correctness and flexibility. Venter (1983)’s Exhibit 3 reproduces in all three columns, including the transformed gamma, which is scipy.stats.gengamma under a different name. Bruno et al. (2006)’s Tables 6, 7 and 8 come back exact at every printed decimal. Bodoff (2017)’s Exhibit 4 reproduces cell for cell. Homer and Clark (2003) reproduces every published cell of sections 2 and 4, and 136 of the 147 cells of section 3. Bear and Nemlick (1990)’s Table I reproduces in both of its reproducible columns, six adjustable features priced by six DecL clauses. Pittarello et al. (2024)’s accuracy table measures where the grid ends rather than how finely it is spaced, and on a long enough grid the error falls as \(h^2\), in the opposite direction to the one the paper reports. Gollini and Rougier (2016)’s 32,060 event hurricane loss table, where the authors’ Panjer recursion stopped at a $100 rounding after 4,651 seconds, is computed unrounded in under a second, and the moment bound they recommend turns out to be four to ten times the exact tail probability.
The exercise uncovered small errors in six papers. Mata and Verheyen (2005)’s Table 1 limited expected values are not the exact lognormal ones, running 0.04% high at $250,000 and drifting to 0.24% low at $5M, while the closed form matches to eight figures. Homer and Clark (2003)’s Table 3.3 does not sum to one, carrying 99.97% of the mass. Asmussen et al. (2016) has a row where its two published saddlepoint columns and its Monte Carlo column disagree, and aggregate, being neither a saddlepoint expansion nor a sampler, can determine which one to trust. Denuit, Dhaene, et al. (2025)’s Figure 1 cannot hold at its right hand edge: agent 3’s loss is capped at 30, so adding up forces agents 1 and 2 above 5 at a pool loss of 40, and their panels are drawn on an axis whose top tick is 4. Pittarello et al. (2024)’s Table 3 prints the same reference value in all four rows for four and five dimensions, a distribution function that would have to be flat from 10 to 10,000; Arbenz et al. (2011), where the values come from, has the right ones. Gollini and Rougier (2016) report Monte Carlo samples of \(10^5\) in the text, while their own design calculation, their summary and the script that drew their figures all use \(10^4\).
Finally, it reveals the FFT computational resolution floor. Ben Rached et al. (2024)’s Theorem 3 predicts that FFT convolution error is bounded absolutely rather than relatively, so below roughly \(10^{-13}\) the output is noise regardless of grid refinement. aggregate sits on that predicted floor to within a factor of a few. Wilson and Keich (2016) shows the same failure on adversarial inputs and, more usefully, shows that it is not always detectable: a wrong answer can come back positive and plausible. Wilson and Keich (2017) exhibits it on a benign input and demonstrates the repair. Together they turned a vague caveat into a measured property, and explain why the far tail now has documented limits rather than optimistic ones.
Computational accuracy is discussed further in two other recent blog posts. Dice of Dice: The FFT Meets Exact Arithmetic compares FFT computed probabilities with an exact arithmetic probability calculation, and Dice of Dice: Exact Finite FFTs discusses using number-theoretic transforms which give exact rational number answers with no floating point noise. The latter distinguishes between the Fourier method which is exact and different numerical implementations. That method now ships in the library as the nttagg kind, and the Wilson and Keich (2016) and Wilson and Keich (2017) chapters use it: their exact reference values, previously hand algebra in one case and trusted naive convolution in the other, are computed in-library as exact rational numbers, and the float pipeline’s error against them is itself computed exactly, per bucket. An optional compiled engine, the Rust add-in aggregate-ntt working over 64-bit primes, extends the reach of the exact method; this book runs the pure-Python reference path.
