Asset Pricing Models and Market Efficiency
Using Machine Learning to Explain Stock Market Anomalies
- 2026
- Book
- Authors
- James W. Kolari
- Wei Liu
- Jianhua Z. Huang
- Huiling Liao
- Publisher
- Springer Nature Switzerland
About this book
This book shows that the stock market returns of hundreds of anomaly portfolios discovered by researchers in finance over the past three decades can be explained by a recent asset pricing model dubbed the ZCAPM. Anomaly portfolios are long/short portfolio returns on stocks that cannot be explained by asset pricing models, and their number has been steadily increasing into the hundreds. Since asset pricing models cannot explain them, behavioral theories have become popular to account for anomalies. Unlike the efficient market hypothesis that assumes rational investors, these human psychology-based theories emphasize irrational investor behavior.
This book collects and analyzes a large database of U.S. stock returns for anomaly portfolios over a long sample period spanning approximately 60 years. The authors overview different asset pricing models that have attempted to explain anomalous portfolio returns in the stock market. They then provide a theoretical and empirical discussion of a new asset pricing model dubbed the ZCAPM and report compelling empirical evidence that reveals the ZCAPM can explain hundreds of anomalies. Implications to the efficient-markets/behavioral-finance controversy are discussed. The book will be of particular interest to researchers, students, and professors of capital markets, asset management, and financial economics alongside professionals.
Advertisement
Table of Contents
-
Frontmatter
-
Introduction
-
Frontmatter
-
Chapter 1. The Rise of Anomalies: Challenging Theory and Practice in Finance
James W. Kolari, Wei Liu, Jianhua Z. Huang, Huiling LiaoThe chapter delves into the debate between the efficient markets hypothesis (EMH) and behavioral finance, focusing on the role of anomalies in stock market returns. It begins by discussing the efficient markets hypothesis (EMH) and the Capital Asset Pricing Model (CAPM), which posits that security prices fully reflect all available information. The chapter then explores how anomalies, such as the size and value effects, challenge the CAPM and lead to the development of multifactor models by Fama and French. The text also examines the rise of behavioral finance, which argues that human psychology plays a significant role in explaining stock market anomalies. The chapter introduces the ZCAPM model, which uses machine learning to explain a large number of anomaly portfolios, providing strong support for the EMH. The chapter concludes by discussing the implications of these findings for the debate between efficient markets and behavioral finance, highlighting the need for further research to replicate and expand on these results.AI Generated
This summary of the content was generated with the help of AI.
AbstractThis chapter introduces the basic issues and problems of anomalies, asset pricing, and behavioral finance. Long/short portfolios of stocks not explained by an asset pricing model are considered to be anomalies. However, other important characteristics of anomalies are abnormally high return performance as well as return predictability. Anomalies with high returns that are persistent over time are particularly attractive to investors. According to the efficient market hypothesis (EMH), anomalies with high returns should be arbitraged away by investors and therefore are only transitory in nature. Unfortunately, despite some controversy, empirical evidence has been accumulating over the past 40 years or more that long/short portfolio anomalies are real and not going away. If anomaly portfolios are not efficiently priced, we need to consider the possibility that inefficiencies exist in the market. Behavioralists argue that human psychology allows for irrational investor behavior, which accounts for inefficiencies in the pricing of anomaly portfolios. Other potential sources of anomalies are information asymmetries and market structure. These sources of abnormal returns should be arbitraged away by profit-seeking investors. However, the effects of irrational investor behavior driven by human behavior are likely more difficult to eliminate via arbitrage. The main purpose of this book is to present new stock market evidence on the ability of asset pricing models to explain large datasets of long/short anomaly portfolios. Also, we review published studies on stock market anomalies and asset pricing models. In this chapter, we provide background discussion on the EMH-behavioralist debate. Later chapters review previous empirical tests and report new evidence using open source (online) datasets of long/short anomaly portfolios. Which school of thought does the market evidence support? Efficient markets? Behavioral finance? Or are both schools useful to our understanding of stock prices? As we will see, based on a new asset pricing model dubbed the ZCAPM by Kolari et al. (2021), our empirical evidence in forthcoming chapters strongly favors the EMH. -
Chapter 2. Anomaly Stock Portfolios
James W. Kolari, Wei Liu, Jianhua Z. Huang, Huiling LiaoThis chapter delves into the fascinating world of anomaly stock portfolios, examining why many anomalies seem to disappear over time. Key topics include the impact of statistical bias and market arbitrage on anomaly returns, the role of publication bias, and the replicability of anomalies across different studies. The chapter also explores the economic significance of anomalies and their potential as tradable investment strategies. Recent research suggests that anomalies are not merely a product of data mining but persist when considering risk-adjusted returns and accurate information. The chapter concludes that while anomalies are real, their practical application in investment strategies is limited due to trading costs and post-publication effects.AI Generated
This summary of the content was generated with the help of AI.
