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on Risk Management |
| By: | Kan Chen; Tuoyuan Cheng |
| Abstract: | Value at risk (VaR) and expected shortfall (ES) are common high quantile-based risk measures adopted in financial regulations and risk management. In this paper, we propose a tail risk measure based on the most probable maximum size of risk events (MPMR) that can occur over a length of time. MPMR underscores the dependence of the tail risk on the risk management time frame. Unlike VaR and ES, MPMR does not require specifying a confidence level. We derive the risk measure analytically for several well-known distributions. In particular, for the case where the size of the risk event follows a power law or Pareto distribution, we show that MPMR also scales with the number of observations $n$ (or equivalently the length of the time interval) by a power law, $\text{MPMR}(n) \propto n^{\eta}$, where $\eta$ is the scaling exponent. The scale invariance allows for reasonable estimations of long-term risks based on the extrapolation of more reliable estimations of short-term risks. The scaling relationship also gives rise to a robust and low-bias estimator of the tail index (TI) $\xi$ of the size distribution, $\xi = 1/\eta$. We demonstrate the use of this risk measure for describing the tail risks in financial markets as well as the risks associated with natural hazards (earthquakes, tsunamis, and excessive rainfall). |
| Date: | 2022–09 |
| URL: | http://d.repec.org/n?u=RePEc:arx:papers:2209.07092&r= |
| By: | Gabriel Bruneau; Thibaut Duprey; Ruben Hipp |
| Abstract: | We develop a corporate default model to forecast corporate loan losses of the Canadian banking sector under stress. First, we tackle a data gap by reconstructing historical default probabilities for banks’ loan portfolios. Second, we estimate tail elasticities to capture non-linear relationships between macrofinancial conditions and default probabilities. By explicitly modelling default probabilities associated with macroeconomic tail events, this model significantly improves the Bank of Canada’s stress-testing infrastructure. |
| Keywords: | Economic models; Financial institutions; Financial stability; Financial system regulation and policies |
| JEL: | C22 C53 G17 G28 |
| Date: | 2022 |
| URL: | http://d.repec.org/n?u=RePEc:bca:bocatr:122&r= |
| By: | Del Vecchio, Leonardo; Giglio, Carla; Shaw, Frances; Spanò, Guido; Cappelletti, Giuseppe |
| Abstract: | One important source of systemic risk can arise from asset commonality among financial institutions. This indirect interconnection may occur when financial institutions invest in similar or correlated assets and is also described as overlapping portfolios. In this work, we propose a methodology to quantify systemic risk derived from asset commonality and we apply it to assess the degree of indirect interconnection of banks due to their financial holdings. Based on granular information of asset holdings of European significant banks, we compute the sensitivity based ∆ CoVaR which captures the potential sources of systemic risk originating from asset commonality. The novel indicator proves to be consistent with other indicators of systemic importance, yet it has a more transparent foundation in terms of the source of systemic risk, which can contribute to effective macroprudential supervision. JEL Classification: C58, E32, G01, G12, G18, G20, G32 |
| Keywords: | CoVaR, Financial networks, Financial regulation, Overlapping portfolios, Systemic risk |
| Date: | 2022–09 |
| URL: | http://d.repec.org/n?u=RePEc:ecb:ecbwps:20222725&r= |
| By: | Pablo Garcia Sanchez |
| Abstract: | I model two ways climate tail risk could threaten the resources available for consumption in an otherwise standard cake-eating problem. I show that precautionary behaviour is optimal, no matter how low the probability of catastrophic climate outcomes. |
| Keywords: | Climate change, Tail Risk, Tipping Point |
| JEL: | E20 Q50 |
| Date: | 2022–08 |
| URL: | http://d.repec.org/n?u=RePEc:bcl:bclwop:bclwp164&r= |
| By: | D Barrera (UNIANDES); S Cr\'epey (LPSM, UPCit\'e); E Gobet (CMAP, X); Hoang-Dung Nguyen (LPSM, UPCit\'e); B Saadeddine (UPS) |
| Abstract: | We propose a non-asymptotic convergence analysis of a two-step approach to learn a conditional value-at-risk (VaR) and expected shortfall (ES) in a nonparametric setting using Rademacher and Vapnik-Chervonenkis bounds. Our approach for the VaR is extended to the problem of learning at once multiple VaRs corresponding to different quantile levels. This results in efficient learning schemes based on neural network quantile and least-squares regressions. An a posteriori Monte Carlo (non-nested) procedure is introduced to estimate distances to the ground-truth VaR and ES without access to the latter. This is illustrated using numerical experiments in a Gaussian toy-model and a financial case-study where the objective is to learn a dynamic initial margin. |
