Quantitative methods in economics and finance

Kliestik, Tomas (Ed.); Valaskova, Katarina (Ed.); Kovacova, Maria (Ed.) Book — Published Version Quantitative methods in economics and finance Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Kliestik, Tomas (Ed.); Valaskova, Katarina (Ed.); Kovacova, Maria (Ed.) (2021) : Quantitative methods in economics and finance, ISBN 978-3-0365-0537-4, MDPI, Basel, https://doi.org/10.3390/books978-3-0365-0537-4 This Version is available at: https://hdl.handle.net/10419/237809 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/ Quantitative Methods in Economics and Finance Printed Edition of the Special Issue Published in Risks www.mdpi.com/journal/risks Tomas Kliestik, Katarina Valaskova and Maria Kovacova Edited by Quantitative Methods in Economics and Finance • Tomas Kliestik, Katarina Valaskova and Maria Kovacova Quantitative Methods in Economics and Finance Quantitative Methods in Economics and Finance Editors Tomas Kliestik Katarina Valaskova Maria Kovacova MDPI •Basel •Beijing •Wuhan •Barcelona •Belgrade •Manchester •Tokyo •Cluj •Tianjin Editors Tomas Kliestik University of Zilina Slovakia Katarina Valaskova University of Zilina Slovakia Maria Kovacova University of Zilina Slovakia Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Risks (ISSN 2227-9091) (available at: https://www.mdpi.com/journal/risks/special issues/Quantitative Methods Economics Finance). For citation purposes, cite each article independently as indicated on the article page online and as indicated below: LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. Journal Name Year,Volume Number, Page Range. ISBN 978-3-0365-0536-7 (Hbk) ISBN 978-3-0365-0537-4 (PDF) © 2021 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND. Contents About the Editors ..............................................vii Preface to ”Quantitative Methods in Economics and Finance” ................... ix Sergey A. Vasiliev and Eugene R. Serov Omnichannel Banking Economy Reprinted from: Risks 2019,7, 115, doi:10.3390/risks7040115 ..................... 1 Adam Marszk and Ewa Lechman Application of Diffusion Models in the Analysis of Financial Markets: Evidence on Exchange Traded Funds in Europe Reprinted from: Risks 2020,8, 18, doi:10.3390/risks8010018 ...................... 13 Pavol Durana, Katarina Valaskova, Darina Chlebikova, Vladislav Krastev and Irina Atanasova Heads and Tails of Earnings Management: Quantitative Analysis in Emerging Countries Reprinted from: Risks 2020,8, 57, doi:10.3390/risks8020057 ...................... 37 Bernd Engelmann and Ha Pham A Raroc Valuation Scheme for Loans and Its Application in Loan Origination Reprinted from: Risks 2020,8, 63, doi:10.3390/risks8020063 ...................... 59 Long Hai Vo and Duc Hong Vo Modelling Australian Dollar Volatility at Multiple Horizons with High-Frequency Data Reprinted from: Risks 2020,8, 89, doi:10.3390/risks8030089 ...................... 79 Zbigniew Palmowski and Tomasz Serafin A Note on Simulation Pricing of π-Options Reprinted from: Risks 2020,8, 90, doi:10.3390/risks8030090 ...................... 95 Dmitrii Rodionov, Olesya Perepechko and Olga Nadezhina Determining Economic Security of a Business Based on Valuation of Intangible Assets according to the International Valuation Standards (IVS) Reprinted from: Risks 2020,8, 110, doi:10.3390/risks8040110 .....................115 Zuzana Rowland, George Lazaroiu and Ivana Podhorsk´a Use of Neural Networks to Accommodate Seasonal Fluctuations When Equalizing Time Series for the CZK/RMB Exchange Rate Reprinted from: Risks 2021,9, 1, doi:10.3390/risks9010001 ......................129 v About the Editors Tomas Kliestik (prof., Ph.D.) is a professor and the head of the Department of Economics, Faculty of Operation and Economics of Transport and Communications, University of Zilina. His application, technical, and scientific activities are focused mainly on the issue of the application of quantitative mathematical-statistical methods in financial management and decision-making process of companies, data envelopment analysis, neural networks, genetic algorithms, fuzzy logic, multivariate statistical methods, risk quantification and analysis, etc. His findings are published in domestic and foreign scientific monographs, academic publications, lecture notes, and the outputs of his research are published in indexed and impact scientific journals (Q1–Q2). More than 1600 citations have been recorded for his publications. His current Hirsch index is 19 in the Web of Science database and 18 in Scopus. He is also a member of the editorial boards of several journals and a member of scientific committees of international scientific conferences, he is also a guarantor and editor of the conference Globalization and its Socio-Economic Consequences. Katarina Valaskova (Ph.D.) is an associate professor at the Department of Economics, Faculty of Operation and Economics of Transport and Communication, University of Zilina. Her research activities are mostly focused on financial and investment management, risk management, and business economics. She is an author of domestic and foreign monographs, academic publications, lecture notes, and the outputs of her research are published in indexed and impact scientific journals (Q1–Q2). More than 800 citations have been recorded for her publications. Her current Hirsch index is 13 in the Web of Science database and 12 in Scopus. She is also a member of the editorial boards of several journals and a member of scientific committees of international scientific conferences. Maria Kovacova (Ph.D.) is an assistant professor at the Department of Economics, Faculty of Operation and Economics of Transport and Communication, University of Zilina. Her research activities are mostly focused on financial and investment management and financial markets. She is an author of domestic and foreign monographs, academic publications, lecture notes, and the outputs of her research are published in indexed and impact scientific journals (Q1–Q2). More than 900 citations have been recorded for her publications. Her current Hirsch index is 13 in the Web of Science database and 12 in Scopus. She is also a member of the editorial boards of several journals and a member of scientific committees of international scientific conferences. vii Risks 2019,7, 115 Direct administrative and management expenses include all payments related to labor remuneration (including taxes and deductions to state funds), as well as expenses for maintaining workplaces (rent, utility bills, communication channels, security, depreciation, property tax, etc.) If necessary, it is possible to take into account the indirect costs for labor and maintenance of workplaces for the management personnel administering the “sellers” of the channel. The universal sales funnel can be divided into three consecutive stages: (1) Bring the client a proposal with the aim of generating interest. (2) Make a request for a product or service with an interested client. (3) Conclude a contract with the client, with subsequent activation of the use of the product. At each of these stages of the general sales cycle, work can be carried out by an employee, or by an automated “machine algorithm”, in the various channels of interaction with the customer. In the model and formulas, the stage number of the sales funnel is indicated by the lower index (1, 2, 3—see Equation (5)). For example, a customer was called by a call center employee, offering to issue a consumer loan. The client promised to think, and after a week he made an application for a loan and insurance via Internet banking. He then applied to the nearest bank office for a cash loan, or, during the next visit to the office to reissue the deposit, the client was offered a payment card. The client, already on his way home, made an order for the card in the online banking mobile application, having issued its delivery to his home by courier. The total cost of the sales cycle is calculated as the product of the total sales duration of each sales cycle and the cost per minute of the employee of the channel carrying out operations in the cycle (Equation (5)): CSCh =N1×T1×CCh1+N2×T2×CCh2+N3×T3×CCh3(5) It is obvious that the number of actions for one sale depends exclusively on the percentage of conversion of “contacts” into “interests”, “interests” into “bids”, and “bids” into “contracts” within the framework of a universal “sales funnel”. The higher the conversion percentage, the lower the number of operations. In assessing the cost of sales, costs are taken into account not only for those operations that ultimately led to the sale of the product, but also for all outstanding transactions and losses. Thus, the model for assessing the value and profitability of sales serves as a tool for making complex management decisions for a number of interrelated parameters: - target customer flow; - conversion of the target customer flow into sales; - the number of sellers in the channel; - specific sales productivity for one seller in the channel for the period (day, month, quarter); - the cost of 1 min of work channel employee; - the standard time of one operation in the context of products, channels, stages of the sales cycle; - the number of actions/operations required for the implementation of one sale of the product in the channel. Moreover, in the framework of the omnichannel service model, the possibility of separate communication of each channel with customers at different stages of the sales funnel creates the potential to optimize costs by building omnichannel chains that minimize costs, which has been taken into account in the proposed model, which separately estimates the cost of each stage of the sales funnel. 3. Results By modeling the sales process using selected key factors affecting the overall effectiveness of transactions within the framework of a typical sales funnel, the following ways to increase efficiency 4 Risks 2019,7, 115 were proposed. According to the model, in order to optimize the cost of sales, it is necessary to (Serov 2018): (1) Reduce regulatory time for rendering operations, introducing new technologies, and optimizing processes; (2) Reduce the cost of 1 min of work of an employee by selecting channels with the lowest cost of maintaining jobs; (3) Reduce the number of transactions required for a sale, automating the processes and selecting channels or sales scenarios with the highest conversion of target client flow into sales. To test the working capacity in practical conditions, the omnichannel sales cost management model for credit organizations was tested in 2018 at a large Russian bank with a wide branch network and developed alternative sales channels (using conditional figures that were close to reality). In that bank, sales of products were organized through four different channels: branch network, call center, field agent sales to companies (or by courier to a place convenient for the client), and Internet banking or bank website. At the first stage, for each of these channels, the cost of 1 min of work for one “seller” was estimated (see Table 1). So, in this example, the highest cost of 1 min of work (0.37 cu) was from one seller in the branch network channel, and the smallest (0.07 cu) was in the Internet channel. At the second stage of modeling, the cost of sales of one unit of a conditional product in the channel (excluding the costs of developing and maintaining software products, marketing, and promotion) was estimated using Equation (4). A conditional calculation example is given in Table 2. As can be seen from the model, for the sale of one product in Channel 1, the branch network, it was necessary: -tooffer the service to 50 customers, of which 10% (5 customers) will be interested; - to offer to issue an application to 5 interested clients, 40% (2 conditional customers) of which will eventually accept; - only 50% of these applicants (1 client) will reach stage of contract execution, passing the application approval procedure, and wishing to use the product. The total sales conversion of the full cycle in Channel 1 thus amounted to 2% =(10% ×40% ×50%) , i.e., of the 50 customers who were offered the product, only 1 was brought to the conclusion of the contract. According to the time standard for one timed operation, the procedure for the initial offer of Product 1 in Channel 1 lasted 2 min, with filling out an application at 15 min, and checking, concluding a contract, and issuing taking 20 min. Multiplying the number of operations at the time of each operation and the cost of 1 min of work of the seller, management can obtain the cost of sales of Product 1 in the channel at direct costs: 79 cu, of which the main costs fall in stages I and II of the sales funnel (37 cu and 27 cu), because it was at these stages that the main losses in conversion of the flow into transactions occurred. The costs of developing and maintaining software, as well as marketing and promotion, were allocated to products and channels in the proportions agreed upon within the bank. First, there was a distribution of the total cost item for individual products, then within each product into the channels, and finally within each channel in proportion to the actual sales for the period in units. An example distribution is shown in Table 3. 