nep-rmg New Economics Papers
on Risk Management
Issue of 2021‒04‒12
24 papers chosen by
Stan Miles
Thompson Rivers University

  1. Artificial intelligence applied to bailout decisions in financial systemic risk management By Daniele Petrone; Neofytos Rodosthenous; Vito Latora
  2. COVID-19: How Will European Banks Fare? By Shekhar Aiyar; Mai Chi Dao; Andreas A. Jobst; Aiko Mineshima; Srobona Mitra; Mahmood Pradhan
  3. FRM Financial Risk Meter for Emerging Markets By Souhir Ben Amor; Michael Althof; Wolfgang Karl H\"ardle
  4. The behaviour of U.S. stocks to financial and health risks By Salissu, Afees; Raheem, Ibrahim; Eigbiremolen, Godstime
  5. Star-shaped Risk Measures By Erio Castagnoli; Giacomo Cattelan; Fabio Maccheroni; Claudio Tebaldi; Ruodu Wang
  6. A reality check on the GARCH-MIDAS volatility models By Virk, Nader; Javed, Farrukh; Awartani, Basel
  7. Bertram's Pairs Trading Strategy with Bounded Risk By Vladim\'ir Hol\'y; Michal \v{C}ern\'y
  8. Some results on the risk capital allocation rule induced by the Conditional Tail Expectation risk measure By Nawaf Mohammed; Edward Furman; Jianxi Su
  9. Industry-Level Baseline Risk of COVID-19 Infection By Lowery, Richard; Canann, Taylor; Carvalho, Carlos
  10. Measuring Systemic Risk in South African Banks By Somnath Chatterjee; Marea Sing
  11. Variational Autoencoders: A Hands-Off Approach to Volatility By Maxime Bergeron; Nicholas Fung; John Hull; Zissis Poulos
  12. Uncertainty, sentiments and time-varying risk premia By Berardi, Michele
  13. Quantum crypto-economics: Blockchain prediction markets for the evolution of quantum technology By Peter P. Rohde; Vijay Mohan; Sinclair Davidson; Chris Berg; Darcy Allen; Gavin K. Brennen; Jason Potts
  14. A deep learning model for gas storage optimization By Nicolas Curin; Michael Kettler; Xi Kleisinger-Yu; Vlatka Komaric; Thomas Krabichler; Josef Teichmann; Hanna Wutte
  15. Influence of risk tolerance on long-term investments: A Malliavin calculus approach By Hyungbin Park
  16. Dual theory of choice with multivariate risks By Alfred Galichon; Marc Henry
  17. Perpetual callable American volatility options in a mean-reverting volatility model By Hsuan-Ku Liu
  18. Identifying structural shocks to volatility through a proxy-MGARCH model By Fengler, Matthias; Polivka, Jeannine
  19. Valuing Switching Options with the Moving-Boundary Method By Dabadghao, Shaunak S.; Chockalingam, Arun; Soltani, Taimaz; Fransoo, Jan C.
  20. JDOI Variance Reduction Method and the Pricing of American-Style Options By Auster Johan; Mathys Ludovic; Maeder Fabio
  21. Insurance Business and Sustainable Development By Dietmar Pfeifer; Vivien Langen
  22. Analytic formula for option margin with liquidity costs under dynamic delta hedging By Kyungsub Lee; Byoung Ki Seo
  23. Benefits of Enterprise Risk Management: A Systematic Review of Literature By Ch V V S N V Prasad
  24. Risk assessment for micro companies belonging to selected economic branches of the professional, scientific and technical services sector in Mexico through the Beta coefficient. By Flores Sánchez, Edgar Mauricio; Rodríguez Batres, Axel; Varela Espidio, Joaquín Bernardo

  1. By: Daniele Petrone; Neofytos Rodosthenous; Vito Latora
    Abstract: We describe the bailout of banks by governments as a Markov Decision Process (MDP) where the actions are equity investments. The underlying dynamics is derived from the network of financial institutions linked by mutual exposures, and the negative rewards are associated to the banks' default. Each node represents a bank and is associated to a probability of default per unit time (PD) that depends on its capital and is increased by the default of neighbouring nodes. Governments can control the systemic risk of the network by providing additional capital to the banks, lowering their PD at the expense of an increased exposure in case of their failure. Considering the network of European global systemically important institutions, we find the optimal investment policy that solves the MDP, providing direct indications to governments and regulators on the best way of action to limit the effects of financial crises.