| Date: | 2022–09 |
| URL: | http://d.repec.org/n?u=RePEc:arx:papers:2209.06476&r= |
| By: | Nathan Foley-Fisher; Nathan Heinrich; Stéphane Verani |
| Abstract: | In this note, we describe U.S. life insurers’ liquidity management when the COVID-19 pandemic broke. We show that life insurance companies immediately created cash buffers to manage potential liquidity shocks. |
| Date: | 2022–08–23 |
| URL: | http://d.repec.org/n?u=RePEc:fip:fedgfn:2022-08-23&r= |
| By: | Ian Dew-Becker |
| Abstract: | This paper describes the response of the economy to large shocks in a nonlinear production network. While arbitrary combinations of shocks can be studied, it focuses on a sector's tail centrality, which quantifies the effect of a large negative shock to the sector – a measure of the systemic risk of each sector. Tail centrality is theoretically and empirically very different from local centrality measures such as sales share – in a benchmark case, it is measured as a sector's average downstream closeness to final production. The paper then uses the results to analyze the determinants of total tail risk in the economy. Increases in interconnectedness in the presence of complementarity can simultaneously reduce the sensitivity of the economy to small shocks while increasing the sensitivity to large shocks. Tail risk is strongest in economies that display conditional granularity, where some sectors become highly influential following negative shocks. |
| JEL: | D24 D57 E13 E32 |
| Date: | 2022–09 |
| URL: | http://d.repec.org/n?u=RePEc:nbr:nberwo:30479&r= |
| By: | Aur\'elien Alfonsi; Nerea Vadillo |
| Abstract: | This paper develops a new stochastic volatility model for the temperature that is a natural extension of the Ornstein-Uhlenbeck model proposed by Benth and Benth (2007). This model allows to be more conservative regarding extreme events while keeping tractability. We give a method based on Conditional Least Squares to estimate the parameters on daily data and estimate our model on eight major European cities. We then show how to calculate efficiently the average payoff of weather derivatives both by Monte-Carlo and Fourier transform techniques. This new model allows to better assess the risk related to temperature volatility. |
| Date: | 2022–09 |
| URL: | http://d.repec.org/n?u=RePEc:arx:papers:2209.05918&r= |
| By: | Wenliang Hou |
| Abstract: | Planning for retirement has always been hard, because retirees face numerous risks – including outliving their money (longevity risk), investment losses (market risk), unexpected health expenses (health risk), the unforeseen needs of family members (family risk), and even retirement benefit cuts (policy risk). The questions are: 1) How important are these risks? and 2) Do retirees properly perceive these risks when making their consumption and investment decisions? To answer these questions, this brief, which is based on an earlier paper, systematically values and ranks the impacts of these various risks from both the objective and subjective perspectives. That is, it quantifies the magnitude of the objective risks that retirees face, repeats the exercise for retirees’ subjective perceptions of the risks, and then compares the two. The analysis, which uses data from the Health and Retirement Study, involves constructing a lifecycle optimization model to quantify each risk by estimating how much wealth retirees are willing to give up to insure against it. The discussion proceeds as follows. The first section presents the background on risks in retirement. The second section discusses the data and methodology. The third section presents the results, showing a significant disconnect between actual and perceived risk. The biggest risk in the objective ranking is longevity risk, followed by health risk and market risk. At the top of the subjective ranking is market risk, which reflects retirees’ exaggerated assessments of market volatility. Perceived longevity risk and health risk rank lower, because retirees are pessimistic about their survival probabilities and often underestimate their health costs in late life. The final section concludes that retirees’ misunderstanding of the importance of various retirement risks highlights the need for more education and provides unique insight into the need for lifetime income, either through Social Security or annuities, which hedge both longevity and market risks. |
| Date: | 2022–07 |
| URL: | http://d.repec.org/n?u=RePEc:crr:issbrf:ib2022-10&r= |
| By: | Leigh Wolfrom |