5 Risks 2019,7, 115 Table 1. Calculation of the cost of 1 min of work channel employee. Name of Sales Channel Number of Sellers Payroll with Deductions (Thousand/Month) Other Direct Costs * (Thousand/Month) The Cost of 1 Minute of Work for One Channel Seller (from Labor Costs, cu) cu The Cost of 1 Minute of Work for One Channel Seller (from Total Direct Costs, cu) 1234=2/1/FWT ** 5 =(2 +3)/1/FWT ** Channel 1 (Branch) 500 800 1000 0.6 0.37 Channel 2 (Call center) 150 180 60 0.12 0.16 Channel 3 (Direct Sales by Agents) 40 56 20 0.14 0.19 Channel 4 (Internet Banking) 10 24 8 0.06 0.07 Channel 1 (Branch) 500 800 1000 0.16 0.37 * the cost of sellers takes into account the direct costs of rent, utilities, depreciation, and taxes, as well as allocated payroll management staff. It does not include software development/maintenance and marketing costs. ** FWT—working time fund. Table 2. Calculation of the cost of sales at direct costs. Name of Sales Channel The Cost of 1 Min of the Channel Seller, cu Get the Client’s Interest Accept an Application from the Client by Interest Checkout Service at the Request of the Client Get the Client’s Interest Accept an Application from the Client by Interest Checkout Service at the Request of the Client Duration of the operation (minutes) Number of operations per sale (based on % conversion) Branch 0.37 2 15 20 50 5 2.0 Call center 0.16 2 10 1 31 6 2.5 Sales by Agents 0.19 8 5 10 38 29 2.9 Internet Bank 0.07 0.5 1 1 185 3,3 3.3 Name of sales channel Get the Client’s Interest Accept an Application from the Client by Interest Checkout Service at the Request of the Client Full Cycle Get the Client’s Interest Accept an Application from the Client by Interest Checkout Service at the Request of the Client % conversion by sales funnel (to the previous stage) Cost of sales at direct costs, cu Branch 10% 40% 50% 2% 37 27 1 Call center 20% 40% 40% 3% 10 10 0.4 Sales by Agents 75% 10% 35% 3% 59 28 6 Internet Bank 2% 100% 30% 1% 7 0.2 0.2 6 Risks 2019,7, 115 Table 3. Distribution to products and channels of software costs and marketing. Name of Product/Sales Channel Channel 1 (Branch) Channel 2 (Call Centre) Channel 3 (Direct Sales by Agents) Channel 4 (Internet Banking) Total Product/channel share in software development and maintenance costs Product 1 (consumer loans): 1% 1% 1% 7% 10% Product 2 (deposits): 0% 1% 0% 4% 5% Distribution of monthly average costs for software development and maintenance based on the share of the product/channel (million cu) 0.40 Product 1 (consumer loans): 0.004 0.004 0.004 0.028 0.04 Product 2 (deposits): 0.004 - 0.016 0.02 Share of product/channel in marketing and promotion costs Product 1 (consumer loans): 5% 6% 1% 8% 20% Product 2 (deposits): 10% 0% 0.1% 5% 15% Distribution of average monthly expenses for marketing and promotion based on the share of the product/channel (million cu) 0.8 Product 1 (consumer loans): 0.04 0.05 0.01 0.06 0.2 Product 2 (deposits): 0.08 - 0.001 0.04 0.1 Average monthly sales of products in channels (pcs) Product 1 (consumer loans): 10,000 5,000 500 10,000 25,500 Product 2 (deposits): 20,000 1,000 300 30,000 51,300 The cost of software development/maintenance, marketing and promotion based on 1 pc of sales (cu) Product 1 (consumer loans): 4.4 10.4 24.0 9.2 Product 2 (deposits): 4.0 4.0 2.7 1.8 In this example, 10% of the average monthly expenses for software development and maintenance (0.04 million out of 0.4 million cu) were allocated to Product 1. In the context of sales channels, the main emphasis in financing was placed on the Internet channel (0.028 million cu or 70% of the total). Similarly, the costs of marketing and promotion can be attributed to products and channels. For example, 20% of the total amount of 0.8 million cu on Product 1 (of which 40% =0.06 million cu per Internet banking channel). As a result, based on the units of product sold, in the context of sales channels, the impact of the costs of software development and maintenance, and marketing and promotion ranged from 1.8 cu (deposits in online channels) to 24 cu (consumer loans in direct agent sales). As can be seen from the calculation, the above specific costs decreased the greater the scale of sales of the product channel. Summing up the previously calculated cost of sales of the product at direct costs with the additional unit costs for software development and maintenance, as well as marketing and promotion, it was possible to calculate the total cost (see Table 4). Thus, for example, the total cost of sales of one unit of product in the branch network channel is: 78.6 +4.4 =83 cu Due to the distribution of operations, conversion, and cost between the stages of the sales funnel, the model allowed calculation of the cost not only of sales of the full cycle in a single channel, but also of omnichannel sales chains. For example, if, instead of selling a product at all stages through one channel (branch network full cycle chain: Br–Br–Br), the first stage, “interest the customer”, happens through live communication in the branch network, and then the client navigates to apply for and receive a loan to his account via the digital Internet banking channel (stages 2 and 3 of the sales funnel), then the cost of the received omnichannel chain (Br–IB–IB) for the bank could decrease 2-fold: 40.4 +0.4 +0.4 =41.2 cu instead of 40.4 +27.8 +14.8 =83.0 cu. The cost could be reduced by optimizing the use of the resource of branch network sellers participating only in the first stage of interaction with the client. This, despite a slight decrease in the overall percentage of conversion of the target client flow into transactions (from 2.0% to 1.5% with 7 Risks 2019,7, 115 a loss of human contact), would lead to an increase in bank profits both per unit of sold products (from 217 up to 259 cu) and per seller per month (from 4.3 to 7.3 thousand cu) (see Table 5). Table 4. Calculation of cost of sales, taking into account the cost of software and marketing. Name of Sales Channel Cost of Sales at Direct Costs (cu) Get the Client’s Interest Accept an Application from the Client by Interest Checkout Service at the Request of the Client Full Cycle Get the Client’s Interest Accept an Application from the Client by Interest Checkout Service at the Request of the Client Full Cycle Unit costs for software development and maintenance, marketing, and promotion (cu) * The total cost of sales of 1 unit of product in the channel (cu) Branch 78.6 3.9 0.4 0.2 4.4 40.4 27.8 14.8 83.0 Call Center 20.7 8.1 1.6 0.7 10.4 18.3 11.8 1.1 31.1 Direct Sales by Agents 91.9 13.2 9.9 1.0 24.0 71.9 37.4 6.5 116 Internet Bank 7.4 8.9 0.2 0.2 9.2 15.7 0.4 0.4 16.6 * distribution at sales stages is based on the ratio of sales funnel conversions. Table 5. Omnichannel sales chain scenario parameters. Name of the Indicator Br–Br–Br Br–IB–IB Target client flow per month, thousand clients 500 417 Conversion of target flow to sales (%) 2.0% 1.2% Specific sales productivity (pcs. per day for one employee) 1.0 1.3 The number of sellers 500 177 Omnichannel chain sales per month, thousand pieces 10 5 The cost of one sale (cu) 83.0 41.2 Omnichannel chain profit per month (thousand cu) 2170 1294 Profit on 1 unit of sales (cu) 217 259 Profit per one seller per month (thousand cu) 4.3 7.3 Similarly, the model was tested in other omnichannel sales chain optimization scenarios with a call center and direct sales agents. This made it possible to calculate the break-even points for each chain and economically justify investment in the development of these channels with the redistribution of the target client flow along with the resource of sellers to more profitable channels. Based on the practical testing results of the model, the most optimal (with business process parameters that existed at the time of testing) omnichannel sales chain for development was the process wherein the service was offered and the application was filled out (by voice) via the call center, and the conclusion of the contract with money transfer was made via Internet banking (the cost of sales of one loan was $30, with the conversion of the target customer flow to sales at 3.2%). 4. Discussion One of the issues debated in building the model was the choice of method by which to allocate the costs of IT, marketing, and promotion per product. Due to the fact that it is practically impossible to accurately determine the proportions of the distribution of these expenses in proportion to the time spent and advertising budgets in the context of individual products and stages of the sales funnel, it was proposed that the above costs be allocated in proportion to the real structure of sales of banking products (either from the previous period or the plan for the next period). As the accuracy of statistics for assessing sales and processes in various dimensions (time, units, financial result) increases, the approaches to allocation and the model can be improved. Another point of discussion in the process of testing the model was the question of correctly taking into account the specifics of the direct sales channel by agents. This was due to the need to choose an algorithm to distribute costs “on the road” to customers and then, if necessary, again to the bank office, between the stages of the sales funnel. As a result, an agreement was reached that these costs would be entirely allocated to the first stage of the sales funnel (to bring to the client a proposal with the aim of generating interest) in proportion to the share of the product in the product package offered to the client. The time for simultaneous voicing of the bank’s proposals to a group of clients 8 Risks 2019,7, 115 during presentations to enterprises was normalized based on the average number of participants in a group presentation, as well as the above on-the-road time allocated to the product. The third aspect discussed during the implementation of the model was the question of the completeness and frequency of accounting for all customer contact activities within the sales funnel. One proposed strategy was the creation of a unified information system for recording the above activities on a monthly basis. However, according to the testing results, this approach was found to be very costly, since it required significant time costs for the employees of the analytical department, or huge investments in IT. Instead, the project management decided to use a ready-made analytical factor analysis of the phased transformation of customer contacts into transactions, which determined the percentage of customers who were transferred to the next stage of the sales funnel. Based on the available percentage of factor analysis, management can present a “countdown” of the number of actions at each stage necessary to conclude a deal with a client, which was applied in the model, updated at least quarterly. As the integrated analytics of the omnichannel sales model develops, it will be possible to move to direct accounting of operations at each stage of the sales funnel. In the process of analyzing the theoretical base and practical application cases, the following limitations (barriers) were identified that impeded the implementation of an omnichannel sales model and cost management of an omnichannel sales chain in banking: A large number of products and processes needed to be reengineered and automatized during the implementation of the omnichannel approach, both from our own company and from partner companies in the sales process. Significant capital expenditures on the development and maintenance of software and the purchase of equipment can be quickly paid for only with large-scale work on a product or project. These product and process upgrades include: (1) A large number of IT systems are needed to account for various products. For example, even in one bank, sales of even the bank’s own products might be counted in different information systems. The exchange of information on the non-bank products sold by partner companies with the IT systems of these partners is carried out, as a rule, in offline mode with a certain frequency. Thus, support for the omnichannel model when outsourcing part of the functions is also significantly hampered. (2) The need to ensure a high degree of protection of information and customer accounts, especially with remote identification and services. This requires a centralized anti-fraud system covering all channels (Terrasoft 2019). The conservatism of certain clients and client segments (for example, Russian pensioners) using digital services wishing to receive documents on paper with live signatures must also be considered. (3) Product-centric (instead of customer-centric) cultures (Maat 2017) are needed. (4) Employees of different channels must be motivated to obtain results from sales in the implementation of their own KPI to achieve the planned targets and to receive a bonus. With omnichannel sales, it is important not only to take into account the contribution of each participant in the sales chain, but also, if possible, to avoid a double and triple accounting of bonuses for the same transaction, without going beyond the required overall profitability of the product’s business. These aforementioned restrictions will determine future areas of scientific and practical research on the topic of increasing efficiency of sales of banking (and not only banking, but also other retail products) via the introduction of an omnichannel approach. 