0.4 The chapters
Table 1 and Table 2 list the chapters in book order, split at the part boundary between the actuarial chapters and the numerical ones: the paper each chapter reproduces, where it was published and how often it has been cited, what the chapter targets, and how it came out.
| Part | Paper | Year | Journal | Citations | Target | Outcome |
|---|---|---|---|---|---|---|
| Actuarial models | Venter (1983) | 1983 | Proceedings of the Casualty Actuarial Society | 109 | Exhibit 3, three methods over 34 rows | Reproduced. The bucket size selects which of the two exact columns you land on |
| Mack and Fackler (2003) | 2003 | Blätter der DGVFM | 10 | Table 1, and the threshold above which Riebesell is exact | Reproduced. All 26 cells within 8e-4, and the threshold is \(5.186\,\mathsf E[X]\) against the paper's “about 5” | |
| Mata and Verheyen (2005) | 2005 | CAS Forum | 10 | Tables 1 to 14, trend and exposure adjustment by layer | Reproduced, and a discrepancy found in the paper's Table 1 | |
| Bodoff (2017) | 2017 | Variance | 0 | Exhibit 4, excess of policy limits | Reproduced. Two unstated model choices inferred and confirmed by fit | |
| Homer and Clark (2003) | 2003 | Proceedings of the Casualty Actuarial Society | 14 | All three worked examples of the bivariate FFT | Reproduced. Every construction is a shipped DecL form | |
| Bear and Nemlick (1990) | 1990 | Proceedings of the Casualty Actuarial Society | 10 | Table I, six treaties with adjustable features | Reproduced. The lognormal column exactly, the collective risk column to the printed digit in five of six | |
| Distortion pricing | Wang (2000) | 2000 | Journal of Risk and Insurance | 676 | Table 1, risk load by layer under two distortions | Reproduced to 0.31%, and the basic limit expected loss is misprinted |
| Wang (2002) | 2002 | ASTIN Bulletin | 268 | Table 2, weather derivative calls off 22 observations | Reproduced. Prices to under half a cent | |
| Risk sharing | Denuit (2019) | 2019 | ASTIN Bulletin | 63 | Figure 1 and the shares at \(s = 1\), four participant pool | Reproduced. The horizontal line resolves a table that contradicts its own text |
| Denuit et al. (2022) | 2022 | ASTIN Bulletin | 18 | Example 4.5, Figure 6, mortality credits at three pool sizes | Reproduced. Shares add up exactly at every pool size | |
| Denuit, Dhaene, et al. (2025) | 2025 | Insurance: Mathematics and Economics | 5 | Figure 1, the comonotonic repair of a conditional mean allocation | Reproduced, and the published figure cannot hold at its right hand edge | |
| Denuit, Ortega-Jiménez, et al. (2025) | 2025 | ASTIN Bulletin | 2 | Figures 2, 5 and 6, conditional expectation given the sum | Reproduced. The turning point lands at 64.46, which is what settles the index convention |
| Part | Paper | Year | Journal | Citations | Target | Outcome |
|---|---|---|---|---|---|---|
| Exact computation | Gerber (1982) | 1982 | Insurance: Mathematics and Economics | 69 | Tables 1 to 3, the exact stop loss premium columns | Reproduced. All 28 values inside the printed precision |
| Robertson (1992) | 1992 | Proceedings of the Casualty Actuarial Society | 25 | Table 4, the piecewise uniform algorithm | Partial. Exact for the paper, \(O(h)\) for a lattice, so the printed precision is out of reach | |
| Embrechts et al. (1993) | 1993 | Statistica Neerlandica | 42 | Figure 1, discretization error and Richardson extrapolation | Reproduced. The \(O(h^2)\) rate measured at 3.998 | |
| Grübel and Hermesmeier (1999) | 1999 | ASTIN Bulletin | 50 | Tables 1 and 2, the L1 sums, the Theorem 2 bound | Reproduced, including the bound at 0.4641 | |
| Pittarello et al. (2024) | 2024 | Annals of Actuarial Science | 1 | Section 3.5 examples, Tables 1 to 3 | Reproduced. Table 1 is a truncation study, Table 2 re-timed on one machine, Table 3's reference values misprinted | |
| Embrechts and Frei (2009) | 2009 | Mathematical Methods of Operations Research | 94 | Table 1, three arithmetization designs | Reproduced exactly, all twelve values | |