AbstractThis chapter surveys the finance literature on anomalous stock portfolios. These portfolios have long positions in selected stocks and short positions in other stocks. Early examples are the size and value anomaly stock portfolios. To explain these anomalies, Fama and French (1992), Fama and French (1993) famously created a three-factor model comprised of a market factor, a size factor that is long small capitalization stocks and short big capitalization stocks, and a value factor that is long high book-to-market equity ratio stocks and short low book-to-market equity ratio stocks. Many subsequent research papers have been published on a wide variety of other long/short anomaly portfolios over the past three decades. In the last few years, work on anomaly stock portfolios has accelerated with some excellent comprehensive studies of hundreds of such anomalies in top finance journals. To encourage research on anomalies, authors of some of recent studies are making available their stock return data series on the internet for hundreds of anomaly portfolios—for example, Chen and Zimmermann (2022) and Jensen et al. (2023). We have downloaded data from these studies via their online websites. This open source data are publicly available to researchers. In forthcoming chapters of this book, we employ this data to conduct empirical tests on the ability of alternative asset pricing models to explain almost 300 anomaly stock portfolios.
-
-
Anomalies Literature and Asset Pricing Models
-
Frontmatter
-
Chapter 3. Prominent Asset Pricing Models and Anomaly Portfolio Returns
James W. Kolari, Wei Liu, Jianhua Z. Huang, Huiling LiaoThis chapter delves into the prominent asset pricing models and their ability to explain anomaly portfolio returns. It begins with the Fama-French three-factor model, which introduced size and value factors to better explain stock returns compared to the Capital Asset Pricing Model (CAPM). The chapter then explores the Carhart four-factor model, which adds a momentum factor to account for momentum anomalies. Further, it discusses the Fama-French five-factor model, which incorporates profitability and capital investment factors, and the subsequent six-factor model that includes a momentum factor. The text also covers alternative models like the Hou-Xue-Zhang four-factor model and the Stambaugh-Yuan four-factor model, which introduce different factors such as management and performance. Additionally, it touches on the Lettau-Pelger latent five-factor model, which uses Principal Component Analysis to identify asset pricing factors. The chapter concludes by highlighting the ongoing challenges in asset pricing models and the potential for future developments, including the ZCAPM model by Kolari, Liu, and Huang. Readers will gain insights into the evolution of asset pricing models, the role of various factors in explaining stock returns, and the current debates and advancements in the field.AI Generated
This summary of the content was generated with the help of AI.
AbstractThe Capital Asset Pricing Model (CAPM) of Treynor (1961, 1962), Sharpe (1964), Lintner (1965), and Mossin (1966) established a new field in finance known as asset pricing. The first general equilibrium model of asset prices, it rapidly became popular with academics and spread to professionals in the financial markets. Even though early empirical evidence was weaker than predicted by theory in terms of the relation between market beta risk and real world stock returns, the CAPM dominated asset pricing for almost three decades and yielded a Nobel Prize in Economics for Sharpe in 1990. But in the 1990s, Fama and French (1992, 1993, 1995, 1996, 1998) published a series of papers that documented extensive empirical evidence using U.S. stock returns over decades that the CAPM did not work—that is, average stock returns were not related to beta. In its place, Fama and French proposed a three-factor model that augmented the market factor with size and value factors. These new factors were intended to explain market anomalies wherein small firms tended to have higher returns than big firms over time and value firms had higher returns than growth firms. To explain these anomalies, they developed innovative long/short portfolios of stocks with small market capitalization minus those with large market capitalization as well as value stocks’ returns minus growth stocks’ returns. Upon adding these long/short portfolios as factors in the three-factor model, the ability to explain stock returns of anomaly portfolios was much improved. Subsequently, as other market anomalies were identified by researchers, asset pricing models advanced to include more long/short portfolio factors. So-called emphmultifactor models appeared to be able to explain stock market anomalies. As discussed in Chapter 2, what happened next was surprising. The number of anomalies identified by researchers exploded into the hundreds and overwhelmed the ability of asset pricing models to keep