{"type":"script","doc_desc":{"producers":[{"name":"Reproducibility WBG","abbr":"DECDI","affiliation":"World Bank - Development Impact Department","role":"Verification and preparation of metadata"}],"prod_date":"2026-10-20","version":"1"},"project_desc":{"authoring_entity":[{"name":"Michal Certik","affiliation":"World Bank","email":"mcertik@worldbank.org"}],"title_statement":{"title":"Reproducibility package for Exact and Approximate Component VaR and Expected Shortfall for Quadratic Normal Portfolios: Fourier, Saddlepoint, Cornish\u2013Fisher, and Frozen\u2013Multiplier Allocations, with Marginal Higher\u2013Moment Attribution and a Convexity\u2013Gradient Correction","idno":"RR_WLD_2026_782"},"data_statement":"The package does not rely on external data. All inputs are defined in the code using parameters and formulas printed in the paper (Section 10.2, Examples 1 and 2, Section 7.4 and Appendix H).","software":[{"name":"Python","version":"3.12"}],"scripts":[{"title":"Reproducibility package for Exact and Approximate Component VaR and Expected Shortfall for Quadratic Normal Portfolios: Fourier, Saddlepoint, Cornish\u2013Fisher, and Frozen\u2013Multiplier Allocations, with Marginal Higher\u2013Moment Attribution and a Convexity\u2013Gradient Correction","date":"2026-10","notes":"Computational reproducibility verified by Development Impact (DECDI) Analytics team, World Bank.","instructions":"See README in reproducibility package.","file_name":"RR_WLD_2026_782","zip_package":"RR_WLD_2026_782.zip"}],"repository_uri":[{"name":"Reproducible Research Repository (World Bank)","uri":"https:\/\/reproducibility.worldbank.org"}],"production_date":"2026-10-20","abstract":"We study analytic Value\u2013at\u2013Risk (VaR) and its decomposition for portfolios whose profit and loss is a quadratic (delta\u2013gamma) function of jointly Gaussian risk\u2013factor changes,\u0394P = \u03b4\u22a4\u0394x + \u0394x\u22a4\u0393\u0394x with \u0394x \u223c N(0,\u03a3). Any risk functional positively homogeneous of degree one in the sensitivity pair (\u03b4,\u0393) is Euler\u2013additive across the deltas and the ordered gamma pairs, cross\u2013gammas included, and we compare four allocations on this common footing. Within the quadratic\u2013Gaussian model the exact VaR allocation consists of conditional moments of the factors given the loss, and the exact Expected Shortfall allocation of tail conditional moments, both computed by one\u2013dimensional Fourier inversion. The saddlepoint allocation is exactly additive because the Lugannani\u2013Rice quantile is scale\u2013equivariant. The fourth\u2013order Cornish\u2013Fisher allocation reduces to trace and matrix expressions evaluable in O(n3) arithmetic, and near a linear book its quantile agrees with the exact quantile to second order in the convexity, which requires the squared\u2013skewness term. Frozen\u2013multiplier schemes, which scale variance shares by one portfolio multiplier, are additive but omit the marginal skewness and kurtosis terms; these form a zero\u2013sum redistribution. Using the trace\u2013square gradient tr(\u0393\u03a3)\u03a3 in place of \u03a3\u0393\u03a3 breaks additivity by a closed\u2013form amount, diagonal gamma matrices included. Finally, a moment constraint on the exact marginals detects allocations that add up correctly yet give hedge signals of the wrong sign.","geographic_units":[{"name":"World","code":"WLD"}],"keywords":[{"name":"Value-At-Risk"},{"name":"Expected Shortfall"},{"name":"Risk Contributions"},{"name":"Euler Allocation"},{"name":"Delta-Gamma Approximation"},{"name":"Cornish-Fisher Expansion"},{"name":"Saddlepoint Approximation"},{"name":"Quadratic Forms In Normal Variables"}],"topics":[{"id":"C46","uri":"https:\/\/www.aeaweb.org\/econlit\/jelCodes.php?view=jel","vocabulary":"Journal