| Schaller and Temnov (2008) | 2008 | Computational Methods in Applied Mathematics | 23 | Table 4.3, operational risk quantiles | Reproduced to 2.2e-4, all twelve distinct values | |
| Bruno et al. (2006) | 2006 | Applied Mathematics and Computation | 8 | Tables 6, 7 and 8 | Reproduced exact at six decimals. 92,832 convolutions in seconds against the paper's 1,950 | |
| Gollini and Rougier (2016) | 2016 | The Journal of Operational Risk | 1 | Figures 1 to 3 and 5 to 7, bounds on a 32,060 event hurricane ELT | Reproduced. The exact answer the authors ruled out takes under a second, and the moment bound is 4 to 10 times high | |
| Wilson and Keich (2016) | 2016 | Computational Statistics and Data Analysis | 8 | Table 2, the FFT-C columns | Reproduced. The finding is detectability, not error size. The Section 1 example is now computed exactly in-library | |
| Wilson and Keich (2017) | 2017 | Journal of Computational and Graphical Statistics | 5 | The worst case, and its repair | Reproduced to the digit and the repair demonstrated, both now certified against exact arithmetic | |
| Rare events | Asmussen et al. (2016) | 2016 | Scandinavian Journal of Statistics | 36 | Twelve tables, CDF and PDF | Reproduced from five build calls, and one bad row adjudicated |
| Ben Amar et al. (2023) | 2023 | Statistics and Computing | 7 | The seven probabilities the paper actually quotes | Reproduced. The whole left tail curve costs what one estimate costs them | |
| Ben Rached et al. (2024) | 2024 | Statistics and Computing | 0 | Table 6, and Theorem 3's floor | Reproduced, and the floor confirmed on the library itself | |
| Liu et al. (2026) | 2026 | Operations Research Letters | 0 | Tables 1, 2 and C.1 | Three of four reference points reproduced. The fourth is below the floor | |
| Commentary | The rare-event group as a whole | Where these methods earn their keep, and where they do not | ||||
| Did not reproduce | Jin et al. (2016) | 2016 | Scandinavian Actuarial Journal | 14 | Nothing | Not a reproduction. The evidence, and the feature that would change it |
Counts are from OpenAlex, retrieved 2026-08-08 (for Pittarello et al. (2024), 2026-10-11, and for Gollini and Rougier (2016), 2026-10-21), except for the Casualty Actuarial Society (CAS) literature. The CAS papers are not indexed and their counts come from a manual Google Scholar lookup. The counts are a snapshot and do not auto-update on rebuild. They should be read relative to the year of publication, and they say how widely each result has been taken up, which is a different question from whether it reproduces.
The two most cited papers here by an order of magnitude, Wang (2000) and Wang (2002), are the two whose reproductions are the shortest and least eventful. The papers that changed the library, Ben Rached et al. (2024), Wilson and Keich (2016) and Wilson and Keich (2017), have 0, 8 and 5 citations between them. Thus, being useful to an implementation and being cited are close to independent!
The final chapter is reference rather than reproduction: it records where each published severity curve shipped in library.agg came from, what the source paper uses it for, and every transcription decision behind the DecL.
0.5 Reading and rebuilding
Each chapter stands alone and they can read independently. Two pairs reward being read together: Homer and Clark (2003) and Jin et al. (2016) sit on opposite sides of the same feature boundary, dependence through a joint severity under a shared count against dependence through a common shock on the counts, and Wilson and Keich (2016) and Wilson and Keich (2017) are companion papers on the same failure, the first on adversarial inputs and the second on ordinary ones.
Rendering requires a Python environment with aggregate and its documentation extras installed, since every chapter executes. The two working documents in this folder, README.md and plan-examples.md, are notes rather than chapters and are deliberately outside the book: the first is a folder index and the second is the candidate assessment, which carries the tier ratings for the papers that were considered and rejected.