| Abstract: | The COVID-19 pandemic led to significant economic disruptions and revenue losses for business impacted by workplace closure measures aimed at restraining the spread of the virus. Governments provided extensive monetary and fiscal support to address liquidity risks and mitigate the potential for mass insolvencies as few businesses had applicable insurance coverage for these types of losses. This paper examines the fiscal and insurance sector responses to the economic disruptions resulting from COVID-19 workplace closures, the challenges to the availability of insurance coverage for this risk and some of the challenges and risks related to large-scale fiscal support for businesses. It also includes a discussion of the potential contribution of a loss-sharing arrangement between governments and insurance markets for pandemic-related business interruption losses as a means of enhancing the contribution of insurance markets to providing financial protection in the context of future pandemics. |
| Keywords: | crisis management, fiscal federalism, insurance |
| JEL: | H12 H51 G22 |
| Date: | 2022–09–30 |
| URL: | http://d.repec.org/n?u=RePEc:oec:ctpaab:40-en&r= |
| By: | Capotă, Laura-Dona; Grill, Michael; Molestina Vivar, Luis; Schmitz, Niklas; Weistroffer, Christian |
| Abstract: | The market turmoil in March 2020 highlighted key vulnerabilities in the EU money market fund (MMF) sector. This paper assesses the effectiveness of the EU's regulatory framework from a financial stability perspective, based on a panel analysis of EU MMFs at a daily frequency. First, we find that investment in private debt assets exposes MMFs to liquidity risk. Second, we find that low volatility net asset value (LVNAV) funds, which invest in non-public debt assets while offering a stable NAV, face higher redemptions than other fund types. The risk of breaching the regulatory NAV limit may have incentivised outflows among some LVNAV investors in March 2020. Third, MMFs with lower levels of liquidity buffers use their buffers less than other funds, suggesting low levels of buffer usability in stress periods. Our findings suggest fragility in the EU MMF sector and call for a strengthened regulatory framework of private debt MMFs. JEL Classification: G11, G15, G23, G28 |
| Keywords: | COVID-19, financial fragility, money market funds, regulation |
| Date: | 2022–10 |
| URL: | http://d.repec.org/n?u=RePEc:ecb:ecbwps:20222737&r= |
| By: | Bagnara, Matteo; Jappelli, Ruggero |
| Abstract: | It is well established that investors price market liquidity risk. Yet, there exists no financial claim contingent on liquidity. We propose a contract to hedge uncertainty over future transaction costs, detailing potential buyers and sellers. Introducing liquidity derivatives in Brunnermeier and Pedersen (2009) improves financial stability by mitigating liquidity spirals. We simulate liquidity option prices for a panel of NYSE stocks spanning 2000 to 2020 by fitting a stochastic process to their bidask spreads. These contracts reduce the exposure to liquidity factors. Their prices provide a novel illiquidity measure reflecting cross-sectional commonalities. Finally, stock returns significantly spread along simulated prices. |
| Keywords: | Asset Pricing,Market Liquidity,Liquidity Risk |
| JEL: | G12 G13 G17 |
| Date: | 2022 |
| URL: | http://d.repec.org/n?u=RePEc:zbw:safewp:358&r= |
| By: | Walch, Florian; Breitenstein, Miriam; Ciummo, Stefania |
| Abstract: | Climate change can be a source of financial risk. This paper examines how credit rating agencies accepted by the Eurosystem incorporate climate change risk in their credit ratings. It also analyses how rating agencies disclose their assessments of climate change risks to rating users. The paper develops an analytical framework to compare the agencies’ definitions, methodologies, assessment models, data usage and disclosure practices. The paper reveals large differences in methodologies and disclosure practices across rating agencies and asset classes. The authors identify three main areas for improvement with respect to climate-related disclosures. These areas concern the level of granularity of definitions of climate change risk, the transparency around models and methods used to estimate the exposure to climate change risk and the disclosure of the magnitude of the impact of material climate change risk on credit ratings. JEL Classification: E52, E58, G24, G32, Q54 |
| Keywords: | climate change, credit rating agencies., credit risk, monetary policy, risk management |
| Date: | 2022–09 |
| URL: | http://d.repec.org/n?u=RePEc:ecb:ecbops:2022303&r= |
| By: | Ruodu Wang; Qinyu Wu |