5. Conclusions Centralized analysis and control of interaction with customers at all stages and in all channels allows banks to significantly increase their targeting and service flexibility, reducing the time to market for products. A complete comprehensive analysis of the entire sales funnel makes it possible to control their cost, selecting the most optimal omnichannel chains. In calculating the cost of sales, it is necessary 9 Risks 2019,7, 115 to take into account not only the direct costs of sales to customers who have made purchase transactions, but also all the losses throughout the whole sales funnel cycle, the costs of developing and maintaining information systems, and the costs of marketing and promotion. As a result of this study, the main advantages of introducing an omnichannel model were identified, with the aim of improving management efficiency; the key factors for inclusion in the model were also identified, as well as the main barriers to implementation. The omnichannel model of interaction with customers will enable banks to simultaneously achieve several key goals of increasing their own business efficiency: (1) Sales growth due to an increase in the frequency of interaction with customers while minimizing the loss of “unsatisfied” customers at the stage of the transaction life cycle; (2) Improving the quality of customer service by providing access to products and services 24/7 and saving customers’ time; (3) Reduction of the specific costs of service per client (per product sold/service provided, as well as contact with the client during the interaction). The “robo-advisors” used by banks are much cheaper than their human counterparts, and are accessible whenever and wherever the user needs their services. Introducing the omnichannel model, banks can accumulate and analyze information about customer behavior using big data technology. By managing data collected in various forms, such as text, audio, and video, banks seek to give customers valuable advice and provide customized suggestions. Customer relationship management (CRM) solutions can be used to integrate heterogeneous data into a single system. One of the key tasks in introducing omnichannel approaches in the banking business is the integration of all IT platforms and solutions into a single centralized data repository (operations) that will allow for seamless interaction with the client, regardless of the product and channel. That is, the client should be able to carry out any purchase or service operation at a convenient time and place, and at any stage of communication. For this, the bank must ensure the availability of a unified accounting base of products, customers, accounts, and operations (for example, by CRM), monetizing the value of its analytics and increasing the value of its brand. Another important point in the implementation of the model is the issue of interconnecting sales plans with a motivation system for employees in different channels. Accounting and incentive systems should motivate employees to work as a team while observing the strategic interests of the bank, which will avoid seller conflict of interests when initiating transactions and contacts with a client base. The proposed omnichannel sales cost management model can be used not only in banking, but also in other retail business formats, where management can collect detailed statistics and set up a factor analysis of the conversion through the sales funnel. Author Contributions: Both authors contributed equally. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflict of interest. References Centric Digital. 2017. 5 Realistic Strategies in Omni Channel Banking. July 21. Available online: https://centricdigital.com/blog/customer-experience/5-realistic-strategies-in-omni-channel-banking (accessed on 8 April 2019). Dube, Apramey, and Anu Helkkula. 2015. Service experiences beyond the direct use: Indirect customer use experiences of smartphone apps. Journal of Service Management 26: 224–48. [CrossRef] Jaakkola, Elina, Anu Helkkula, and Leena Aarikka-Stenroos. 2015. Service experience co-creation: Conceptualization, implications, and future research directions. Journal of Service Management 26: 182–205. [CrossRef] 10 Risks 2019,7, 115 Klaus, Phil, and Bang Nguyen. 2013. Exploring the role of the online customer experience in firms’ multi-channel strategy: An empirical analysis of the retail banking services sector. Journal of Strategic Marketing 21: 429–42. [CrossRef] Komulainen, Hanna, and Hannu Makkonen. 2018. Customer experience in omni-channel banking services. Journal of Financial Services Marketing 23: 99–190. [CrossRef] Komulainen, Hanna, Saila Saraniemi, Pauliina Ulkuniemi, and Marianne Ylilehto. 2018. End-customer value restructuring the financial service supply chain. Marketing Intelligence & Planning 36: 709–20. Koneva, Anna. 2019. Chto znachit “prodavat’ vezde”: Omnikanal’nyye prodazhi s pomoshch’yu Ekvida. July 30. Available online: https://www.ecwid.ru/blog/how-to-make-ecwids-omnichannel-potential-work-for-you (accessed on 7 August 2019). Maat, Jenny. 2017. Omni-Channel Banking Begins with an Outside-in Approach. August 22. Available online: https://backbase.com/2017/08/22/omni-channel-banking (accessed on 1 September 2019). Marketing Science Institute. 2016. Research Priorities 2016–2018. Available online: www.msi.org (accessed on 10 February 2019). Obilisetty, Sridhar. 2019. Omnichannel Banking Goes Beyond Seamless Transactions. Available online: https://thefinancialbrand.com/73488/omnichannel-banking-data-customer-journey-experience (accessed on 3 April 2019). Okorokov, Semon. 2016. Omnikanal’nost’: Panatseya ot krizisa ili modnyy trend. June 1. Available online: https://roem.ru/01-06-2016/225473/omnikalnost (accessed on 9 April 2019). Retail Pragmatist. 2019. The Economics of Omni-Channel Retail—What Does Omni-Channel Mean? January 5. Available online: http://www.retailpragmatist.com/economics-omni-channel-retail (accessed on 12 April 2019). Rosman, Tobias. 2015. Investigating Omni-Channel Banking Opportunities in Sweden: From a User Perspective. Stockholm: KHT Royal Institute of Technology, School of Computer Science and Communication. Saghiri, Soroosh, Richard Wilding, Carlos Mena, and Michael Bourlakis. 2017. Toward a three-dimensional framework for omni-channel. Journal of Business Research 77: 53–67. [CrossRef] Sahoo, Debjani, and Sreejesh S. Pillai. 2017. Role of mobile banking servicescape on customer attitude and engagement: An empirical investigation in India. International Journal of Bank Marketing 35: 1115–32. [CrossRef] Sberbank. 2018. Annual Report, 2017. Available online: https://www.sberbank.com/common/img/uploaded/files/ pdf/yrep/en/sberbank_annual_report_2017_eng.pdf (accessed on 1 April 2019). Serov, Eugene R. 2018. Managing of the net cost of omni-channel sales in credit organizations. Problems of Modern Economics 3: 188–92. Shaikh, Aijaz A., and Heikki Karjaluoto. 2015. Mobile banking adoption: A literature review. Telematics and Informatics 32: 129–42. [CrossRef] Skan, Julian, James Dickerson, and Samad Masood. 2015. The Future of Fintech and Banking: Digitally Disrupted or Reimagined? Available online: www.accenture.com (accessed on 15 February 2019). Tam, Carlos, and Tiago Oliveira. 2017. Literature review of mobile banking and individual performance. International Journal of Bank Marketing 35: 1044–67. [CrossRef] Terrasoft. 2019. Omni-Channel Banking. Available online: https://www.terrasoft.ru/financial-services/insights/ omnikanalny-banking (accessed on 11 April 2019). Tinkoff. 2018. Strategy Day, June. Available online: http://finside.ru/wp-content/uploads/tinkoff-strategy-day- 2018.pdf (accessed on 7 April 2019). © 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). 11 risks Article Application of Diffusion Models in the Analysis of Financial Markets: Evidence on Exchange Traded Funds in Europe Adam Marszk * and Ewa Lechman Faculty of Management and Economics, Gdansk University of Technology, Narutowicza 11/12, 80-233 Gdansk, Poland; [email protected] *Correspondence: [email protected] Received: 14 January 2020; Accepted: 10 February 2020; Published: 14 February 2020 Abstract: Exchange traded funds (ETFs) are financial innovations that may be considered as a part of the index financial instruments category, together with stock index derivatives. The aim of this paper is to explore the trajectories and formulates predictions regarding the spread of ETFs on the financial markets in six European countries. It demonstrates ETFs’ development trajectories with regard to stock index futures and options that may be considered as their substitutes, e.g., in risk management. In this paper, we use mathematical models of the diffusion of innovation that allow unveiling the evolutionary patterns of turnover of ETFs; the time span of the analysis is 2004–2015, i.e., the period of dynamic changes on the European ETF markets. Such an approach has so far rarely been applied in this field of research. Our findings indicate that the development of ETF markets has been strongest in Italy and France and weaker in the other countries, especially Poland and Hungary. The results highlight significant differences among European countries and prove that diffusion has not taken place in all the cases; there are also considerable differences in the predicted development paths. Keywords: financial innovations; diffusion; exchange traded funds; stock index futures; stock index options; stock market indexes 1. Introduction Over the last decades, dynamic changes across financial markets have included the introduction of innovative financial instruments that contribute to global financial diversity. The category of innovative financial instruments is highly heterogeneous, e.g., in terms of the rate of their expansion; exchange-traded funds (ETFs) are among the most rapidly expanding financial instruments. ETFs are funds structured to mimic the performance of selected financial assets, usually stock indexes. The difference between ETFs and conventional investment products (such as mutual funds) is that units of ETFs resemble, financial instruments such as listed equities or bonds because they are purchased and sold through stock exchanges. The growing popularity of ETFs, the increase in the sums involved and the rate of turnover are predominantly enhanced by low trading costs, low tracking errors, high liquidity and (in some countries) high tax efficiency (Agapova 2011;Madhavan 2016; Ben-David et al. 2017;Lettau and Madhavan 2018). Until recently, ETFs were mainly considered as substitutes for index funds in passive investing strategies because of their similar features and users. However, a rising recognition and complexity of the products offered has resulted in increasing demand from different types of players in the financial markets. As a result, the shares of innovative funds have become substitutes not only for index funds but also for derivatives. To the best of our knowledge, there have been almost no empirical works covering the subject of switching between ETFs and stock index derivatives, although a theoretical background was provided by the framework suggested by Gastineau (2010). Risks 2020,8, 18; doi:10.3390/risks8010018 www.mdpi.com/journal/risks 13 Risks 2020,8,18 Thus, the modified specification of Equation (6) is: ETFi(t)=κETF i 1+exp−αETF it−βETF i, (8) with notation as explained above. The parameters in Equation (8) can be estimated using not only ordinary least squares (OLS) but also maximum likelihood (MLE), algebraic estimation (AE) or nonlinear least squares (NLS). Nonetheless, as Satoh (2001) suggests, NLS returns the best predictions, as its estimates of standard errors (of κETF i , αETF i , βETF i) are more valid than those returned using the other methods. Adopting NLS allows time-interval biases, which occur in the case of OLS estimates (Srinivasan and Mason 1986), to be avoided. However, NLS has the disadvantage that estimates of the parameters may be sensitive to the initial values of the time-series adopted. Finally, it should be emphasized that the construction of the utilized model hinders inclusion of the explanatory variables. However, the issue of the factors that affect the diffusion of ETFs was analyzed using different methodologies—the results were presented in, inter alia, Lechman and Marszk (2015) and Marszk et al. (2019). 