Arbenz, P., Embrechts, P., & Puccetti, G. (2011). The AEP algorithm for the fast computation of the distribution of the sum of dependent random variables. Bernoulli, 17(2), 562–591. https://doi.org/10.3150/10-BEJ284
Asmussen, S., Jensen, J. L., & Rojas-Nandayapa, L. (2016). Exponential Family Techniques for the Lognormal Left Tail. Scandinavian Journal of Statistics, 43(3), 774–787. https://doi.org/10.1111/sjos.12203
Bear, R. A., & Nemlick, K. J. (1990). Pricing the impact of adjustable features and loss sharing provisions of reinsurance treaties. Proceedings of the Casualty Actuarial Society, 77(147), 86–87. https://doi.org/10.1016/0167-6687(93)91078-9
Ben Rached, N., Hoel, H., & Meo, J. V. (2024). A fast and accurate numerical method for the left tail of sums of independent random variables. Statistics and Computing. https://doi.org/10.1007/s11222-024-10514-5
Bodoff, N. M. (2017). An Actuarial Model of Excess of Policy Limits Losses. Variance, 11. https://doi.org/10.66573/001c.116059
Bruno, M. G., Camerini, E., Manna, A., & Tomassetti, A. (2006). A new method for evaluating the distribution of aggregate claims. Applied Mathematics and Computation. https://doi.org/10.1016/j.amc.2005.09.038
Denuit, M. M., Dhaene, J., Ghossoub, M., & Robert, C. Y. (2025). Comonotonicity and Pareto optimality, with application to collaborative insurance. Insurance: Mathematics and Economics, 120, 1–16. https://doi.org/10.1016/j.insmatheco.2024.11.001
Denuit, M. M., Ortega-Jiménez, P., & Robert, C.-Y. (2025). Conditional expectations given the sum of independent random variables with regularly varying densities. ASTIN Bulletin. https://doi.org/10.1017/asb.2025.11
Gollini, I., & Rougier, J. (2016). Rapidly bounding the exceedance probabilities of high aggregate losses. The Journal of Operational Risk, 1–33. https://doi.org/10.21314/JOP.2016.179
Homer, D. L., & Clark, D. R. (2003). Insurance Applications of Bivariate Distributions. Proceedings of the Casualty Actuarial Society, 90(iid), 274–307. http://www.casact.org/pubs/proceed/proceed03/03274.pdf
Jin, T., Provost, S. B., & Ren, J. (2016). Moment-based density approximations for aggregate losses. Scandinavian Actuarial Journal, 2016(3), 216–245. https://doi.org/10.1080/03461238.2014.921640
Mack, T., & Fackler, M. (2003). Exposure-rating in liability reinsurance. Blätter der DGVFM, 26(2), 229–238.
Mata, A. J., & Verheyen, M. A. (2005). An Improved Method for Experience Rating Excess of Loss Treaties using Exposure Rating Techniques. CAS Forum.
Pittarello, G., Luini, E., & Marchione, M. M. (2024). GEMAct: a Python package for non-life (re) insurance modeling. Annals of Actuarial Science, 1–37. https://doi.org/10.1017/S1748499524000022
Venter, G. G. (1983). Transformed beta and gamma distributions and aggregate losses. Proceedings of the Casualty Actuarial Society, 70(133), 289–308.
Wang, S. S. (2000). A class of distorion operators for pricing financial and insurance risks. Journal of Risk and Insurance, 67(1), 15–36.
Wang, S. S. (2002). A Universal Framework for Pricing Financial and Insurance Risks. ASTIN Bulletin, 32(2), 213–234. https://doi.org/10.2143/AST.32.2.1027
Wilson, H., & Keich, U. (2016). Accurate pairwise convolutions of non-negative vectors via FFT. Computational Statistics and Data Analysis, 101, 300–315. https://doi.org/10.1016/j.csda.2016.03.010
Wilson, H., & Keich, U. (2017). Accurate small tail probabilities of sums of iid lattice-valued random variables via FFT. Journal of Computational and Graphical Statistics, 26(1), 223–229.