    Date: 2021–02
    URL: http://d.repec.org/n?u=RePEc:arx:papers:2102.02121&r=all
  2. By: Shekhar Aiyar; Mai Chi Dao; Andreas A. Jobst; Aiko Mineshima; Srobona Mitra; Mahmood Pradhan
    Abstract: This paper evaluates the impact of the crisis on European banks’ capital under a range of macroeconomic scenarios, using granular data on the size and riskiness of sectoral exposures. The analysis incorporates the important role of pandemic-related policy support, including not only regulatory relief for banks, but also policies to support businesses and households, which act to shield the financial sector from the real economic shock.
    Keywords: Corporate profits;Bank risk management;Bank capital;Stress testing;Europe;Bank capital;stress test;COVID-19;bank profitability; corporate risk
    Date: 2021–03–26
    URL: http://d.repec.org/n?u=RePEc:imf:imfdps:2021/008&r=all
  3. By: Souhir Ben Amor; Michael Althof; Wolfgang Karl H\"ardle
    Abstract: The fast-growing Emerging Market (EM) economies and their improved transparency and liquidity have attracted international investors. However, the external price shocks can result in a higher level of volatility as well as domestic policy instability. Therefore, an efficient risk measure and hedging strategies are needed to help investors protect their investments against this risk. In this paper, a daily systemic risk measure, called FRM (Financial Risk Meter) is proposed. The FRM-EM is applied to capture systemic risk behavior embedded in the returns of the 25 largest EMs FIs, covering the BRIMST (Brazil, Russia, India, Mexico, South Africa, and Turkey), and thereby reflects the financial linkages between these economies. Concerning the Macro factors, in addition to the Adrian and Brunnermeier (2016) Macro, we include the EM sovereign yield spread over respective US Treasuries and the above-mentioned countries currencies. The results indicated that the FRM of EMs FIs reached its maximum during the US financial crisis following by COVID 19 crisis and the Macro factors explain the BRIMST FIs with various degrees of sensibility. We then study the relationship between those factors and the tail event network behavior to build our policy recommendations to help the investors to choose the suitable market for in-vestment and tail-event optimized portfolios. For that purpose, an overlapping region between portfolio optimization strategies and FRM network centrality is developed. We propose a robust and well-diversified tail-event and cluster risk-sensitive portfolio allocation model and compare it to more classical approaches
    Date: 2021–02
    URL: http://d.repec.org/n?u=RePEc:arx:papers:2102.05398&r=all
  4. By: Salissu, Afees; Raheem, Ibrahim; Eigbiremolen, Godstime
    Abstract: This article examines the hedging effectiveness of U.S. stocks against uncertainties due to equity market (financial risk) and pandemics (health risk), including Covid-19 pandemic. Consequently, we consider two categories of U.S. stocks—defensive and non-defensive stocks drawn from 10 different sectors and distinctly analysed over two data samples—pre- and post-Covid periods. We construct a predictive panel data model that simultaneously accounts for both heterogeneity and common correlated effects and also complementarily determine the predictive power of accounting for uncertainties in the valuation of U.S. stocks. We find that hedging effectiveness is driven by the types of stocks and measures of uncertainty. Defensive stocks provide a good hedge for pandemic-induced uncertainty, and the hedging effectiveness is higher during calm market conditions as compared to turbulent conditions, while both categories lack hedging capability in the face of equity-induced uncertainty. Finally, we find that the inclusion of uncertainty in the predictive model of U.S. stock returns improves its forecasts and this conclusion is robust to alternative measures of uncertainty and multiple forecast horizons.
    Keywords: Covid-19, defensive stocks, forecast evaluation, non-defensive stocks, pandemics, panel data, uncertainty, U.S. stocks
    JEL: F15 G10
    Date: 2020–12
    URL: http://d.repec.org/n?u=RePEc:pra:mprapa:105354&r=all
  5. By: Erio Castagnoli; Giacomo Cattelan; Fabio Maccheroni; Claudio Tebaldi; Ruodu Wang
    Abstract: In this paper monetary risk measures that are positively superhomogeneous, called star-shaped risk measures, are characterized and their properties studied. The measures in this class, which arise when the controversial subadditivity property of coherent risk measures is dispensed with and positive homogeneity is weakened, include all practically used risk measures, in particular, both convex risk measures and Value-at-Risk. From a financial viewpoint, our relaxation of convexity is necessary to quantify the capital requirements for risk exposure in the presence of competitive delegation or robust aggregation mechanisms. From a decision theoretical perspective, star-shaped risk measures emerge from variational preferences when risk mitigation strategies can be adopted by a rational decision maker.