3.2. Data Our research covers stock exchanges in six European countries: two countries in the Central and Eastern Europe (CEE) region—Poland and Hungary; another four EU countries with the longest history of ETF trading—France, Italy, Germany and Spain. Our analysis covers the Euronext exchange considered as a whole (due to data availability), and thus (in addition to France) also includes The Netherlands, Belgium and Portugal. However, most of the turnover is reported to be in the French segment and so we decided to consider this exchange as if it was located in France. Consequently, we also used other indicators for France. The time span of the analysis is 2004–2015. It was selected due to the high rate of changes on the ETF markets in Europe, following their launch in the analyzed countries. The beginning of this period was chosen due to the fact that there were almost no ETFs traded in Europe in 2003 or earlier. The selected end of the time period of analysis is 2015 as since 2016, the changes have been less significant. The time coverage is also a result of data availability. For the period 2004–2015 a balanced data set is available for most of the countries included in the analysis, while for the CEE countries, the time span of the analysis is shorter as ETFs were launched there later than in the advanced European economies. The financial instrument databases used in the study are the dataset provided by the World Federation of Exchanges (World Federation of Exchanges 2017), datasets provided by the selected stock exchanges and reports published by these institutions. The most important financial indicators used are the turnover values (in USD millions) on the stock exchanges of the instruments selected: ETFs, stock index options and stock index futures. Monthly data are used. Due to a lack of reliable data on the turnover of stock index futures and options on the main stock exchange in the United Kingdom (caused by changes in the organizational structure of the London Stock Exchange Group), it was excluded from the analysis. 4. Results 4.1. Exchange Traded Fund Market Development: Preliminary Evidence Our investigation of the development of the ETF markets starts with an analysis of summary statistics on the key changes in two measures: the turnover value and the percentage share of the total turnover of index financial instruments (see Table 2). 20 Risks 2020,8,18 Table 2. Summary statistics for exchange traded funds, stock index options, stock index futures and total index financial instruments. Monthly data for 2004–2015. For ETFs, the periods of analysis are: Poland, 2010m9–2015m12; Hungary, 2007m1–2015m12; Italy, 2004m1–2015m12; Spain, 2006m7–2015m12; Germany, 2004m1–2015m12; and France, 2004m1–2015m12. The number of ETFs varies across time periods and countries. Statistics Poland Hungary Turnover on Local Stock Exchanges (in million USD) ETFs Stock Index Options Stock Index Futures Total Index Financial Instruments ETFs Stock Index Options Stock Index Futures Total Index Financial Instruments # obs. 64 144 144 144 132 132 132 132 Min 0.90 29.14 1041.05 1070.2 0.0 0.0 9.2 10.08 (2012m11) (2004m7) (2004m7) (2004m7) (2015m5) (2015m7) Max 15.79 1187.1 15,820.6 16,274.5 7.6 9.2 1015.6 1015.6 (2011m8) (2011m3) (2008m1) (2008m1) (2007m4) (2006m8) (2006m5) (2006m5) Mean 5.5 336.2 6422.6 6761.30 0.60 0.10 244.9 245.7 Absolute change in value (pp) 0.76 184.05 3220.6 3408.8 −5.6 0.0 −172.7 −0.25 Average monthly dynamic 100.3 101.1 100.7 100.7 48.6 66.8 97.9 97.9 Share of Total Turnover of Index Financial Instruments on Local Stock Exchanges [%] ETFs Stock Index Options Stock Index Futures - ETFs Stock Index Options Stock Index Futures - # obs. 64 144 144 - 132 132 132 - Min 0.02 1.6 86.01 −0.0 0.0 77.9 - (2012m11) (2008m4) (2008m4) (multiple periods) (multiple periods) (2015m) Max 0.39 13.9 98.4 −22.01 1.4 100 - (2015m4) (2013m8) (2013m8) (2015m5) (2006m8) (multiple periods) Mean 0.08 5.05 94.9 - 0.55 0.02 99.4 - Absolute change in share (pp) 0.05 1.9 −2.00 - −0.25 0.0 −0.89 - Average monthly dynamic 101.4 100.4 99.9 - 39.7 60.7 99.9 - Italy Spain Turnover on Local Stock Exchanges (in million USD) ETFs Stock Index Options Stock Index Futures Total Index Financial Instruments ETFs Stock Index Options Stock Index Futures Total Index Financial Instruments # obs. 144 144 144 144 114 144 144 144 Min 246.4 9038.99 32,881.4 48,685.7 131.2 1134.1 28,439.7 32,317.4 (2004m5) (2011m12) (2009m2) (2009m2) (2012m8) (2012m1) (2012m2) (2004m8) Max 13,435.2 53,337.8 178,067.9 234,279.7 3397.5 16,971.7 169,693.3 183,875.7 (2015m3) (2007m3) (2007m3) (2007m3) (2008m1) (2008m1) (2007m11) (2007m10) Mean 5669.7 22,329.0 78,508.4 106,507.1 630.5 5764.6 74,557.7 80,821.5 Absolute change in value (pp) 8306.9 13,222.2 64,909.3 86,438.5 779.1 3950.7 33,361.5 38,333.1 Average monthly dynamic 102.3 100.5 100.6 100.6 101.3 100.6 100.4 100.5 21 Risks 2020,8,18 Table 2. Cont. Share in Total Turnover of Index Financial Instruments on Local Stock Exchanges [%] ETFs Stock Index Options Stock Index Futures - ETFs Stock Index Options Stock Index Futures - # obs. 144 144 144 114 144 144 - Min 0.27 10.7 63.6 0.14 81.6 81.6 - (2004m3) (2014m12) (2007m10) (2007m9) (2011m1) (2012m12) Max 13.7 32.3 83.3 2.2 96.8 96.8 - (2012m1) (2007m10) (2014m12) (2015m7) (2012m12) (2011m1) Mean 5.6 20.9 73.4 0.75 7.1 92.3 - Absolute change in share (pp) 5.2 −3.07 −2.1 1.04 1.6 −2.9 - Average monthly dynamic 101.7 99.8 99.9 101.3 100.1 99.9 - Germany France Turnover on Local Stock Exchanges (in million USD) ETFs Stock Index Options Stock Index Futures Total Index Financial Instruments ETFs Stock Index Options Stock Index Futures Total Index Financial Instruments # obs. 144 144 144 144 144 144 144 144 Min 2549.1 259,180.4 550,142.9 823,511.8 810.6 44,421.9 204,785.8 269,320.8 (2004m9) (2004m12) (2004m2) (2004m7) (2004m9) (2015m11) (2004m8) (2015m11) Max 44,323.2 2,830,918 3,993,353 6,852,531 26,980.5 726,885.4 1,059,843 1,725,926 (2011m8) (2008m1) (2008m1) (2008m1) (2011m8) (2007m11) (2008m1) (2008m1) Mean 14,351.3 1,161,163 1,781,525 2,957,040 9170.3 266,211.7 485,718.8 761,100.9 Absolute change in value (pp) 14,556.7 979,971.1 1,747,945 2,742,473 16,803.99 −168,972 − 33,997.2 −186,165 Average monthly dynamic 101.0 100.9 100.9 100.9 101.7 98.9 99.9 99.6 Share of Total Turnover of Index Financial Instruments on Local Stock Exchanges [%] ETFs Stock Index Options Stock Index Futures - ETFs Stock Index Options Stock Index Futures - # obs. 144 144 144 - 144 144 144 - Min 0.22 24.06 49.4 −0.13 16.5 49.5 - (2004m9) (2004m12) (2015m7) (2004m9) (2015m11) (2004m8) Max 0.89 50.06 75.6 −6.4 50.3 78.7 - (2011m7) (2015m7) (2004m12) (2015m12) (2004m8) (2015m5) Mean 0.48 38.6 60.9 - 1.4 33.9 64.5 - Absolute change in share (pp) 0.03 −0.31 0.27 - 6.00 −28.7 22.7 - Average monthly dynamic 100.0 99.9 100.0 - 102.0 99.3 100.2 - In the time period analyzed, there were only two countries in the CEE region where ETFs were listed on the local stock exchanges: Poland and Hungary. In Poland, the highest values of ETF turnover were reached several months after their launch in September 2010, in August 2011 (see Table 2and Figure A1 in the Appendix A). However, in 2012 turnover severely declined and reached a minimum level of only $0.9 million USD in November 2012. From 2013 to 2015, trading in ETFs was still at a rather low level. However, in April 2015, ETFs reached their maximum share of the total market: 0.39%, which was mostly caused by a one-month spike in ETF trading (yet it was still one of the lowest shares among the countries considered). In Hungary, ETFs were launched much earlier than in Poland (in 2007) but their turnover was significantly lower (a mean monthly value of $0.6 million USD compared to $5.5 million USD in Poland). As in Poland, the highest turnover values were observed soon after their introduction. However, in terms of ETF market share, the highest value in Hungary was reached 22 Risks 2020,8,18 (as in the Polish market) near the end of the time period analyzed, in May 2015 (see Figure 1). In contrast to the Polish exchange, the turnover of other related financial instruments (stock index futures and options) on the Hungarian market was extremely low: in most months there were almost no transactions in options and the value of futures trading was steadily declining. As a result, the mean turnover values in Hungary were minimal in comparison to the other stock exchanges considered. The very low turnover of ETFs in both Poland and Hungary was mostly caused by the low number of such financial products. In Poland, the number of ETFs grew from 1 to 3 (yet only one of them was listed exclusively in Poland and it accounted for the majority of turnover; the other two ETFs were cross-listed). In Hungary, there was only one ETF listed between 2007 and 2015 and it had a minimal turnover. The lack of further development was caused by a number of factors, including a lack of awareness of ETF features among market participants and the relatively small size of the financial markets, which limited the possibility of gaining benefits from the larger scale of offerings provided by ETF managers. 85 90 95 100 % 0 5 10 15 % 2004m1 2006m1 2008m1 2010m1 2012m1 2014m1 2016m1 Poland 80 85 90 95 100 % 0 5 10 15 20 % 2004m1 2006m1 2008m1 2010m1 2012m1 2014m1 2016m1 Spain 20 40 60 80 % 0 2 4 6 % 2004m1 2006m1 2008m1 2010m1 2012m1 2014m1 2016m1 France 50 55 60 65 70 75 % 0 10 20 30 40 50 % 2004m1 2006m1 2008m1 2010m1 2012m1 2014m1 2016m1 Germany 0 20 40 60 80 100 % 0 5 10 15 20 % 2004m1 2006m1 2008m1 2010m1 2012m1 2014m1 2016m1 Hungary 0 20 40 60 80 % 0 5 10 15 % 2004m1 2006m1 2008m1 2010m1 2012m1 2014m1 2016m1 ETF share OPT share FUT share Italy ETF/stock index options/stock index futures share in total index financial instruments. 2004-2015. Figure 1. ETFs, stock index options and stock index futures—share of total turnover of index financial instruments. 2004–2015 (monthly time series). Left-hand Y axis—ETF share of total index financial instruments; right-hand Y axis—stock index options and stock index futures share of total index financial instruments. In the four advanced EU countries selected, the only country with no ETFs listed at the beginning of the time period analyzed was Spain (ETFs were launched in Spain in July 2006). In terms of ETF turnover, in Italy growth of the ETF market was somewhat stable and the highest values were reached near the end of the time period. In France and Germany, ETF turnover grew until 2011, when it sharply declined, which may be explained by the eurozone crisis and falling stock prices (in the other three advanced EU economies a decline in ETF turnover in 2011 was also observed but it was relatively weaker (see Figure A1 in the Appendix A)). After 2011, turnover in France began to grow, whereas in Germany it was stable. The Spanish ETF market developed in a different way. After much variability until 2011, it entered a stage of stability between 2012 and 2013, and from the end of 2013 it started growing. This shows that the development of the ETF markets in these countries was to some extent shaped by similar determinants (e.g., the euro-zone crisis), although there were also some country-specific factors despite the high level of financial market integration. 23 Risks 2020,8,18 Regarding ETF market shares, some substantial differences between the four countries can be noticed (see Table 2and Figure 1). In Spain and Germany, the market share of ETFs was very low over the whole period. The case of Germany is particularly interesting. The mean value of ETF turnover in this country was the highest among all the countries analyzed and one of the highest in the world. Nevertheless, their average market share was the second-lowest (it was only lower in Poland), which shows that the role of ETFs in Germany was negligible compared to that of other index financial instruments. In both France and Italy, the market share of ETFs increased considerably: in France particularly from 2014, and in Italy from 2009. The mean market share value of ETFs in Italy was the highest of all the countries under study (5.6%), yet was still much lower than the shares of the other index instruments. The rapid development of the Italian ETF market may to a large extent be explained by the acquisition of the Italian stock exchange by its British counterpart, which is one of the largest in Europe in terms of the number and turnover of ETFs. The two markets have been integrated in some areas, which considerably boosted the Italian stock exchange’s growth opportunities. In the remainder of this study, we will use the market share as the indicator of ETF market development, as changes occurring in ETF markets should not be viewed in isolation but instead put in a broader context, thus showing the position of these innovative financial products in the financial system. Our preliminary analysis of changes occurring in the ETF markets will be expanded in the next sections—we will attempt to analyze the main features of the ETF diffusion process and predict its trajectories. 