| Abstract: | Quasi-convexity in probabilistic mixtures is a common and useful property in decision analysis. We study a general class of non-monotone mappings, called the generalized rank-dependent functions, which include the preference models of expected utilities, dual utilities, and rank-dependent utilities as special cases, as well as signed Choquet integrals used in risk management. As one of our main results, quasi-convex (in mixtures) signed Choquet integrals precisely include two parts: those that are convex (in mixtures) and the class of scaled quantile-spread mixtures, and this result leads to a full characterization of quasi-convexity for generalized rank-dependent functions. Seven equivalent conditions for quasi-convexity in mixtures are obtained for dual utilities and signed Choquet integrals. We also illustrate a conflict between convexity in mixtures and convexity in risk pooling among constant-additive mappings. |
| Date: | 2022–09 |
| URL: | http://d.repec.org/n?u=RePEc:arx:papers:2209.03425&r= |
| By: | Aglasan, Serkan; Rejesus, Roderick M. |
| Abstract: | This study examines whether cover crop adoption reduces production risk. A crop insurance loss measure is used as the main measure of downside production risk. To achieve this objective, we utilize a county-level panel data set with information on cover crop adoption rate, crop insurance production losses, and weather variables. The data covers the main corn and soybean production regions in the Midwestern United States (US) for the period 2005 to 2018. We employ linear fixed effects econometric models and a number of robustness checks in the empirical analysis (i.e., including a fractional regression approach, recently developed instrumental variable procedures, and alternative empirical specifications). The estimation methods used take advantage of the panel nature of the data to address various specification and endogeneity issues. Our estimation results suggest that counties with higher cover crop adoption tend to have lower crop insurance losses (and thus have lower downside production risk). |
| Keywords: | Crop Production/Industries |
| Date: | 2022 |
| URL: | http://d.repec.org/n?u=RePEc:ags:aaea22:324776&r= |
| By: | Budnik, Katarzyna; Panos, Jiri; Boucherie, Louis |
| Abstract: | This paper proposes a methodology for measuring the macroprudential policy stance based on a distance-to-tail metric perspective. This approach employs a large-scale semi-structural model reflecting the dynamics of 91 significant euro area banks and 19 euro area economies and is presented through an assessment of the stance evolution for the aggregate euro area economy and for the individual euro area countries. Our results uncover mild tightening of the macroprudential policy stance before the end of 2019. This trend is abruptly interrupted at the onset of the Covid-19 pandemic but reappears at the end of 2020 before picking up again over the first half of 2021. Our assessment also reveals a marginal impact of the macro-financial policies applied, which is particularly notable throughout 2020. JEL Classification: E37, E58, G21, G28 |
| Keywords: | distance-to-tail metric, growth-at-Risk, lending-at-Risk, macroprudential policy, macroprudential policy stance |
| Date: | 2022–09 |
| URL: | http://d.repec.org/n?u=RePEc:ecb:ecbops:2022301&r= |
| By: | Ms. Burcu Hacibedel; Ritong Qu |
| Abstract: | In this paper, we study systemic non-financial corporate sector distress using firm-level probabilities of default (PD), covering 55 economies, and spanning the last three decades. Systemic corporate distress is identified by elevated PDs across a large portion of the firms in an economy. A machine-learning based early warning system is constructed to predict the onset of distress in one year’s time. Our results show that credit expansion, monetary policy tightening, overvalued stock prices, and debt-linked balance-sheet weaknesses predict corporate distress. We also find that systemic corporate distress events are associated with contractions in GDP and credit growth in advanced and emerging markets at different degrees and milder than financial crises. |
| Keywords: | Nonfinancial sector; Probability of default; Early warning systems; Macroprudential policy; balance-sheet weakness; appendix B constructing predictor; distress events; appendix C machine learning model; PD indices; Corporate sector; Banking crises; Credit; Financial statements; Global |
| Date: | 2022–07–29 |
| URL: | http://d.repec.org/n?u=RePEc:imf:imfwpa:2022/153&r= |
| By: | Francesco Caloia; David-Jan Jansen |
| Abstract: | If climate change continues unabated, extreme weather events are expected to occur more frequently. Rising flood incidence will especially affect low-lying countries, both through property damage and macro-financial adversity. Using a stress test framework and geocoded data on real-estate exposures for Dutch banks, we study when floods would start impairing financial stability. We find that the banking sector is capitalised sufficiently to withstand floods in unprotected areas, where there is relatively little real estate. However, capital depletions would increase quickly in case more severe floods hit the densely-populated western part of the Netherlands. These findings have possible implications for various policy areas, including macroprudential policy. |