    Date: 2021–03
    URL: http://d.repec.org/n?u=RePEc:arx:papers:2103.15790&r=all
  6. By: Virk, Nader (Plymouth Business School); Javed, Farrukh (Örebro University School of Business); Awartani, Basel (Westminster Business School)
    Abstract: We employ a battery of model evaluation tests for a broad-set of GARCH-MIDAS models and account for data snooping bias. We document that inferences based on standard tests for GM variance components can be misleading. Our data mining free results show that the gains of macro-variables in forecasting total (long run) variance by GM models are overstated (understated). Estimation of different components of volatility is crucial for designing differentiated investing strategies, risk management plans and pricing of derivative securities. Therefore, researchers and practitioners should be wary of data mining bias, which may contaminate a forecast that may appear statistically validated using robust evaluation tests.
    Keywords: GARCH-MIDAS models; component variance forecasts; macro-variables; data snooping
    JEL: C32 C52 G11 G17
    Date: 2021–03–30
    URL: http://d.repec.org/n?u=RePEc:hhs:oruesi:2021_002&r=all
  7. By: Vladim\'ir Hol\'y; Michal \v{C}ern\'y
    Abstract: Finding Bertram's optimal trading strategy for a pair of cointegrated assets following the Ornstein-Uhlenbeck price difference process can be formulated as an unconstrained convex optimization problem for maximization of expected profit per unit of time. We generalize this model to the form where the riskiness of profit, measured by its per-time-unit volatility, is controlled (e.g. in case of existence of limits on riskiness of trading strategies imposed by regulatory bodies). The resulting optimization problem need not be convex. In spite of this undesirable fact, we demonstrate that it is still efficiently solvable. We also investigate the problem critical for practice that parameters of the price difference process are never known exactly and are imprecisely estimated from an observed finite sample. We show how the imprecision affects the optimal trading strategy and quantify the loss caused by the imprecise estimate compared to a theoretical trader knowing the parameters exactly.
    Date: 2021–02
    URL: http://d.repec.org/n?u=RePEc:arx:papers:2102.04160&r=all
  8. By: Nawaf Mohammed; Edward Furman; Jianxi Su
    Abstract: Risk capital allocations (RCAs) are an important tool in quantitative risk management, where they are utilized to, e.g., gauge the profitability of distinct business units, determine the price of a new product, and conduct the marginal economic capital analysis. Nevertheless, the notion of RCA has been living in the shadow of another, closely related notion, of risk measure (RM) in the sense that the latter notion often shapes the fashion in which the former notion is implemented. In fact, as the majority of the RCAs known nowadays are induced by RMs, the popularity of the two are apparently very much correlated. As a result, it is the RCA that is induced by the Conditional Tail Expectation (CTE) RM that has arguably prevailed in scholarly literature and applications. Admittedly, the CTE RM is a sound mathematical object and an important regulatory RM, but its appropriateness is controversial in, e.g., profitability analysis and pricing. In this paper, we address the question as to whether or not the RCA induced by the CTE RM may concur with alternatives that arise from the context of profit maximization. More specifically, we provide exhaustive description of all those probabilistic model settings, in which the mathematical and regulatory CTE RM may also reflect the risk perception of a profit-maximizing insurer.
    Date: 2021–02
    URL: http://d.repec.org/n?u=RePEc:arx:papers:2102.05003&r=all
  9. By: Lowery, Richard; Canann, Taylor; Carvalho, Carlos (Mercury Publication)
    Abstract: Abstract not available.
