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) J Bus 36:394-419] and Fama [Fama EF (1965) J Bus 38:34-105] proposed a symmetric Lévy probability distribution function (PDF) to describe the stochastic properties of commodity changes and price changes. We find that an asymmetric Lévy PDF, L, characterized by infinite variance, models several multiple credit ratios used in financial accounting to quantify a firm's financial health, such as the Altman [Altman EI (1968) J Financ 23:589-609] Z score and the Zmijewski [Zmijewski ME (1984) J Accounting Res 22:59-82] score, and models changes of individual financial ratios, ΔX i . We thus find that Lévy PDFs describe both the static and dynamics of credit ratings. We find that for the majority of ratios, ΔX i scales with the Lévy parameter α ≈ 1, even though only a few of the individual ratios are characterized by a PDF with power-law tails X −1−α i with infinite variance. We also find that α 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 Z score, we find that PðΔZÞ follows a Lévy PDF with power-law exponent α ≈ 1, consistent with changes of individual financial ratios. Estimating the conditional PðΔZjZÞ versus Z, we demonstrate how this continuous credit rating approach and its dynamics can be used to evaluate credit risk.complex systems | econophysics | rating migrations M ost tests and tools used in statistics assume that any errors in a financial model are Gaussian distributed, and it is a common practice in economics to use a Gaussian distribution to approximate empirical data. Mandelbrot (1) and Fama (2) were among the first to notice that the logarithm of cotton price fluctuations and common stock price fluctuations have fatter tails than those produced by a Gaussian distribution, and they proposed a stable Lévy distribution to model the stochastic properties of the fluctuations. Analyzing high-frequency data, Mantegna and Stanley (3) reported that the stable Lévy distribution accurately models only a broad central region of the probability distribution function (PDF) of stock price changes, whereas Gopikrishnan et al. reported that a power law with an exponent value beyond the Lévy regime is needed to describe the tails (4, 5).The central limit theorem (CLT) implies that the mean of a sufficiently large number of independent random variables, each with finite variance, will approximately follow a normal distribution (6). A generalization of the CLT shows that the mean of a sufficiently large number of independent random variables, each with infinite variance, approximately follows a stable Lévy distribution L α;γ ðxÞ ¼ ð1∕πÞ∫ ∞ 0 dq expð−γq α Þ cosðqxÞ, wher...

Using a large sample of small private companies, we show incremental influence of economic incentives over prescriptions from accounting standards by financial statement preparers in a code-law setting with high alignment between financial and tax reporting and no agency problems. Contrary to predictions from standards, more profitable companies are more likely to write-off and the write-off magnitude is greater, reflecting tax minimisation. Larger companies are more likely to write-off, but the magnitude decreases with size, reflecting increasing political costs due to greater visibility to tax authorities. Previous write-off patterns and magnitudes are persistent, reflecting institutional learning linked to regulatory changes. Copyright (c) 2008 The Authors Journal compilation (c) 2008 Blackwell Publishing Ltd.

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This paper finds that private firms make the decision to write off, and write off more in terms of total amount, if they are: (i) more profitable, (ii) have more financial debt, and (iii) pay dividends. Our findings are contrary to expectations based on accounting standards and the existing revaluation literature. They are, however, consistent with the codified, high booktax alignment economic setting in which sample private firms operate. This includes agency problems faced by private firms' stakeholders. We use a comprehensive sample of German SMEs reporting in local GAAP, based on the German commercial code (Handelsgesetzbuch) in [2003][2004][2005][2006]. We view write-offs as corrections of departures of book values from their underlying economic values, in contrast to upward asset revaluations. This governs our choice of estimation -the tobit regression.