4.2. Exchange Traded Funds: Diffusion Models As an aim of this study is to provide in-depth insight into the development process of ETFs across countries, we adopt a logistic growth model (for details, see Section 3) because use of this type of model allows the development trajectories of different variables in economic systems to be approximated and evaluated. Moreover, it allows the characteristic phases of the process of diffusion to be distinguished, such as the early diffusion phase, take-off, the exponential growth phase and saturation (maturity phase). Through the early diffusion stage, the number of contacts between adopters and non-adopters of a given innovation is still small, which may hinder its dissemination and so in this stage of diffusion the process is still reversible. However, under favorable conditions, easy contacts allow a domino effect to come into play and hence diffusion may speed up. Driven by various market forces, reductions in the cost of adopting innovations and multiple applications and uses of them, the number of new-users can rapidly increase and the curve takes off. It then enters a fast diffusion phase, when the diffusion process usually proceeds exponentially. Finally, a maturity (stabilization) stage is reached, during which the pace of diffusion again becomes slow and no substantial growth in the number of new users of the innovation is reported. In addition to revealing these phases, a simple logistic growth model returns good forecasts of future development (Kucharavy and Guio 2011). Following the above-mentioned approach and using monthly time series for the period 2004–2015, we develop logistic growth patterns and estimate parameters (see Section 3) representing the ETF share of total index financial instrument turnover for each country individually. The results of our analysis are presented in Figure A2 in the Appendix A, which shows that the current and predicted ETF share diffusion paths, and in Table 3, which summarizes the country-wise logistic growth model estimates. The graphical evidence presented in Figure 1suggests that ETF diffusion patterns in some countries (i.e., growing ETF shares) may be well described by the logistic (sigmoid) growth trajectory. In some cases, the characteristic phases of the S-shaped path can be distinguished (also see the analysis for other countries in Lechman and Marszk (2015)). Initially slow changes in the ETF share of total turnover of index financial instruments are followed by a sudden take-offand then the ETF share pattern enters the rapid growth phase. However, it is important to note that the shapes of the ETF share diffusion paths across the countries examined are different and so they need special attention. 24 Risks 2020,8,18 Table 3. Diffusion of exchange traded funds (as share of total turnover of index financial instruments). Logistic growth model estimates. 2004–2015 (monthly time series). Poland—data from 2010m9; Hungary—data from 2005m1; Spain—data from 2006m7. Parameter Poland Hungary Italy κETF i(ceiling/upper asymptote) 0.087 97,207 8.56 TmETF i(βETF i) (midpoint) 397,133.9 1062 52.9 αETF i(rate of diffusion) −2606.7 0.013 0.113 ΔtETF i(specific duration) −0.002 339.2 38.7 R2of the model 0.00 0.075 0.76 # of obs. 64 130 (outliers excluded) 144 Parameter Spain Germany France κETF i(ceiling/upper asymptote) 500,100.2 0.585 7,755,333.6 TmETF i(βETF i) (midpoint) 1,175,6.2 −3.94 777.3 αETF i(rate of diffusion) 0.012 0.026 0.023 ΔtETF i(specific duration) 354.4 169.8 194.2 R2of the model 0.411 0.27 0.789 # of obs. 114 144 144 The picture which emerges from analysis of the ETF share in the two selected CEE countries—Poland and Hungary—differs radically from that for other countries. As already mentioned in the previous section, in neither Poland nor Hungary did ETFs gain much popularity and their share of total turnover remained extremely low over the time period analyzed. In Hungary, the growth of the ETF share of total turnover was minimal and its role in shaping the financial market was negligible. In Poland, a diffusion of ETFs across the domestic financial market was reported but still their role and share of the total turnover was marginal. It should be noted that between 2004 and 2015, the ETF share of the total turnover was close to zero. This leads to the conclusion that in both Hungary and Poland adiffusion of ETFs did not take place and so logistic growth models should not be applied. Table 3 presents the estimates of logistic growth models for Poland and Hungary, but as in both cases the R 2 of the models is zero, the parameters returned are misleading and inconclusive. Finally, we discuss the results of the analysis of ETF diffusion for the four developed financial markets selected: Italy, Germany, France and Spain. In Germany, a diffusion of ETFs on the domestic financial market was not observed and ETF market penetration remained below 1%. As in the cases of Hungary and Poland, the logistic growth model estimates are not reliable. Despite the fact that the R 2 of the model is 0.27 (see Table 3), the value returned for the midpoint (Tm) is negative and so cannot be treated as valid. The situation in Spain is analogous, with a very low ETF share of total turnover during the time period examined. At the end of 2015, Spain was still located in the early diffusion stage, and as a result reliable estimates of a logistic growth model are not possible (the logistic growth parameters returned cannot be treated as valid). In the other two advanced European economies—France and Italy—the ETF share was relatively high between 2004 and 2015. In both cases, the ETF diffusion patterns take offinto self-sustaining growth after the early diffusion stage, during which increases in the ETF share were slow. In the case of Italy, the specific take-offoccurred relatively early compared to the other economies examined. It should be noted that between June and July 2008 the ETF share almost doubled (from 1.8% to 3.4%) and the take-offtook place shortly afterwards—between the middle of December 2008 and January 2009, when the ETF share increased from 3.7% to 8.0%. All the parameters returned from the logistic growth model estimates for Italy are statistically significant. The R 2 of the model is about 0.76, which implies a good fit between the empirical data and the theoretical model. Even though the R 2 of the model is low, there are no obvious misspecifications as the diffusion of ETFs is relatively well described by the logistic growth trajectory. The upper asymptote is estimated as κETF i =8.56%. The estimated midpoint 25 Risks 2020,8,18 is TmETF i =52.9. The rate of diffusion is αETF i =0.113 and ΔtETF i =38.7, which can be interpreted as the number of months required to pass from 10% to 90% of κETF i. For France too, the diffusion of ETFs is well described by the logistic growth trajectory, despite the fact that in this case the early diffusion stage was relatively long. The take-offinto the exponential growth phase did not happen until between the middle of December 2013 and January 2014, when the ETF share of total turnover grew abruptly. Even though the diffusion of ETFs (in terms of market share) on the French financial market is well approximated by the logistic growth pattern, the parameters estimated for the logistic growth model are not valid. The upper asymptote (ceiling) is reported as κETF i=7,755,333, which is a definite overestimation. Regarding the process of ETF diffusion in our country sample, the eight economies can be divided into two groups. The first group encompasses two countries—France and Italy—where an early diffusion stage was followed by a take-offinto an exponential growth phase along a sigmoid trajectory. These two countries managed to leave the early diffusion stage, during which ETF share growth was slow and spasmodic, and take offinto rapid growth. In the other four countries, the ETF share did not leave the early diffusion stage and remained virtually locked at a low level. This empirical analysis of ETF diffusion trajectories can be enriched by providing additional specifications of the predicted development of ETFs across the economies selected. Table 4summarizes the predicted country-specific ETFs diffusion paths, and Figure A2 in the Appendix Aportrays them graphically. Fixing the critical level of the upper asymptote ( κETF i) at 5%, 7.5%, 10%, 15%, 20%, 25% and 30%, we predict logistic growth model parameters under the strict assumption that ETF market development will follow an S-shaped trajectory. For Hungary, with κETF i fixed at 5% the predicted TmETF i is June 2027 and the ‘specific duration’ forecast is about 320 months, i.e., more than 26 years. The predicted rate of diffusion is 0.014, which implies that the speed of ETF diffusion will be rather low in Hungary. The forecasts for higher κETF i show even more distant midpoints and they cannot be treated as being very reliable (and also because of the low R2of the models). Italy has already reached the levels of κETF i =5%, 7.5% and 10%. With κETF i fixed at 15%, the predicted TmETF iis April 2012 if the Italian ETF market follows an S-shaped trajectory. The predicted rate of diffusion is similar to that in Hungary, i.e., much lower than in, e.g., France. Regarding Spain, with κETF i fixed at 5% the predicted TmETF i is July 2021 (considerably sooner than in the case of Hungary) and the ‘specific duration’ forecast is about 300 months. The rate of diffusion predicted is 0.015, which is consistent with the results obtained for Hungary and Italy. Finally, for France with κETF i fixed at 7.5%, the predicted TmETF i is July 2015 if the French ETF market follows the S-shaped trajectory. The rate of diffusion predicted for this level of κETF i is 0.028, but for higher levels it is slightly lower, which suggests that the diffusion of ETFs on the French market will be much faster than in other European countries. The ETF diffusion paths predicted for Germany and Poland are not valid and so they will not be discussed. It should be emphasized that all these forecasts are tentative and should be treated with caution. The projected future diffusion paths are not entirely random but rather assume an S-shaped trajectory and all the predictions show a high level of sensitivity to historical data. Special caution is urged regarding the predictions referring to relatively high fixed ceilings like 20%, 25% and 30%, where the accuracy of the forecasts is questionable and they are to some extent misleading and inconclusive. 