up with them. While anomalies could be grouped into categories and then factors developed to explain different categories of anomalies, at the time of this writing, the large number of anomalies discovered by researchers are causing major gaps in asset pricing models. Behavioralists propose that psychological explanations can close these major gaps in asset pricing models. However, a fundamental rift is occurring in finance as behavioralists argue investors are not always rational and financial markets not always efficient, whereas asset pricing model researchers assume rational investors and efficient markets. In this chapter we review prominent asset pricing models that help to explain anomaly portfolio returns. In forthcoming chapters, we show that that widely accepted multifactor models do not explain hundreds of anomalies’ returns. By contrast, a new model dubbed the ZCAPM by Kolari, Liu, and Zhang (2021) well explains virtually all of these anomalies. -
Chapter 4. The ZCAPM and Previous Tests of Anomaly Portfolio Returns
James W. Kolari, Wei Liu, Jianhua Z. Huang, Huiling LiaoThe chapter delves into the ZCAPM, a groundbreaking asset pricing model derived from Black's zero-beta CAPM. It introduces two key factors: the market factor and the cross-sectional market volatility factor, both estimable from daily market data. The ZCAPM's theoretical framework is explored, highlighting its unique geometry and the role of market dispersion in reaching the efficient frontier. Empirical tests demonstrate the ZCAPM's impressive performance, outperforming well-known multifactor models in predicting anomaly portfolio returns. The chapter also discusses the innovative use of machine learning techniques, such as the expectation-maximization algorithm, to estimate hidden variables and improve predictive accuracy. The ZCAPM's ability to explain a wide range of anomaly returns is thoroughly examined, providing compelling evidence for its validity as an asset pricing model. The chapter concludes with a summary of the ZCAPM's strengths and its potential to revolutionize asset pricing in the finance literature and profession.AI Generated
This summary of the content was generated with the help of AI.
AbstractBlack (1972) proposed the zero-beta CAPM in an effort to reconcile empirical evidence on tests of the famed Capital Asset Pricing Model (CAPM) by Treynor (1961, 1962), Sharpe (1964), Lintner (1965), and Mossin (1966). The CAPM is a general equilibrium model that can be graphically represented by the Security Market Line (SML) defined by excess expected returns over the riskless rate of assets on the Y-axis and associated estimated betas on the X-axis. A positive linear relationship is hypothesized between expected returns and beta risk. Using U.S. stock market returns, early evidence by Black, Jensen, and Scholes (1972) showed that the SML was flatter with a higher intercept (or \(\alpha \) parameter) than predicted by the CAPM. More specifically, low beta stocks had higher returns than theorized by the CAPM and vice versa for high beta stocks. They inferred that, instead of using the riskless rate as in the CAPM, a higher borrowing rate should be used in line with investor practice in financial markets. Black extended this potential solution by proposing the possible existence of a zero-beta portfolio uncorrelated with the market portfolio in the CAPM. In effect, there are two factors—namely, a market factor and zero-beta factor. Unfortunately, he did not provide any guidance on how to construct a zero-beta portfolio in the real world. For this reason, no empirical model of the zero-beta CAPM is available.
-
-
Explaining Anomaly Portfolio Returns
-
Frontmatter
-
Chapter 5. The ZCAPM and Large Online Datasets of Anomaly Portfolio Returns
James W. Kolari, Wei Liu, Jianhua Z. Huang, Huiling LiaoThis chapter delves into the performance of the Zero-Crossing Asset Pricing Model (ZCAPM) in explaining anomaly portfolio returns, comparing it to various multifactor models. The study uses large online datasets of anomaly portfolio returns and conducts out-of-sample cross-sectional regression tests. The results show that the ZCAPM significantly outperforms other models, with higher goodness-of-fit and lower mispricing errors. The chapter also explores the implications of these findings for market efficiency and asset pricing theories. It suggests that the ZCAPM can be a unifying model that links general equilibrium models to empirically designed multifactor models. The analysis highlights the importance of market return dispersion in explaining anomaly returns and supports the efficient market hypothesis. The chapter concludes with recommendations for future research, including replication studies, exploring unpriced anomalies, event studies, corporate finance applications, portfolio management, determinants of market return dispersion, and extending tests to other asset classes.AI Generated
This summary of the content was generated with the help of AI.
