{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,4]],"date-time":"2026-10-04T21:34:47Z","timestamp":1791149687685,"version":"4.1.0"},"reference-count":37,"publisher":"National Academy of Sciences","issue":"44","content-domain":{"domain":["www.pnas.org"],"crossmark-restriction":true},"short-container-title":["Proc. Natl. Acad. Sci. U.S.A."],"published-print":{"date-parts":[[2011,11]]},"abstract":"\n Because financial crises are characterized by dangerous rare events that occur more frequently than those predicted by models with finite variances, we investigate the underlying stochastic process generating these events. In the 1960s Mandelbrot [Mandelbrot B (1963)\n J Bus<\/jats:italic>\n 36:394\u2013419] and Fama [Fama EF (1965)\n J Bus<\/jats:italic>\n 38:34\u2013105] proposed a symmetric L\u00e9vy probability distribution function (PDF) to describe the stochastic properties of commodity changes and price changes. We find that an asymmetric L\u00e9vy PDF,\n \n \n <\/jats:inline-formula>\n , characterized by infinite variance, models several multiple credit ratios used in financial accounting to quantify a firm\u2019s financial health, such as the Altman [Altman EI (1968)\n J Financ<\/jats:italic>\n 23:589\u2013609]\n Z<\/jats:italic>\n score and the Zmijewski [Zmijewski ME (1984)\n J Accounting Res<\/jats:italic>\n 22:59\u201382] score, and models changes of individual financial ratios, \u0394\n X<\/jats:italic>\n \n i<\/jats:italic>\n <\/jats:sub>\n . We thus find that L\u00e9vy PDFs describe both the static and dynamics of credit ratings. We find that for the majority of ratios, \u0394\n X<\/jats:italic>\n \n i<\/jats:italic>\n <\/jats:sub>\n scales with the L\u00e9vy parameter\n \u03b1<\/jats:italic>\n \u00a0\u2248\u00a01, even though only a few of the individual ratios are characterized by a PDF with power-law tails\n \n \n <\/jats:inline-formula>\n with infinite variance. We also find that \u03b1 exhibits a striking stability over time. A key element in estimating credit losses is the distribution of credit rating changes, the functional form of which is unknown for alphabetical ratings. For continuous credit ratings, the Altman\n Z<\/jats:italic>\n score, we find that\n P<\/jats:italic>\n (\u0394\n Z<\/jats:italic>\n ) follows a L\u00e9vy PDF with power-law exponent\n \u03b1<\/jats:italic>\n \u00a0\u2248\u00a01, consistent with changes of individual financial ratios. Estimating the conditional\n P<\/jats:italic>\n (\u0394\n Z<\/jats:italic>\n |\n Z<\/jats:italic>\n ) versus\n Z<\/jats:italic>\n , we demonstrate how this continuous credit rating approach and its dynamics can be used to evaluate credit risk.\n <\/jats:p>","DOI":"10.1073\/pnas.1113330108","type":"journal-article","created":{"date-parts":[[2011,10,18]],"date-time":"2011-10-18T01:23:13Z","timestamp":1318900993000},"page":"17883-17888","update-policy":"https:\/\/doi.org\/10.1073\/pnas.cm10313","source":"Crossref","is-referenced-by-count":69,"title":["Asymmetric L\u00e9vy flight in financial ratios"],"prefix":"10.1073","volume":"108","author":[{"given":"Boris","family":"Podobnik","sequence":"first","affiliation":[{"name":"Center for Polymer Studies and Department of Physics, Boston University, Boston, MA 02215;"},{"name":"School of Civil Engineering, University of Rijeka, 51000 Rijeka, Croatia;"},{"name":"Department of Finance, Zagreb School of Economics and Management, 10000 Zagreb, Croatia;"},{"name":"Department of Money and Finance, Faculty of Economics, University of Ljubljana, 1000 Ljubljana, Slovenia; and"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Aljo\u0161a","family":"Valentin\u010di\u010d","sequence":"additional","affiliation":[{"name":"Department of Money and Finance, Faculty of Economics, University of Ljubljana, 1000 Ljubljana, Slovenia; and"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Davor","family":"Horvati\u0107","sequence":"additional","affiliation":[{"name":"Faculty of Natural Sciences, University of Zagreb, 10000 Zagreb, Croatia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"H. Eugene","family":"Stanley","sequence":"additional","affiliation":[{"name":"Center for Polymer Studies and Department of Physics, Boston University, Boston, MA 02215;"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"341","published-online":{"date-parts":[[2011,10,17]]},"reference":[{"key":"e_1_3_2_1_2","doi-asserted-by":"publisher","DOI":"10.1086\/294632"},{"key":"e_1_3_2_2_2","doi-asserted-by":"publisher","DOI":"10.1086\/294743"},{"key":"e_1_3_2_3_2","doi-asserted-by":"publisher","DOI":"10.1038\/376046a0"},{"key":"e_1_3_2_4_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.60.5305"},{"key":"e_1_3_2_5_2","doi-asserted-by":"publisher","DOI":"10.1080\/096031096333917"},{"key":"e_1_3_2_6_2","volume-title":"An Introduction to Probability Theory and Its Applications","author":"Feller W","year":"1971","unstructured":"W Feller An Introduction to Probability Theory and Its Applications (Wiley, New York, 1971)."},{"key":"e_1_3_2_7_2","volume-title":"Th\u00e9orie de l\u2019Addition des Variables Al\u00e9atoires","author":"L\u00e9vy P","year":"1937","unstructured":"P L\u00e9vy Th\u00e9orie de l\u2019Addition des Variables Al\u00e9atoires (Gauthier-Villars, Paris, 1937)."},{"key":"e_1_3_2_8_2","doi-asserted-by":"publisher","DOI":"10.2298\/PIM0694171J"},{"key":"e_1_3_2_9_2","doi-asserted-by":"publisher","DOI":"10.1146\/annurev.economics.050708.142940"},{"key":"e_1_3_2_10_2","doi-asserted-by":"publisher","DOI":"10.1111\/j.1540-6261.1968.tb00843.x"},{"key":"e_1_3_2_11_2","doi-asserted-by":"publisher","DOI":"10.2307\/2490859"},{"key":"e_1_3_2_12_2","doi-asserted-by":"publisher","DOI":"10.1086\/209665"},{"key":"e_1_3_2_13_2","unstructured":"LV Carty Moody\u2019s Rating Migration and Credit Quality Correlation 1920\u20131996. 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