of Economic Literature (JEL)","name":"Specific Distributions \u2022 Specific Statistics","parent_id":"C4"},{"id":" C58","uri":"https:\/\/www.aeaweb.org\/econlit\/jelCodes.php?view=jel","vocabulary":"Journal of Economic Literature (JEL)","name":"Financial Econometrics","parent_id":"C5"},{"id":" C63","uri":"https:\/\/www.aeaweb.org\/econlit\/jelCodes.php?view=jel","vocabulary":"Journal of Economic Literature (JEL)","name":"Computational Techniques \u2022 Simulation Modeling","parent_id":"C6"},{"id":" G17","uri":"https:\/\/www.aeaweb.org\/econlit\/jelCodes.php?view=jel","vocabulary":"Journal of Economic Literature (JEL)","name":"Financial Forecasting and Simulation","parent_id":"G1"},{"id":" G32","uri":"https:\/\/www.aeaweb.org\/econlit\/jelCodes.php?view=jel","vocabulary":"Journal of Economic Literature (JEL)","name":"Financing Policy \u2022 Financial Risk and Risk Management \u2022 Capital and Ownership Structure \u2022 Value of Firms \u2022 Goodwill","parent_id":"G3"}],"output":[{"type":"Working Paper","description":"Policy Research Working Paper (PRWP)","title":"Exact and Approximate Component VaR and Expected Shortfall for Quadratic Normal Portfolios: Fourier, Saddlepoint, Cornish\u2013Fisher, and Frozen\u2013Multiplier Allocations, with Marginal Higher\u2013Moment Attribution and a Convexity\u2013Gradient Correction"}],"language":[{"name":"English","code":"EN"}],"technology_requirements":"Run time: ~3 minutes. ","disclaimer":"The materials in the reproducibility packages are distributed as they were prepared by the staff of the International Bank for Reconstruction and Development\/The World Bank. The findings, interpretations, and conclusions expressed in this event do not necessarily reflect the views of the World Bank, the Executive Directors of the World Bank, or the governments they represent. The World Bank does not guarantee the accuracy of the materials included in the reproducibility package.","license":[{"name":"MIT License","uri":"https:\/\/opensource.org\/license\/mit"},{"name":"World Bank IGO Rider","uri":"https:\/\/github.com\/worldbank\/metadata-editor\/blob\/main\/WB-IGO-RIDER.md"}],"contacts":[{"name":"Michal Certik","affiliation":"The World Bank Group","email":"mcertik@worldbank.org"},{"name":"Reproducibility WBG","affiliation":"World Bank","email":"reproducibility@worldbank.org"}],"reproduction_instructions":"To reproduce the findings in the paper, restore the environment in `requirements.txt` and execute the python script \"main\".","technology_environment":"Paper exhibits were reproduced on a computer with the following specifications:\n\u2022 OS: Windows 11 Enterprise\n\u2022 Processor: Intel(R) Xeon(R) Gold 5218 CPU @ 2.30GHz (2.30 GHz) (2 processors)\n\u2022 Memory available: 16.0 GB"},"datacite":{"creators":[{"givenName":"Michal","familyName":"Certik","nameType":"Personal","affiliation":[{"name":"World Bank","affiliationIdentifier":"https:\/\/ror.org\/00ae7jd04","affiliationIdentifierScheme":"ROR","schemeUri":"https:\/\/ror.org"}]}],"titles":[{"lang":"en","title":"Reproducibility package for Exact and Approximate Component VaR and Expected Shortfall for Quadratic Normal Portfolios: Fourier, Saddlepoint, Cornish\u2013Fisher, and Frozen\u2013Multiplier Allocations, with Marginal Higher\u2013Moment Attribution and a Convexity\u2013Gradient Correction"},{"title":"RR_WLD_2026_782","titleType":"Other"}],"publisher":"World Bank","publicationYear":"2026","types":{"resourceType":"Reproducibility package","resourceTypeGeneral":"Other"},"url":"https:\/\/reproducibility.worldbank.org\/index.php\/catalog\/study\/RR_WLD_2026_782","language":"en"},"tags":[{"tag":"DOI"},{"tag":"Open Code"},{"tag":"Open Data"}],"schematype":"script"}