Extremal quantiles and stock price crashes

Abstract

AbstractWe employ extreme value theory to identify stock price crashes, featuring low-probability events that produce large, idiosyncratic negative outliers in the conditional distribution. Traditional methods employ approximations under Gaussian assumptions and central moments. This is inherently imprecise and susceptible to misspecifications, especially for tail events. We instead propose new definitions and measures for crash risk based on conditional extremal quantiles (CEQ) of idiosyncratic stock returns. CEQ provide information on quantile-specific impact of covariates, and shed light on prior empirical puzzles and shortcomings in identifying crashes. Additionally, to capture the magnitude of crashes, we provide an expected shortfall analysis of the losses due to crash. Our findings have important implications for a burgeoning literature in financial economics that relies on traditional approximations.KEYWORDS: Extremal quantilesextreme value theoryquantile regressionstock price crashesJEL Classification: C14D81G11G12G32 AcknowledgmentsThe views expressed in this article are those of the authors and not necessarily reflect those of the Bank of Greece or the Eurosystem.Notes1 Some notable examples, inter alia, are: Chen et al. (Citation2001); Jin and Myers (Citation2006); Hutton et al. (Citation2009); Kim et al. (Citation2011); Callen and Fang (Citation2015); Andreou et al. (Citation2016); Kim et al. (Citation2016); Andreou et al. (Citation2017); Chang et al. (Citation2017); Ertugrul et al. (Citation2017); Cheng et al. (Citation2020); Li and Zeng (Citation2019).2 The nonclustering condition is of the Meyer (Citation1973) type and states that the probability of two extreme events co-occurring at nearby dates is much lower than the probability of just one extreme event. This assumption is convenient because it leads to limit distributions of extremal quantile regression estimators as if independent sampling had taken place. The plausibility of the nonclustering assumption is an empirical matter.3 Due to data limitation issues, we cannot perform our analysis on a per-firm basis. However, we performed also the analysis by (a) pooling per-year-industry and (b) pooling data per-year and then take the average over all years. We find that our findings are not sensitive to the way we split the data. All robustness checks are available upon request.4 Excess return is typically computed as deviation from a given risk free return. Here, idiosyncratic weekly return is computed as deviation from a statistically determined, stable, weekly market, and industry return. An interpretation is that we are removing a linear projection expected value of market and/or industry returns. This is a partialling out of returns that accounts for the expected value of market and common industry factors, before a quantile regression is conducted on other conditioning covariates. An alternative approach would be a single step estimation of quantiles, controlling for quantile effects of market, and industry weekly returns. Another approach may first estimate the conditional distribution of weekly returns, controlling for all desired covariates simultaneously, by a method such as distribution regressions.5 Examples of such papers include Hutton et al. (Citation2009); Kim et al. (Citation2011); Callen and Fang (Citation2015); Andreou et al. (Citation2016); Kim et al. (Citation2016); Andreou et al. (Citation2017); Chang et al. (Citation2017); Ertugrul et al. (Citation2017); Kim et al. (Citation2019); Li and Zeng (Citation2019); Andreou et al. (Citation2022).6 The quantile regressions are based on the 52 idiosyncratic weekly returns pooled over all stocks within a given industry, in a given year. We have repeated all computations using only the industry returns (no market returns) in the index model; the results are quite robust.7 We estimate the conditional extreme quantile following in step the code as per the Koenker (Citation2016) quantreg package, as well as the code from Chernozhukov and Du (Citation2006) and Chernozhukov and Fernández-Val (2011).8 All robustness checks are available upon request.9 Note that we have checked whether excluding firm-years with a stock price less than $1 at the end of the fiscal year artificially creates the upward-trending frequency of crashes observed and the number of crashes is fairly steady.10 https://www.bis.org/publ/bcbs265.pdf11 For the needs of this analysis, following Andreou et al. (Citation2022), we impose additional filtering rules, particularly, keeping common stocks (i.e., share codes 10 and 11) traded in NYSE, AMEX, and NASDAQ, excluding firm-years with a stock price less than $1 at the end of the fiscal year and having fewer than 26 weeks of stock returns in a fiscal year, and dropping firm-year observations without available information in Compustat for computing the financial variables.12 The nonclustering condition is of the Meyer (Citation1973) type and states that the probability of two extreme events co-occurring at nearby dates is much lower than the probability of just one extreme event. For example, it assumes that a large market crash is not likely to be immediately followed by another large crash.Additional informationFundingESADE Business School, Ramon LLull University, Avenida de Torreblanca 59, 08172, Sant Cugat, Barcelona, Spain; e-mail: carlo.sala@esade.edu. Financial support from the AGAUR - SGR 2017-640 grant and from the Spanish Ministry of Science and Innovation - PID2019-1064656GBI00/AEI/10.13039/501100011033 are gratefully acknowledged.