26 Risks 2020,8,18 Table 4. Predicted ETFs diffusion patterns (as share of total turnover of index financial instruments). Hungary—outliers excluded. Italics =misspecifications. κETF i(Upper Asymptote)—Fixed TmETF i (Midpoint)—Refers to a Specific Date ΔtETF i(Specific Duration)—Number of Months αETF i(Rate of Diffusion) R2of the Model Poland 5% −229,626,799 −249,867,896 0.00 0.016 7.5% 981.7 857.4 0.005 0.018 10% 1,010.6 859.5 0.005 0.018 15% −258,875,673 −220,949,493 0.00 0.016 20% 1,181,1 862.7 0.005 0.018 25% 1,225.8 863.3 0.005 0.018 30% 1,262.3 863.8 0.005 0.018 Hungary 5% 282.4 (2027m6) 319.5 0.014 0.073 7.5% 318.3 (2030m6) 326.0 0.013 0.073 10% 343.2 (2032m7) 329.3 0.013 0.073 15% 377.4 (2035m5) 332.6 0.013 0.074 20% 401.2 (2037m5) 334.2 0.013 0.074 25% 419.4 (2038m11) 335.2 0.013 0.074 30% 434.2 (2040m2) 335.9 0.013 0.074 Italy 5% Already achieved 7.5% Already achieved 10% Already achieved 15% 100.24 (2012m4) 246.9 0.018 0.485 20% 141.7 (2015m9) 318.7 0.014 0.454 25% 175.7 (2018m7) 359.9 0.012 0.445 30% 204.1 (2020m12) 386.9 0.011 0.435 Spain 5% 211.4 (2021m7) 300.3 0.015 0.41 7.5% 252.4 (2024m12) 318.2 0.014 0.41 10% 280.6 (2027m4) 327.2 0.013 0.41 15% 318.9 (2030m6) 336.3 0.013 0.41 20% 345.4 (2032m9) 340.8 0.013 0.41 25% 365.6 (2034m5) 343.5 0.013 0.41 30% 381.7 (2035m9) 345.4 0.013 0.41 Germany 5% 1,402.6 1426.9 0.003 0.02 7.5% 892.6 1349.1 0.003 0.02 10% 1003.8 1375.0 0.003 0.02 15% 1156.0 1401.0 0.003 0.02 20% 1261.3 1413.9 0.003 0.02 25% 1341.7 1421.7 0.003 0.02 30% 730.8 1297.3 0.003 0.02 France 5% Already achieved 7.5% 139.9 (2015m7) 157.6 0.028 0.73 10% 156.8 (2016m12) 165.6 0.027 0.75 15% 179.7 (2018m11) 174.5 0.025 0.76 20% 195.3 (2020m3) 179.2 0.025 0.77 25% 207.2 (2021m3) 182.1 0.024 0.77 30% 216.6 (2021m12) 184.0 0.024 0.77 27 Risks 2020,8,18 5. Conclusions This extensive research was designed to analyze the development paths and dynamics of financial innovations introduced on stock exchanges in France, Germany, Spain, Italy—which have been treated as economies with relatively well developed stock exchanges—Hungary and Poland—two European economies where financial innovations have relatively short histories. We examined the development of ETF markets using descriptive statistics and diffusion models. Graphical evidence on the ETF markets shows that ETF diffusion patterns in some countries may be described as a logistic growth trajectory—characteristic phases of the S-shaped path can be distinguished, which justifies the application of diffusion models. In Hungary and Poland, the level of ETF market development was very low and no significant changes are expected in the future unless the market environment is deeply transformed (which cannot be predicted). The trajectory of ETF market development in the more advanced European economies differed considerably. In Spain and Germany, the ETF market share remained very low and no meaningful predictions could be obtained using diffusion models. In France and Italy, significant development of ETF markets was identified and the predictions indicate potential further growth. In our research we claimed that ETFs are financial innovations and thus it would be justifiable to analyze their development paths analogously to the process of diffusion of other tangible or intangible innovations. Following the latter, we have proposed to use the mathematical diffusion models, traditionally used to approximation of diffusion patterns of innovations, to draw the trajectories of ETFs diffusion across the financial markets. Our empirical evidence has demonstrated applicability of these diffusion models to the numerical analysis of ETFs diffusion process, allowing for detecting their in-time behavior, case-specific dynamics and development patterns, as well as providing long-term predictions. We believe that this approach to the analysis of financial markets development paves avenues for further and more profound research in this field (similar in-kind conclusions were reached in Marszk et al. (2019)). Notably, ETFs as innovative financial instruments are still a poorly explored area, including our knowledge on what determines their development, or simply—what enhances or hinders their fast diffusion across financial markets. Apparently, in some countries, ETFs have rapidly gained popularity, while in other their development is negligible. The main limitation of the research method used in this study is that it is derived from the logistic growth function that is based on S-shaped trajectory of the diffusion of innovation which may be inconsistent with the attributes of the financial innovations. Moreover, our analysis has not addressed (with the exception of some preliminary suppositions) the factors that have influenced the diffusion processes. Detecting major determinants of ETFs diffusion in relation to other stock index instruments, including legal and institutional regulations that enable or stop this process, constitutes the direction of possible future research. Author Contributions: Both authors contributed equally. Conceptualization, A.M. and E.L.; methodology, A.M. and E.L.; software, A.M. and E.L.; formal analysis, A.M. and E.L.; investigation, A.M. and E.L.; writing—original draft preparation, A.M. and E.L.; writing—review and editing, A.M. and E.L.; project administration, A.M. and E.L.; funding acquisition, A.M. and E.L. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by project no. 2015/19/D/HS4/00399 financed by the National Science Centre, Poland. It was also supported by a grant from the CERGE-EI Foundation under a program of the Global Development Network. Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results. 28 Risks 2020,8,18 Appendix A 0 5 10 15 mln USD 2004m1 2006m1 2008m1 2010m1 2012m1 2014m1 2016m1 Exchange traded funds 0 5000 10000 15000 mln USD 2004m1 2006m1 2008m1 2010m1 2012m1 2014m12016m1 Stock index futures 0 500 1000 1500 mln USD 2004m1 2006m1 2008m1 2010m1 2012m1 2014m1 2016m1 Stock index options 0 5000 10000 15000 mln USD 2004m1 2006m1 2008m1 2010m1 2012m1 2014m12016m1 Total index financial instruments Poland 0 2 4 6 8 mln USD 2004m1 2006m1 2008m1 2010m1 2012m1 2014m1 2016m1 Exchange traded funds 0 200 400 600 800 1000 mln USD 2004m1 2006m1 2008m1 2010m1 2012m1 2014m1 2016m1 Stock index futures 0 2 4 6 8 10 mln USD 2004m1 2006m1 2008m1 2010m1 2012m1 2014m1 2016m1 Stock index futures 0 200 400 600 800 1000 mln USD 2004m1 2006m1 2008m1 2010m1 2012m1 2014m1 2016m1 Total index financial instruments Hungary 0 5000 10000 15000 mln USD 2004m1 2006m1 2008m1 2010m1 2012m12014m1 2016m1 Exchange traded funds 0 50000 100000 150000 200000 mln USD 2004m12006m12008m12010m12012m12014m12016m1 Stock index futures 10000 20000 30000 40000 50000 mln USD 2004m1 2006m1 2008m1 2010m1 2012m12014m1 2016m1 Stock index options 50000 100000 150000 200000 250000 mln USD 2004m12006m12008m12010m12012m12014m12016m1 Total index financial instruments Italy Figure A1. Cont. 29 risks Article Heads and Tails of Earnings Management: Quantitative Analysis in Emerging Countries Pavol Durana 1,*, Katarina Valaskova 1, Darina Chlebikova 1, Vladislav Krastev 2 and Irina Atanasova 2 1Faculty of Operation and Economics of Transport and Communications, Department of Economics, University of Zilina, 010 26 Zilina, Slovakia; [email protected] (K.V.); [email protected] (D.C.) 2Department of Economics, South-West University, 2746 Blagoevgrad, Bulgaria; [email protected] (V.K.); [email protected] (I.A.) *Correspondence: [email protected]; Tel.: +421-41-513-3217 Received: 29 April 2020; Accepted: 27 May 2020; Published: 1 June 2020 Abstract: Earnings management is a globally used tool for long-term profitable enterprises and for the apparatus of reduction of bankruptcy risk in developed countries. This phenomenon belongs to the integral and fundamental part of their business finance. However, this has still been lax in emerging countries. The models of detections of the existence of earnings management are based on discretionary accrual. The goal of this article is to detect the existence of earnings management in emerging countries by times series analysis. This econometric investigation uses the observations of earnings before interest and taxes of 1089 Slovak enterprises and 1421 Bulgarian enterprises in financial modelling. Our findings confirm the significant existence of earnings management in both analyzed countries, based on a quantitative analysis of unit root and stationarity. The managerial activities are purposeful, which is proven by the existence of no stationarity in the time series and a clear occurrence of the unit root. In addition, the results highlight the year 2014 as a significant milestone of change in the development of earnings management in both countries, based on homogeneity analyses. These facts identify significant parallels between Slovak and Bulgarian economics and business finance. Keywords: business finance; earnings management; EBIT; financial modelling; homogeneity; stationarity; time series methods; unit root 1. Introduction Theissuesofriskmanagementhavebeenanalyzed and discussed for a long time (Hudakova et al. 2018) . The managements of the enterprises must select the best solutions for future development in any conditions (Kral et al. 2019). Spuchlakova and Cug (2015) argue that a structural approach is necessary to reduce and model their business risk. Meyers et al. (2019) highlight big data-driven algorithmic decision-making related to risk management. Vagner (2017) adds that the practical benefits connected with cost controlling and costs optimization and earnings management may be very beneficial for applying to enterprises to risk. Earnings management is an accounting technique to manage financial reports that shows a mostly positive view of business finance and the financial situation. Earnings management means the transformation into a new accounting regime, in a lot of cases (Hoang and Joseph 2019). The reasons for managers to do a manipulation of earnings is good looking for investors and potential investors (Susanto et al. 2019), moreover, Khanh and Thu (2019) declare a positive correlation between earnings management and leverage management. This phenomenon of earnings modification is an increasingly important topic, obviously in the area of the assessment of efficiency—which is a fundamental part of the corporate rational behavior that aims to survive in a challenging competitive Risks 2020,8, 57; doi:10.3390/risks8020057 www.mdpi.com/journal/risks 37 Risks 2020,8,57 environment in the long term (Balcerzak et al. 2017)—as well as in the areas of financial accounting, financial risk and financial modelling. The most relevant researches have been conducted in the developed markets, but this topic is very rarely investigated in emerging countries, as they still adhere to conventional approaches. The European market significantly varies from other global markets (Rahman et al. 2017). The number of publications concerned with earnings management changes according to the country. From all the countries, earnings management is the most discussed topic in the USA. Almost five thousand research papers have their origins in this country. We may highlight that developed European countries, such as the United Kingdom (Iatridis and Kadorinis 2009;Pina et al. 2012), Spain (Ferrer Garcia and Lainez Gadea 2013;Rodriguez-Perez and van Hemmen 2010), Germany (Christensen et al. 2015;Velte 2019) , Italy (Cimini 2015), the Netherlands (Kempen 2010), Belgium (Andries et al. 2017), France (Bzeouich et al. 2019;Ben Amar and Chakroun 2018) and many others, focus on earnings management phenomenon from different perspectives. However, the issue of earnings management is not developed and investigated properly in emerging countries, with various explanations for this. The identified research gap was analyzed in the case of Slovakia (SK) and Bulgaria (BG). Both countries, former Soviet-controlled Eastern bloc countries, have experienced a massive transformation of their economies over the past decades, and their significant development has been emphasized by their participation in the European Union. Following their economic development (Figure 1) using the gross domestic product (GDP) index, it is evident that the same development trend in the 10-year horizon can be indicated. Ϭ ϮϬ ϰϬ ϲϬ ϴϬ ϭϬϬ ϭϮϬ ϮϬϭϬ ϮϬϭϭ ϮϬϭϮ ϮϬϭϯ ϮϬϭϰ ϮϬϭϱ ϮϬϭϲ ϮϬϭϳ ϮϬϭϴ ϮϬϭϵ ^< ' Figure 1. Gross domestic product (GDP) index. Source: Trading Economics. Moreover, both countries established mutual cooperation in several areas, one of them is focused on the support of local communities and society by the establishment of the Environmental Partnership Association (EPA) Consortium; they expand the bilateral cooperation in the high-tech sector. The development of mutual cooperation in different sectors of economy forced authors to aim the research at these two countries. Slovakia is the largest producer of cars per capita, with highly developed automobile and electronics exports accounting for more than 80 per cent of national gross domestic product. However, Bulgaria changes its sectoral orientation from agricultural to an industrial economy, which makes the situation in the countries easier to compare. Despite the fact that, according to the rating of the World Bank Country and Lending Groups, Bulgaria is an open upper-middle-income market economy, contrary to Slovakia, which is an open high-income market economy, the importance of the research on the earnings management phenomenon is of a vital importance in both countries. The detection and revelation of manipulation with earnings needs to be portrayed, as it is a relevant measure of investors’ and business partners ´ protection against risks which may occur if distorted and 38 Risks 2020,8,57 incomplete information is presented by the enterprises; thus, it is a helpful tool to solve the basic issues of risk management. It is evident that the earnings management phenomenon plays an important role in financial reports managing and should be properly investigated in conditions of national economies. However, the fact is that this issue has not been explored properly in both countries analyzed. In Slovakia, the first researches on earnings management were published in 2019, highlighting the importance of this issue in unique country samples. Nonetheless, no relevant research has been published yet on the conditions of Bulgaria. Thus, this study investigates earnings management in both countries and determines its presence by quantitative methods of time series. The main aim and the essence of the study is to investigate the question of earnings management in Bulgarian and Slovak environments, where the motivation is to detect the existence of earnings management by time series analysis, as this topic is only rarely searched for in emerging countries. The investigation of the presence of manipulation with earnings may help to reveal the reasons for earnings management occurrence. As the issue of manipulation with earnings in both countries is unexplored, the significance of the analysis of unique country samples has to be underlined. The manuscript is structured as follows. In the introduction, the purpose of the study and the significance of the issue of earnings management are provided. Then, the literature review is presented, concentrating on the analyses of different approaches and investigations of the solved topic. The next chapter depicts the materials used and appropriate methods of mathematical statistics to fulfil the aim. These quantitative methods are: the Dixon test, Jarque–Bera test, Box–Pierce test, Dickey–Fuller tests, Kwiatkowski–Phillips–Schmidt–Shin test, Von Neumann’s test, and Standard Normal Homogeneity test. The outcomes of the investigation, as well as the results of the hypotheses, are portrayed in the Results section. This part confirms the existence of the earnings management of Slovak and Bulgarian enterprises and marks the year 2014 as a significant milestone in the development of earnings management in both countries. In the Discussion section, the connection of ease of doing business, annual growth rate of gross domestic product (GDP), long-term unemployment rank and Standard & Poor’s outlook to the results is implicated and previous studies from emerging markets are compared. The limitations and weaknesses of our study are noted, and possible avenues of future research are determined in the conclusions of the research. 