    Date: 2020–06–11
    URL: http://d.repec.org/n?u=RePEc:ajw:wpaper:10671&r=all
  10. By: Somnath Chatterjee; Marea Sing
    Abstract: MeasuringSystemicRiskinSouthAfricanBanks
    Date: 2021–04–06
    URL: http://d.repec.org/n?u=RePEc:rbz:wpaper:11004&r=all
  11. By: Maxime Bergeron; Nicholas Fung; John Hull; Zissis Poulos
    Abstract: A volatility surface is an important tool for pricing and hedging derivatives. The surface shows the volatility that is implied by the market price of an option on an asset as a function of the option's strike price and maturity. Often, market data is incomplete and it is necessary to estimate missing points on partially observed surfaces. In this paper, we show how variational autoencoders can be used for this task. The first step is to derive latent variables that can be used to construct synthetic volatility surfaces that are indistinguishable from those observed historically. The second step is to determine the synthetic surface generated by our latent variables that fits available data as closely as possible. As a dividend of our first step, the synthetic surfaces produced can also be used in stress testing, in market simulators for developing quantitative investment strategies, and for the valuation of exotic options. We illustrate our procedure and demonstrate its power using foreign exchange market data.
    Date: 2021–02
    URL: http://d.repec.org/n?u=RePEc:arx:papers:2102.03945&r=all
  12. By: Berardi, Michele
    Abstract: Why are stock prices much more volatile than the underlying dividends? The excess volatility of prices can in principle be attributed to two different causes: time-varying discount rates for expected future dividends, arising from variation in risk premia; or the irrational exuberance of investors, bidding prices up and down even in the absence of changes in the underlying value of the asset. No consensus has so far emerged among economists as to the prevalence of one or the other source of price variation. I propose in this paper a novel way to approach this problem, by identifying changes in the uncertainty faced by investors regarding the fundamental value of an asset and exploiting the different response in prices that such changes in uncertainty would generate through sentiments or risk premia. I then apply this framework to the S&P 500 index from 1872 till 2019: the positive correlation found between uncertainty and prices (or, equivalently, the negative correlation between uncertainty and implied risk premia) is not compatible with rational investors' behavior and suggests instead the presence of a significant sentiments component in stock prices.
    Keywords: uncertainty, risk premium, sentiments; information, financial markets.
    JEL: D81 D83 G12 G14
    Date: 2021–02–18
    URL: http://d.repec.org/n?u=RePEc:pra:mprapa:106922&r=all
  13. By: Peter P. Rohde; Vijay Mohan; Sinclair Davidson; Chris Berg; Darcy Allen; Gavin K. Brennen; Jason Potts
    Abstract: Two of the most important technological advancements currently underway are the advent of quantum technologies, and the transitioning of global financial systems towards cryptographic assets, notably blockchain-based cryptocurrencies and smart contracts. There is, however, an important interplay between the two, given that, in due course, quantum technology will have the ability to directly compromise the cryptographic foundations of blockchain. We explore this complex interplay by building financial models for quantum failure in various scenarios, including pricing quantum risk premiums. We call this quantum crypto-economics.
    Date: 2021–02
    URL: http://d.repec.org/n?u=RePEc:arx:papers:2102.00659&r=all
  14. By: Nicolas Curin; Michael Kettler; Xi Kleisinger-Yu; Vlatka Komaric; Thomas Krabichler; Josef Teichmann; Hanna Wutte
    Abstract: To the best of our knowledge, the application of deep learning in the field of quantitative risk management is still a relatively recent phenomenon. In this article, we utilize techniques inspired by reinforcement learning in order to optimize the operation plans of underground natural gas storage facilities. We provide a theoretical framework and assess the performance of the proposed method numerically in comparison to a state-of-the-art least-squares Monte-Carlo approach. Due to the inherent intricacy originating from the high-dimensional forward market as well as the numerous constraints and frictions, the optimization exercise can hardly be tackled by means of traditional techniques.
    Date: 2021–02
    URL: http://d.repec.org/n?u=RePEc:arx:papers:2102.01980&r=all
  15. By: Hyungbin Park
    Abstract: This study investigates the influence of risk tolerance on the expected utility in the long run. We estimate the extent to which the expected utility of optimal portfolios is affected by small changes in the risk tolerance. For this purpose, we adopt the Malliavin calculus method and the Hansen--Scheinkman decomposition, through which the expected utility is expressed in terms of the eigenvalues and eigenfunctions of an operator. We conclude that the influence of risk aversion on the expected utility is determined by these eigenvalues and eigenfunctions in the long run.