AbstractThis chapter utilizes open source datasets with over 250 anomaly portfolios constructed by Chen and Zimmerman (2022) and Jensen, Kelly, and Pedersen (2023) to conduct out-of-sample tests of prominent asset pricing models. Chen and Zimmerman found that 161-out-of-319 long-short anomalies are “clear predictors” in terms of the statistical significance of cross-sectional return predictability on an out-of-sample basis. Jensen, Kelly, and Pedersen documented that 153 long-short factors across 93 countries can be replicated over time, survive joint modeling of all factors, hold up out-of-sample after their original publication, are strengthened by the large number of factors including global evidence, and can be grouped into a smaller number of anomaly clusters. Unlike previous studies, the authors employed alphas from the CAPM to measure anomaly returns. These risk-adjusted returns yielded results different from those of raw returns – unlike declining raw returns, alphas have not decreased over time. An important inference from the Jensen et al. study is that anomaly returns contain mispricing errors that likely arise from premiums associated with their risks. As we will see in this chapter, confirming this conjecture, risk premiums associated with systematic risk factors in the ZCAPM well explain large datasets of anomaly portfolio returns. The ZCAPM is a new asset pricing model by Kolari, Liu, and Huang (2021). Because large numbers of anomalies can be explained by ZCAPM systematic risk factors (viz., market returns and market return dispersion), the market efficiency hypothesis is supported. Also, we infer that hundreds of long-short portfolio anomalies yielding relatively high abnormal returns are no longer anomalous or unexplained. If previously published anomalies are explained by the ZCAPM, further research is needed to find long-short portfolios not explained by the ZCAPM. Finally, the consistent outperformance of the ZCAPM compared to prominent multifactor models in our out-of-sample empirical tests, in many cases by large margins, suggests that the ZCAPM is a valid asset pricing model worthy of further study by researchers. Can multifactor models be enhanced to match the out-of-sample performance of the ZCAPM? Or, should extant multifactor models be scrapped by researchers to make way for future progress in the development of the ZCAPM? -
Chapter 6. Further Tests of Asset Pricing Models and Anomaly Portfolio Returns
James W. Kolari, Wei Liu, Jianhua Z. Huang, Huiling LiaoThis chapter delves into the performance of various asset pricing models in explaining the returns of different portfolios, including those based on firm and stock characteristics, industry portfolios, and anomaly portfolios in Japan. The study employs out-of-sample Fama-MacBeth cross-sectional regressions and graphical analyses of average mispricing errors to evaluate models like CAPM, Fama-French three-, five-, and six-factor models, Carhart four-factor model, Hou-Xue-Zhang four-factor q model, and Stambaugh-Yuan four-factor model. A key finding is the consistent outperformance of the ZCAPM model, which demonstrates superior explanatory power and statistical significance in its factor loadings. The ZCAPM's ability to explain both long/short anomaly portfolio returns and their constituent portfolios suggests its potential as a more valid asset pricing model. The chapter also highlights the challenges in pricing industry portfolios and calls for further research to understand their unique dynamics. The empirical evidence supports the efficient markets hypothesis and questions the need for behavioral explanations in finance.AI Generated
This summary of the content was generated with the help of AI.
AbstractThis chapter extends the cross-sectional tests of different asset pricing models in the previous chapter using different test asset portfolios. In Chapter 5, we conducted tests based on 133 U.S. anomaly portfolios constructed by Chen and Zimmerman (2020) and 153 anomaly portfolios by Jensen, Kelly, and Pedersen (Jensen and Kelly (2023)). Here, we report further evidence using other test asset portfolios. First, we employ commonly used portfolios in published asset pricing studies reviewed in Chapter 2. To do this, a variety of size, value, profitability, capital investment, and momentum stock portfolios are downloaded from Kenneth French’s database website. With the exception of momentum portfolios, these test assets are not long/short portfolios as in the case of anomaly portfolios; instead, they are portfolios that are used in the construction of anomaly portfolios. For example, Fama and French (1992, 1993) sorted individual stocks by market capitalization into size deciles. The top/bottom three size deciles are used to construct the long/short size factor in their multifactor asset pricing models. Thus, we take a more granular look at the returns of long and short portfolios that are used in forming anomaly portfolios. Second, we utilize industry portfolios available on French’s website. Industry portfolios’ returns are notorious for being difficult to explain with any asset pricing model and therefore can be considered to be anomalous. Finally, we add to the previous chapter by testing a sample of 86 anomaly portfolio returns in Japan. These long/short portfolios are available online at the website provided by Jensen et al. As in Chapter 5, our goal is to show that the ZCAPM does a good job of explaining stock portfolios’ returns. In turn, the efficient markets hypothesis holds in the sense that rational investors and systematic market risk explain stock returns. Behavioral explanations based on irrational investors subject to psychological biases are not needed to explain long/short portfolio stock returns for the most part.