| Keywords: | financial stability; flood risk; real estate; stress test |
| JEL: | G21 Q54 R30 |
| Date: | 2021–11 |
| URL: | http://d.repec.org/n?u=RePEc:dnb:dnbwpp:730&r= |
| By: | Karalos Arapakis |
| Abstract: | One source of financial risk that older Americans face is high health care expenses. Although the largest share of these expenses is due to predictable insurance premiums,retirees can pay sizable additional out-of-pocket costs as well. Medicare and Medicaid help reduce this risk, but Medicare does not cover every expense and Medicaid covers only households with very low income and assets. Thus, better understanding the extent of this coverage and the remaining burden on individuals is an important issue for retirees and policymakers alike. This brief is based on a recent paper that addresses the following question: How much do retirees pay in lifetime out-of-pocket health costs, excluding premiums and including long-term care? The analysis proceeds in two steps. It first calculates the distribution and evolution of total non-premium health care spending over the lifecycle of retired households. It then determines the amount covered by public and private insurers and subtracts this portion from the total to obtain out-of-pocket spending. By focusing on lifetime totals, this brief captures not only the risk of high expenses in a single year, but also the risk of moderate but persistent expenses that add up to a high cost burden over time. The discussion proceeds as follows. The first section provides a brief background on health care spending and insurance programs. The second section describes the data and methodology. The third section presents the results: total spending on health services, the portion covered by insurance, and what retirees pay out-of-pocket. The final section concludes that lifetime health care spending by retirees above and beyond predictable premiums is high and uncertain. However, Medicare, Medicaid, and other insurers cover a large portion of these expenditures. As a result, 65-year-old households pay, on average, $67,260 in out-of-pocket costs over their remaining lifetime, which is about one-fifth of total non-premium costs. Households at the 90th percentile of the health spending distribution pay twice this amount out-of-pocket, though it represents a similar share of the total. |
| Date: | 2022–08 |
| URL: | http://d.repec.org/n?u=RePEc:crr:issbrf:ib2022-14&r= |
| By: | Minghao Pan |
| Abstract: | This paper studies relations among axioms on individuals' intertemporal choices under risk. The focus is on Risk Averse over Time Lotteries (RATL), meaning that a fixed prize is preferred to a lottery with the same monetary prize but a random delivery time. Though surveys and lab experiments documented RATL choices, Expected Discounted Utility cannot accommodate any RATL. This paper's contribution is two-fold. First, under a very weak form of Independence, we generalize the incompatibility of RATL with two axioms about intertemporal choices: Stochastic Impatience (SI) and No Future Bias. Next, we prove a representation theorem that gives a class of models satisfying RATL and SI everywhere. This illustrates that there is no fundamental conflict between RATL and SI, and leaves open possibility that RATL behavior is caused by Future Bias. |
| Date: | 2022–09 |
| URL: | http://d.repec.org/n?u=RePEc:arx:papers:2209.01790&r= |
| By: | El-Khatib, Youssef; Hatemi-J, Abdulnasser |
| Abstract: | Cryptocurrencies are increasingly utilized by investors and financial institutions worldwide. The current paper proposes a prediction model for a cryptocurrency that encompasses three properties observed in the markets for cryptocurrencies—namely high volatility, illiquidity, and regime shifts. By using Ito calculus, we provide a solution for the suggested stochastic differential equation (SDE) along with a proof. Moreover, numerical simulations are performed and are compared to the real data, which seems to capture the dynamics of the price path of a cryptocurrency in the real markets. |
| Keywords: | Stochastic Modeling, cryptocurrencies, illiquid, high volatility, regime switching, CTMC. |
| JEL: | G1 G12 G17 |
| Date: | 2022–09 |
| URL: | http://d.repec.org/n?u=RePEc:pra:mprapa:114556&r= |
| By: | Larbi Alaoui; Antonio Penta |