    Date: 2021–04
    URL: http://d.repec.org/n?u=RePEc:arx:papers:2104.00911&r=all
  16. By: Alfred Galichon; Marc Henry
    Abstract: We propose a multivariate extension of Yaari's dual theory of choice under risk. We show that a decision maker with a preference relation on multidimensional prospects that preserves first order stochastic dominance and satisfies comonotonic independence behaves as if evaluating prospects using a weighted sum of quantiles. Both the notions of quantiles and of comonotonicity are extended to the multivariate framework using optimal transportation maps. Finally, risk averse decision makers are characterized within this framework and their local utility functions are derived. Applications to the measurement of multi-attribute inequality are also discussed.
    Date: 2021–02
    URL: http://d.repec.org/n?u=RePEc:arx:papers:2102.02578&r=all
  17. By: Hsuan-Ku Liu
    Abstract: This paper investigates problems associated with the valuation of callable American volatility put options. Our approach involves modeling volatility dynamics as a mean-reverting 3/2 volatility process. We first propose a pricing formula for the perpetual American knock-out put. Under the given conditions, the value of perpetual callable American volatility put options is discussed.
    Date: 2021–04
    URL: http://d.repec.org/n?u=RePEc:arx:papers:2104.01127&r=all
  18. By: Fengler, Matthias; Polivka, Jeannine
    Abstract: We extend the classical MGARCH specification for volatility modeling by developing a structural MGARCH model targeting identification of shocks and volatility spillovers in a speculative return system. Similarly to the proxy-sVAR framework, we work with auxiliary proxy variables constructed from news-related measures to identify the underlying shock system. We achieve full identification with multiple proxies by chaining Givens rotations. In an empirical application, we identify an equity, bond and currency shock. We study the volatility spillovers implied by these labelled structural shocks. Our analysis shows that symmetric spillover regimes are rejected.
    Keywords: Givens rotations, identification, news-based measures, proxy-MGARCH, shock labelling, structural innovations, volatility spillovers
    JEL: C32 C51 C58 G12
    Date: 2021–04
    URL: http://d.repec.org/n?u=RePEc:usg:econwp:2021:03&r=all
  19. By: Dabadghao, Shaunak S.; Chockalingam, Arun; Soltani, Taimaz; Fransoo, Jan C. (Tilburg University, School of Economics and Management)
    Date: 2021
    URL: http://d.repec.org/n?u=RePEc:tiu:tiutis:45fe7e78-129f-4d41-ac2f-594a7a66789e&r=all
  20. By: Auster Johan; Mathys Ludovic; Maeder Fabio
    Abstract: The present article revisits the Diffusion Operator Integral (DOI) variance reduction technique originally proposed in Heath and Platen (2002) and extends its theoretical concept to the pricing of American-style options under (time-homogeneous) L\'evy stochastic differential equations. The resulting Jump Diffusion Operator Integral (JDOI) method can be combined with numerous Monte Carlo based stopping-time algorithms, including the ubiquitous least-squares Monte Carlo (LSMC) algorithm of Longstaff and Schwartz (cf. Carriere (1996), Longstaff and Schwartz (2001)). We exemplify the usefulness of our theoretical derivations under a concrete, though very general jump-diffusion stochastic volatility dynamics and test the resulting LSMC based version of the JDOI method. The results provide evidence of a strong variance reduction when compared with a simple application of the LSMC algorithm and proves that applying our technique on top of Monte Carlo based pricing schemes provides a powerful way to speed-up these methods.
    Date: 2021–04
    URL: http://d.repec.org/n?u=RePEc:arx:papers:2104.01365&r=all
  21. By: Dietmar Pfeifer; Vivien Langen
    Abstract: In this study, we will discuss recent developments in risk management of the global financial and insurance business with respect to sustainable development. So far climate change aspects have been the dominant aspect in managing sustainability risks and opportunities, accompanied by the development of several legislative initiatives triggered by supervisory authorities. However, a sole concentration on these aspects misses out other important economic and social facets of sustainable development goals formulated by the UN. Such aspects have very recently come into the focus of the European Committee concerning the Solvency II project for the European insurance industry. Clearly the new legislative expectations can be better handled by larger insurance companies and holdings than by small- and medium-sized mutual insurance companies which are numerous in central Europe, due to their historic development starting in the late medieval ages and early modern times. We therefore also concentrate on strategies within the risk management of such small- and medium-sized enterprises that can be achieved without much effort, in particular those that are not directly related to climate change.