1.1. Literature Review 1.1.1. Graphic Modelling of Specific Accruals The first mention of earnings management is captured in a study of Hepworth (1953), which was focused on balancing periodic income. The author captured several tactics, e.g., methods of balancing income through specific accruals that can be used to move net profit to subsequent accounting periods. Hepworth (1953) did not capture a way to identify the transfer of profits itself. The initial disclosure of corporate earnings management is based on graphical methods based on data set in the time series. Gordon (1964) examines whether managers choose accounting principles and reporting rules that allow them to balance reported earnings. For each of the enterprises examined, he establishes a curve showing the profit calculated in two ways—excluding and including the dependent variables. If the discrepancies in the observations are smaller in the latter case, the earnings adjustment is due to movements in the account. Dopuch and Drake (1966) create a group of enterprises. For each enterprise, they record the total income and income from the given investment shares. The authors argue that adjusting the earnings with this approach does not pose a serious problem for the enterprise in the group, a certain part of the observed enterprise apparently acts purposefully. Archibald (1967) investigates how and why the set of enterprises has shifted from accelerated depreciation of fixed assets to straight-line depreciation for financial and tax reasons. 39 Risks 2020,8,57 1.1.2. Mathematical Modelling of Specific Accruals Gordon et al. (1966) use mathematical modelling to test the profit equalization. The authors choose the investment credit as a variable to test whether enterprises are trying to balance profits. Copeland (1968) empirically tests the use of more than one variable in revealing the existence of earnings management through additional scrutiny of government financial statements. White (1970) applies other tests, using profits from a decade. He includes several dependent variables in the tests and, for the first time, uses regression as a method to detect enterprises that balance earnings. Dascher and Malcom (1970) perform a test applying data from a six- and eleven-year time interval and draw conclusions about the reduction in semi-logarithmic trend variability attributable to discretionary balance variables. Barefield and Comiskey (1972) use data from a ten-year time series to identify variability and average absolute profit increase in enterprises that may use earnings from non-consolidated subsidiaries to balance. 1.1.3. Modelling of Total Discretionary Accruals with Application of Cross-Sectional Data Burgstahler and Dichev (1997) detect earnings management on a cross-sectional analysis. In their research, they verify whether the managers of the tested enterprises are trying to avoid a decline in profits or losses. They choose binomial tests to verify the hypotheses in their research and present the results graphically using histograms. Degeorge et al. (1999) focus on exceeding threshold values. The authors conclude that thresholds artificially evoke specific forms of earnings management, with positive thresholds being the most dominant. 1.1.4. Modelling Using Manipulation Score Beneish (1997) proposes a model detecting earnings manipulation similar to the Altman’s bankruptcy model. Variables called M-score capture both the distortion of financial statements and the factors that can stimulate enterprises to manipulate. Beneish (1997) and Young (1999) independently express doubts about the involvement of depreciation in the measurement of total accruals. 1.1.5. Cross-Sectional Earnings Analysis and Accrual Modelling Peasnell et al. (2000) provide a new approach for approximation of abnormal accruals, labelled as the Margin model, which applied cross-sectional data to mitigate the weaknesses of the Jones model (Jones 1991). The authors take a two-step approach from previous models but use the working capital accrual and different explanatory variables—sales and cash from customers—as an estimate of the total accrual. The authors are criticized for assuming a linear relationship between cash flow and accruals. 1.1.6. Detection of Real Earnings Management Burgstahler and Dichev (1997) find that enterprises often use cash flow gained by operating activities and working capital to earnings management. Headquarters pursuing specific goals thus change their economic performance and their decisions in order to make a profit. Dechow and Skinner (2000) point out that head officers can modify earnings by shifting revenue differentiation, changing the timing of deliveries, or postponing research and development to keep costs at the desired level. Graham et al. (2005) note that the most commonly used earnings management method is the modification of discretionary accruals thanks to its simplicity, inexpensiveness and difficulty to identify by recipients of financial statements. Roychowdhury (2006) finds that many enterprises stop earnings management through discretionary accruals. The author proves that the modification of discretionary accruals is no more the core way of earnings management. Penman and Zhang (2002) argue that enterprises increase earnings by reducing capital investment. Gunny (2010) states that real earnings management involves changes in the underlying operations and activities of the enterprises to increase earnings in the current period. Eldenburg et al. (2011) run their study in the environment of non-profit organizations, proving the existence of real operational decisions in order to manage earnings. 40 Risks 2020,8,57 1.1.7. Modelling Using Neural Networks Hoglund (2012), because of the insufficient results of previous approaches, applies an alternative way to deal with the nonlinearity of accrual processes through neural networks. He designs models based on self-organizing maps, multilayer networks and general regression. 1.1.8. Modelling of Total Discretionary Accruals with Application of Time Series Healy (1985) applies average total accruals as an estimate of discretionary accruals, and thus an estimateofearningsmanagement. Healy’smodelclearlyassumesthenon-existenceofnon-discretionary accruals during estimation periods. The author concludes that the accrual policy of managers is related to incentive bonuses, which are enshrined in their contracts, and thus the shift in accounting practices is related to changes of the extra payment schedule. Kaplan (1985) criticized Healy (1985). DeAngelo (1986) supplemented Healy’s model with an accrual from the previous period. The model does not assume the existence of non-discretionary accruals in the present interval and uses the non-discretionary accruals from the previous period to estimate them. McNichols and Wilson (1988) add to the DeAngelo model capturing discretionary accruals as measures of earnings management, replacing the total accruals applied by Healy (1985) and DeAngelo (1986). Jones (1991) investigates earnings management using two-step models during a government investigation of import relief in the United States. It is used an enterprise-specific model, based on data from at least fourteen-year time series. Discretionary accrual, which represents the remainder, prediction error, calculated as the difference among the current total accruals found in the financial statements and the expected non-discretionary accruals. Dechow et al. (1995) modified the original Jones model by supplementing the year-on-year change in receivables, thus eliminating the error of the discretionary accrual estimate. Guay et al. (1996) criticize both the original and the modified Jones model but does not suggest any other alternatives. Our study also continues in approaches of detecting of earnings management with application of time series. We consider the new gap to disclose the earnings manipulation of the enterprises through unit root and stationarity analysis, supported by homogeneity analyses. The time series analysis allows us to formulate the following hypotheses: •HA. There is a unit root for the series of EBIT. There is a significant existence of the earnings management. •HB.The series of EBIT is not stationary. There is a significant existence of the earnings management. •HC.The series of EBIT is heterogeneous. There is a significant change in the earnings management. •HD. The series of EBIT is heterogeneous. There is a year of a significant change in the earnings management. 2. Materials and Methods The secondary sources are observations of earnings before interest and taxes (EBIT) of the enterprises from the chosen emerging countries (Slovakia and Bulgaria). In the context of historical development, we may add Slovakia to the Soviet-controlled Eastern bloc countries and Bulgaria to the Soviet-controlled Balkans countries. In total, 1347 Slovak enterprises and 1839 Bulgarian enterprises were extracted from the Amadeus database over the period 2010 to 2018 and involved in the analysis. The variable earnings before interests and taxes (EBIT) is selected to eliminate different tax and interest policies of these countries. We require three conditions to be met by the analyzed business units: (a) The amount of total assets is at least EUR 3,000,000; (b) The amount of total sales is at least EUR 2,000,000; (c) The amount of net income is minimally EUR 100,000. These criteria were used to analyze only the companies with stable financial situation and the same financial and economic background to mitigate the problems of the classification of enterprises by their size or the years of their operation. Following methodological steps were used: 41 Risks 2020,8,57 1. The elimination of missing cases. The database Amadeus provides a large sample of data, but there are some missing cases involved. If we have a sufficiently large data file, we may afford a simple solution in the form of removing those units from the file that have missing values (Svabova and Michalkova 2018). Thus, these observations are necessary to be found and eliminated. 2. The removal of inconsistent cases. An outlier in a sample is an observation far away from most or all other observations (Ghosh and Vogt 2012) .Different methods and tests are used to determine the existence of outliers in raw samples. Svabova and Michalkova (2018) recommend in pre-processing of data in earnings management to use Dixon or Grubbs test. Both tests provide satisfying results in identification of the outlying values (Garcia 2012). “Masking phenomenon” (several observations are close together, but the group of observations is still outlying from the rest of data (Berti- É quille et al. 2015) could occur in our case that is why the Dixon statistics r22 is chosen. This test is designed to be used in situations where additional outliers may occur to minimize the effect of these outliers arise because of masking (Garcia 2012). An avoiding of additional outliers allows using conventional Dickey–Fuller tests in further analysis and prevents the spurious rejection of H o of these tests (Leybourne et al. 1998). The test statistic of Dixon is defined as: r22 =y3−y1 yn−2−y1or r22 =yn−yn−2 yn−y3(1) where yis an analyzed variable and numbers mean the places in the order. Nagy (2016) highlights the possibilities after the detection of outliers: do not consider/ignore outliers, exclude outliers or exclude only extreme values (far outliers). We decide to apply the possibility of removal of all inconsistent cases to robust statistics and results insensitive to the outliers which is also supported by the study of Svabova and Durica (2019). They argue that it may be useful to eliminate outlined enterprises from the analyzed group because of the fact that outliers may generate discrepancies of conclusions of statistical tests and procedures. We run test and its p-values are estimated with a Monte Carlo simulation using 1,000,000 replicates. 