-
-
Asset Pricing Model Validity
-
Frontmatter
-
Chapter 7. Empirical Tests on the Validity of Asset Pricing Models
James W. Kolari, Wei Liu, Jianhua Z. Huang, Huiling LiaoThis chapter delves into the empirical testing of asset pricing models, focusing on the validity of models like CAPM and multifactor models. The study introduces a new alpha test based on out-of-sample cross-sectional regression analyses, addressing the limitations of traditional in-sample tests. Key findings include the significance of missing factors in most models and the superior performance of the ZCAPM, which shows minimal mispricing errors. The research also highlights the importance of out-of-sample testing in identifying missing factors and validating asset pricing models. The study concludes that the ZCAPM is a robust model, efficiently capturing systematic risk factors and suggesting market efficiency. The chapter provides valuable insights into the ongoing search for missing factors in asset pricing models and the potential discontinuation of this search if the ZCAPM's performance is confirmed.AI Generated
This summary of the content was generated with the help of AI.
AbstractPrevious chapters have shown that well-known asset pricing models published in top tier finance journals do a poor job of explaining anomaly portfolio returns. In stark contrast, a recent but lesser known model dubbed the ZCAPM by Kolari, Liu, and Huang (2021) almost completely explains anomaly stock returns. These findings call into question the validity of prominent asset pricing models. That is, commonly used multifactor models appear to be missing factors that are needed to explain anomaly returns. To test this possibility, in this chapter we employ a new out-of-sample cross-sectional regression coefficient test of alpha mispricing error terms. According to Jensen (1968), the alpha parameter in the estimation of a time-series regression for an asset pricing model measures mispricing error which arises primarily due to missing factors in the model. Many authors test for the presence of mispricing errors using the Gibbons, Ross, and Shanken (GRS) (1989) test for the joint zero equality of alphas across a sample of test assets (e.g., stock portfolios). Notably, this test is restricted to linear model tests on an in-sample basis. Extending the GRS test of alphas, a recent study by Kolari, Huang, Liu, and Liao (2014) proposed a novel out-of-sample alpha test. More specifically, alpha parameters for tests assets estimated by a time-series regression model are used as a factor loading in a standard out-of-sample (one-month-ahead) Fama and MacBeth (1973) cross-sectional regression. If the alpha loading is significant, it means that missing factors exist in the time-series regression asset pricing model. Kolari et al. found that, even if in-sample GRS tests indicate that alphas are jointly zero with no apparent mispricing error and missing factors, out-of-sample alpha tests can suggest missing factors exist. In this chapter, we compare out-of-sample alpha test results for the CAPM, prominent multifactor models, and the recent ZCAPM. Our analyses use a variety of different test assets. Across these test asset samples, we find that mispricing errors are consistently lower in the ZCAPM compared to other models. For most test assets, the ZCAPM shows evidence of no missing factors. Thus, we conclude that the ZCAPM is not plagued by a missing factor problem as in the case of the CAPM and popular multifactor models. By inference, while our evidence supports the ZCAPM as a valid asset pricing model, well-known multifactor models are not supported. Also, due to the ability of the ZCAPM to well explain the cross-section of average stock returns, our empirical evidence suggests that the stock market is efficient.
-
-
Conclusion
-
Frontmatter
-
Chapter 8. Machine Learning in Asset Pricing: The Dominance of the ZCAPM
James W. Kolari, Wei Liu, Jianhua Z. Huang, Huiling LiaoThis chapter delves into the transformative impact of machine learning on asset pricing, with a focus on the Zero-Beta CAPM (ZCAPM). The ZCAPM, which utilizes the mean market return and cross-sectional return dispersion as its factors, is shown to outperform traditional multifactor models in explaining stock market anomalies. The chapter reviews the theoretical derivation and empirical specification of the ZCAPM, highlighting its use of the expectation-maximization algorithm to estimate hidden variables. Extensive out-of-sample tests with large datasets of anomaly portfolios demonstrate the ZCAPM's exceptional explanatory power, even for challenging momentum portfolios. The chapter also discusses the implications for future asset pricing research and market efficiency, suggesting areas for further development and study. The findings support the efficient market hypothesis and challenge the need for behavioral theories to explain anomaly portfolio returns.AI Generated
This summary of the content was generated with the help of AI.