    Date: 2021–02
    URL: http://d.repec.org/n?u=RePEc:arx:papers:2102.02612&r=all
  22. By: Kyungsub Lee; Byoung Ki Seo
    Abstract: This study derives the expected liquidity cost when performing the delta hedging process of a European option. This cost is represented by an integration formula that includes European option prices and a certain function depending on the delta process. We first define a unit liquidity cost and then show that the liquidity cost is a multiplication of the unit liquidity cost, stock price, supply curve parameter, and the square of the number of options. Using this formula, the expected liquidity cost before hedging can be calculated much faster than when using a Monte Carlo simulation. Numerically computed distributions of liquidity costs in special cases are also provided.
    Date: 2021–03
    URL: http://d.repec.org/n?u=RePEc:arx:papers:2103.15302&r=all
  23. By: Ch V V S N V Prasad (BITS Pilani K K Birla Campus, Zuarinagar, 403726, Goa, India Author-2-Name: Sankalp Naik Author-2-Workplace-Name: BITS Pilani K K Birla Campus, Zuarinagar, 403726, Goa, India Author-3-Name: Author-3-Workplace-Name: Author-4-Name: Author-4-Workplace-Name: Author-5-Name: Author-5-Workplace-Name: Author-6-Name: Author-6-Workplace-Name: Author-7-Name: Author-7-Workplace-Name: Author-8-Name: Author-8-Workplace-Name:)
    Abstract: Objective - In an enhanced climate of risk complexities, the firm's stakeholders desire a risk management framework that promises the benefits of efficiencies, transparencies, and solutions for interrelated risks. Enterprise risk management (ERM) is widely seen as a suitable instrument to address these issues. However, not all are convinced of ERM's benefits. This necessitates a review of extant literature and collating it to generate interrelated insights. This paper reviews articles on ERM from the management and finance domain and catalogs the benefits of ERM. Methodology/Technique: – This paper reviews 129 articles addressing ERM benefits. It examines the academic disciplines of journals publishing ERM studies by looking into their H Indices, SJR scores, and ABDC rankings to assess ERM's impact and acceptability among scholars. The research articles are analyzed for their subject domains, geographic scope, and methodology used in exploring the relationship between ERM adoption and its benefits to the firm. Collating and reviewing these articles enables the mitigation of data gaps. These studies were primarily from accounting, finance, management, corporate governance, and strategy domains. Findings – Improved cost-effectiveness, earnings stability, increased profitability, improved decision making, better risk communication, competitive advantage, better resource allocation, enhanced firm value, and performance are the key benefits of ERM adoption identified in this study. A knowledge gap is presented around assessing ERM benefits and extending ERM research scope to developing countries like India. Novelty – The study catalogs the benefits of ERM and makes a strong case for ERM adoption among firms. Type of Paper - Review
    Keywords: Enterprise risk management (ERM); firm value; firm performance; ERM benefits; Covid19
    JEL: M10 M14 G30 G32
    Date: 2021–03–31
    URL: http://d.repec.org/n?u=RePEc:gtr:gatrjs:jfbr181&r=all
  24. By: Flores Sánchez, Edgar Mauricio; Rodríguez Batres, Axel; Varela Espidio, Joaquín Bernardo
    Abstract: Companies are the economic units that generate economic growth for any country or region by offering the goods and services demanded by society at the same time that they generate a large number and variety of jobs, specifically the micro companies belonging to the sector of the professional, technical and scientific services in Mexico contribute with a significant number of companies and employment generated. Despite this, micro enterprises do not have optimal access to the financing they need to continue growing, mainly due to poor risk management. The valuation of companies is a discipline that offers various methods to determine the risk of an asset, highlighting the Capital Asset Pricing Model which requires the estimation of Beta risk coefficients and which has been applied in a general way to large companies belonging to developed countries. The present work is focused on developing an own index of micro companies in Mexico and determining the respective risk coefficients for nine economic branches of the mentioned subsector. The final results made it possible to identify the nine Beta coefficients for the chosen branches of the professional, scientific and technical services sector, in such a way that said risk indicators can be used to calculate the value and risk associated with the micro-companies of said branches of the economy in Mexico; providing a valuation alternative to improve access to financing.
    Keywords: valuation, risk betas, micro companies, professional, technical and scientific services.
    JEL: G11 G12 L84
    Date: 2021
    URL: http://d.repec.org/n?u=RePEc:pra:mprapa:105727&r=all

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