| Abstract: | Individuals often attach a special meaning to obtaining a certain goal, and getting past a threshold marks the difference between what they consider a success or a failure. In this paper we take a standard von Neumann-Morgenstern Expected Utility setting with an exogenous reference point that separates success from failure, and define attitudes towards success and failure as features of preferences over lotteries. The distinctive feature of our definitions is that they all concern a local reversal of the decision maker's risk attitude between riskaversion and risk-lovingness across the reference point. Our findings provide a unified view of several well-known models of reference-dependent preferences in economics, finance and psychology, and also include novel representations. Moreover, we introduce orderings over the primitive space of preferences to define different attitudes with which each attitudes can be displayed, and characterize them in terms of the representation, with indices analogous to the well-known Arrow-Pratt index of risk aversion. Our findings shed new light on frequently used notions of reference-dependent preferences, and suggest that new comparative statics analyses be conducted in these settings. Finally, we argue that our framework may prove useful to incorporate, within a standard economic model, behavioral manifestations of personality traits that have received increasing attention within the empirical economics literature. |
| Keywords: | expected utility, loss aversion, aspirations, Risk Aversion, reference-dependence |
| JEL: | D01 D81 |
| Date: | 2022–03 |
| URL: | http://d.repec.org/n?u=RePEc:bge:wpaper:1336&r= |
| By: | Mel Win Khaw; Ziang Li; Michael Woodford |
| Abstract: | In an experiment that elicits subjects’ willingness to pay (WTP) for the outcome of a lottery, we confirm the fourfold pattern of risk attitudes described by Kahneman and Tversky. In addition, we document a systematic effect of stake sizes on the magnitude and sign of the relative risk premium, holding fixed both the probability that a lottery pays off and the sign of its payoff (gain vs. loss). We further show that in our data, there is a log-linear relationship between the monetary payoff of the lottery and WTP, conditional on the probability of the payoff and its sign. We account quantitatively for this relationship, and the way in which it varies with both the probability and sign of the lottery payoff, in a model in which all departures from risk-neutral bidding are attributed to an optimal adaptation of bidding behaviour to the presence of cognitive noise. Moreover, the cognitive noise required by our hypothesis is consistent with patterns of bias and variability in judgments about numerical magnitudes and probabilities that have been observed in other contexts. |
| Keywords: | prospect theory, fourfold pattern, endogenous precision, cognitive noise |
| JEL: | C91 D03 D81 D87 |
| Date: | 2022 |
| URL: | http://d.repec.org/n?u=RePEc:ces:ceswps:_9923&r= |
| By: | Nghia Chu; Binh Dao; Nga Pham; Huy Nguyen; Hien Tran |
| Abstract: | Predicting fund performance is beneficial to both investors and fund managers, and yet is a challenging task. In this paper, we have tested whether deep learning models can predict fund performance more accurately than traditional statistical techniques. Fund performance is typically evaluated by the Sharpe ratio, which represents the risk-adjusted performance to ensure meaningful comparability across funds. We calculated the annualised Sharpe ratios based on the monthly returns time series data for more than 600 open-end mutual funds investing in listed large-cap equities in the United States. We find that long short-term memory (LSTM) and gated recurrent units (GRUs) deep learning methods, both trained with modern Bayesian optimization, provide higher accuracy in forecasting funds' Sharpe ratios than traditional statistical ones. An ensemble method, which combines forecasts from LSTM and GRUs, achieves the best performance of all models. There is evidence to say that deep learning and ensembling offer promising solutions in addressing the challenge of fund performance forecasting. |
| Date: | 2022–09 |
| URL: | http://d.repec.org/n?u=RePEc:arx:papers:2209.09649&r= |
| By: | Peng Liu; Yanyan Zheng |
| Abstract: | Systematical and precise analysis on the 1-min datasets over the 17-year period 2005-2021 for both the Shanghai Stock Exchange and the Shenzhen Stock Exchange composite index is conducted in this paper. Here we precisely measure the property of return distributions of composite indices over time scale $\Delta t$ ranging from 1 min up to almost 4,000 min, to reveal the difference between the Chinese stock market and the mature stock market in developed countries. The return distributions of composite indices for both exchanges show similar behavior. Main findings in this paper are as follows. (1) The central part of return distribution is well described by symmetrical L$\acute{e}$vy $\alpha$-stable process with stability parameter comparable with the value of about 1.4 extracted in the U.S. stock market. (2) Distinctively, the stability parameter shows a potential change when $\Delta t$ increases, and thus a crossover region located at 15 $ |
| Date: | 2022–09 |
| URL: | http://d.repec.org/n?u=RePEc:arx:papers:2209.08521&r= |