AbstractThe empirical results in this book have shown that machine learning is a powerful new approach to asset pricing that can boost predictive performance of asset pricing models. The ZCAPM of Kolari, Liu, and Huang (2021) takes advantage of machine learning in estimating the empirical ZCAPM via the expectation-maximization (EM) algorithm in regression analyses. The ZCAPM contains a hidden or latent variable that determines the positive versus negative sensitivity of asset returns to changes in the cross-sectional return dispersion of assets in the market. It is unknown if a stock will be positively or negatively affected the market return dispersion at any given time. In estimating the empirical ZCAPM regression, the EM algorithm incorporates information about asset returns over the past year. From this information, it estimates a probability (or p value) that an asset or portfolio of assets will have a positive sensitivity to market return dispersion. As it turns out, from an artificial intelligence perspective, this hidden probability is crucial to the predictive success of the ZCAPM in explaining stock market anomalies in our out-of-sample cross-sectional regression tests. Without it, the model would not work in predicting anomaly portfolio returns. Here we review the material in previous chapters. Hundreds of long/short portfolios of stock market anomalies have been constructed by researchers. Because asset pricing models cannot explain their abnormally high average stock returns, these portfolios are considered to be anomalies. Importantly, the inability of prominent asset pricing models to explain economically interesting anomalies strikes a major blow to the famous efficient market hypothesis (EMH) of Nobel Prize winner Professor Eugene Fama. Asset prices should be related to their market risks. If systematic market risks in asset pricing models cannot explain their prices, the EMH is rejected. In its place, behavioral theories have proposed irrational investor behavior grounded in inherent biases in human behavior to potentially explain stock market anomalies’ returns. Given the rising tide of anomalies, which are outpacing the ability of multifactor models in asset pricing to keep up with them, behavioralists have gained a solid footing in the anomalies literature. The stock market evidence documented in this book sheds new light on the longstanding yet increasing debate between efficient markets and behavioral theories of asset pricing. We compared the most prominent asset pricing models in the finance literature in terms of their ability to predict in out-of-sample tests the stock market returns of hundreds of anomaly portfolios. Also, we included in our tests the ZCAPM a lesser known asset pricing model that employs expectation-maximization (EM) algorithm optimization methods to estimate a hidden signal variable. Strikingly, the ZCAPM is the only model that does a good job of predicting the returns of large numbers of stock market anomaly portfolios (i.e., long/short portfolios based on accounting and market characteristics of firms). Virtually all of the anomalies tested are explained by the ZCAPM on an out-of-sample basis. Our ZCAPM evidence reverses the above mentioned trends in asset pricing and market efficiency. Given that the ZCAPM can explain anomalies’ abnormally high returns over time, our results strongly support the EMH. In turn, behavioral explanations of anomalies are not needed. Also, since prominent multifactor models cannot predict anomaly returns over time, the ZCAPM dominates these models in terms of asset pricing ability. In further tests of model validity, we confirm that the ZCAPM has lower out-of-sample mispricing errors among anomaly portfolios than commonly-used, popular multifactor models. Important implications of our findings are discussed with respect to asset pricing and market efficiency in particular and the field of finance in general. Lastly, we provide closing remarks on the findings of this book.
-
-
Backmatter
- Title
- Asset Pricing Models and Market Efficiency
- Authors
-
James W. Kolari
Wei Liu
Jianhua Z. Huang
Huiling Liao
- Copyright Year
- 2026
- Publisher
- Springer Nature Switzerland
- Electronic ISBN
- 978-3-031-92901-4
- Print ISBN
- 978-3-031-92900-7
- DOI
- https://doi.org/10.1007/978-3-031-92901-4
PDF files of this book have been created in accordance with the PDF/UA-1 standard to enhance accessibility, including screen reader support, described non-text content (images, graphs), bookmarks for easy navigation, keyboard-friendly links and forms and searchable, selectable text. We recognize the importance of accessibility, and we welcome queries about accessibility for any of our products. If you have a question or an access need, please get in touch with us at accessibilitysupport@springernature.com.