3. The verification of normal distribution. Normally distributed sample is a required assumption in the estimation of attributes of the times series (Bai and Ng 2005). There are nearly 40 tests of normality in the statistical literature (Dufour Jean-Marie et al. 1998) .Bai and Ng (2005) recommend testing the normality of time series of financial data by the Jarque–Bera test. Jarque and Bera (1980) and Bera and Jarque (1981) show their test statistics as follows: JB =n 6S2+1 4(K−3)2(2) S=Skewness =ˆ μ3 ˆ σ3= 1 nn i=1(yi−y)3 1 nn i=1(yi−y)23 2 (3) K=Kurtosis =ˆ μ4 ˆ σ4= 1 nn i=1(yi−y)4 1 nn i=1(yi−y)22(4) whereyisananalyzedvariable, n meansallamountofobservations, ˆ μ3 and ˆ μ4 meantheapproximations of third and fourth central moments, y means the average of the sample, ˆ σ2 is the approximation of the second central moment, the variance. Jarque–Bera is asymptotically χ 2 distributed with two degrees of freedom because test statistics of Jarque–Bera test is just the sum of squares of two asymptotically independent standardized normals (Bowman and Shenton 1975). 42 Risks 2020,8,57 4. The proof of no serial correlation. The occurrence of no serial correlation means that the data are independently distributed, and it is a recommended assumption for financial time series after testing normality. The Box–Pierce and Ljung–Box tests are generally run to test the required independence in time series. Box and Pierce (1970) perform the test of the randomness at each distinct lag in their study. Ljung and Box (1978) modify this test to overall randomness. We prefer the robustness of the Box–Pierce Q statistic to test if the analyzed sample of financial data is uncorrelated without assuming statistical independence. Q=n h  k=1 r2 k(5) Qis the Box–Pierce test statistic, which is compared with the χ 2 distribution; nmeans all amount of observations; his the maximum lag we are considering (Box and Pierce 1970). 5. The determination of unit root and disproof of stationarity. A time series is stationary if its statistical properties do not change in the process of time. A stationary time series means that the mean and variance are constant over time. The white noise is an example of a stationary time series. The determination that a series is not stationary enables to study where the non-stationarity comes from. Stationarity tests may determine whether a series is stationary or not. There are different approaches on how to test stationarity (unit root or stationarity tests). Unit root tests, as the Dickey–Fuller test and its augmented version, for which H 0 is that the series possesses a unit root and thus is not stationary. On the other hand, there are stationarity tests as the parametric Kwiatkowski–Phillips–Schmidt–Shin test or nonparametric Phillips–Perron test, for which H 0 is that the series is stationary. Standard Dickey–Fuller tests can have very low power and can lead to a very serious problem of spurious rejection of the unit root H 0 (Leybourne et al. 1998) and thus we support the tests by Kwiatkowski–Phillips–Schmidt–Shin test. Dickey and Fuller (1979) show three different equations to test the occurrence of unit root: Δyt=γyt−1+εt(6) Δyt=a0+γyt−1+εt(7) Δyt=a0+γyt−1+a2t+εt(8) where Δytis first order linear differential of equation, γis unit root, εtis white noise. The difference between these deterministic elements is a0 and a2t . Under the null hypothesis, Equation (6) represents a pure model of random walk, Equation (7) adds the intercept a0 , and Equation (8) contains both the a0 as well as linear time trend a2t .Hacker and Hatemi-J (2010) argue that it is difficult to choose from the three Dickey–Fuller equations for unit root testing. According to Elder and Kennedy (2001), if the trend is mistakenly included, the strength of the test drops. On the contrary, if the trend is not included, there is only one way to capture the trend—to use the intercept to detect the trend. All three tests are computed to compare their results and strength in our analysis. The Kwiatkowski–Phillips–Schmidt–Shin test verifies if a time series is stationary around a mean or linear trend or is non-stationary due to a unit root (Kwiatkowski et al. 1992). Time series is divided into the sum of the random walk rt, deterministic trend ξt, and stationary errors εt: yt=rt+ξt+εt(9) where rtis random walk: rt=rt−1+ut(10) where utare independent and identically distributed random variables 0, σ2 u. 43 Risks 2020,8,57 It is used ωstatistics for testing: ω=T t=1S2 t T2ˆ σ2 ε (11) where St= t  i=1 ei(12) and ˆ σ2 εis the estimate of long-term variance et: ˆ σ2 ε=lim T→∞ E TT t=1εt2(13) 6. The determination of heterogeneity. Homogeneity tests allow detecting if time series may be considered as homogeneous during the analyzed time period, or if there is any date at which significant change in a mean of data occurred. Kanovsky (2018) and Agha et al. (2017) recommend selecting from von Neumann test, standard normal homogeneity test, Buishand tests, and Pettitt’s test. We apply the von Neumann test to detect the existence of significant changepoint in the earnings management and parametric standard normal homogeneity test to determine a year when a significant change occurs. Von Neumann’s test is a test using the ratio of mean square successive (year to year) difference to the variance (Von Neumann 1941). The test statistic is shown as follows: N=n−1 i=1(yi−yi+1)2 n−1 i=1(yi−y)2(14) The null hypothesis is that the data are dependent. If the value of Nis equal to 2, it means that the sample is homogeneous while the values of Nless than 2 indicate that the sample has a breakpoint (Buishand 1982). This test gives no information about the break point. The standard normal homogeneity test is a method created by Alexandersson (1986) and assumes if a times series is normally distributed (Kang and Yusof 2012). Then the following model with a single change can be proposed according to Pohlert (2016) as: yi=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ μ+ii=1,...,m μ+Δ+ii=m+1,...,n(15) ε≈ N(0, σ ). The null hypothesis Δ= 0 is tested against the alternative hypothesis Δ 0. The test statistic is: Tk=kz2 1+(n−k)z2 2(1≤k≤n)(16) where z1=1 k k  i=1 yi−y σz2=1 n−k n  i=k+1 yi−y σ(17) The critical value is: T0=max 1≤y≤kTk(18) The p-value is estimated by a Monte Carlo simulation using mreplicates. We run test and its p-values are estimated with a Monte Carlo simulation using 1,000,000 replicates. 44 Risks 2020,8,57 3. Results This part consists of pre-processing data, testing of assumptions and processing results. 3.1. Pre-Processing of Data The samples were very wide but consisted of significant amount of missing values. These values of EBIT were found and eliminated from the Slovak sample of enterprises as well as the Bulgarian one. Table 1involves the number of missing values. Table 1. Investigate samples. Samples Slovakia Bulgaria Origin 1347 1839 Missing values 189 358 Outliers 69 60 Final 1089 1421 Source: own research. The detection of inconsistent data (outliers) follows the identification of missing values. Dixon test is used in the analysis. Testing is run for every observation for each year from the analyzed nine-year period. The outlying cases are detected for every year. The enterprise is removed from the analysis for all periods if only one value is detected as an outlier. The Dixon test is created for small sample despite this fact we use it for its robustness. The p-value was computed using 1,000,000 Monte Carlo simulations. The existence of minimal one outlying value of EBIT for Slovak and Bulgarian samples in every analyzed year is confirmed based on p-value computed in Table 2, which portrays the amount of outlying cases of enterprises for both sides and the final sample as well. Table 2. Dixon test. Year Observed Value Critical Value p-Value (Two-Tailed) Alpha Slovakia Bulgaria Slovakia Bulgaria Slovakia Bulgaria 2010 0.621 0.447 0.174 0.168 <0.0001 <0.0001 0.05 2011 0.272 0.597 0.174 0.168 <0.0001 <0.0001 0.05 2012 0.491 0.205 0.174 0.168 <0.0001 <0.012 0.05 2013 0.273 0.411 0.174 0.168 <0.0001 <0.0001 0.05 2014 0.715 0.290 0.174 0.168 <0.0001 0.0002 0.05 2015 0.611 0.434 0.174 0.168 <0.0001 <0.0001 0.05 2016 0.639 0.241 0.174 0.168 <0.0001 <0.003 0.05 2017 0.809 0.540 0.174 0.168 <0.0001 <0.0001 0.05 2018 0.668 0.248 0.174 0.168 <0.0001 0.002 0.05 Source: own research. Based on annual values of EBIT of 1089 Slovak enterprises and 1421 Bulgarian enterprises, annual average EBIT is calculated for the analyzed period from 2010 to 2019 (Table 3). The development of both countries in time is very similar, which is shown in Figure 2. The similarities of the development of EBIT is also supported by 5% error bars (calculated based on standard deviation) which show almost identical coverage of EBIT development in seven years from the nine-year analyzed period. 45 Risks 2020,8,57 Sosnowski (2018) confirms no proof of the existence of private equity fund between the shareholders of the enterprise in the time of preceding first listing of stocks on a market constrains the applying of earnings management prior to the IPO. He does not reject that any significant discrepancy exists between the discretionary accruals in private equity backed and matched enterprises, when controlling for the market value and book-to-market ratio. Lizinska and Czapiewski (2018) disclose positive and significant discretionary accruals in the IPO year that can be considered as an indication of weak earning quality. They depict that analyzed accruals are indirectly depended on the subsequent long-term market value for IPOs realized before the global recession. Istrate (2019) confirms the increased rounding of earnings in a limited amount of units, even if the amplitude of identified gaps is really significant. The development of the accounting regulation tends to the state when it has begun preferring decreased modifications. The International Financial Reporting Standards (IFRS) transition does not tend to a limitation of the gap among the real occurrence and the normal one. This study finds out that smaller enterprises modify the net income not so significantly upward than the larger enterprises. Turlea et al. (2019) provide results from Romania concerned to the impact of granted by the auditor when the value of discretional accruals is encountered and approximate the impact on the mandatory implementation of IFRS. They estimate the value of discretional accruals by the value of residuals from two equations as regression models that calculate and detect the value of total accruals. The paper of Tanchev and Todorov (2019) examines the long-run and short-run tax buoyancies. They empirically test the impact of the buoyancy on income, profit, and consumption increases in Bulgaria. 5. Conclusions The effective business finance is a key core of the success of all enterprises to be profitable in short as well as long-term period. The globally used phenomenon of earnings management allows a legal opportunity for the enterprises to make a purpose-built decision in the profit policy. Earnings management is widely realized in developed countries and the occurrence is comprehensively mapped. However, the aim of this paper was to detect the existence of earnings management in emerging countries by the times series analysis. Our results confirm that also managers of Slovak and Bulgarian enterprises are not static but significantly manage their earnings during the analyzed nine-year period. Earnings management creates an important part of coherent business finance. It supports the annual prosperity of the enterprises and presents a substantial tool of reducing risk in analyzed emerging countries. The weakness of the provided research is the use of annual average values of EBIT. Panel data for the whole analyzed period may be used in further research. The analysis could be extended for all Soviet-controlled countries to disclose a comprehensive view on the issue of earnings management in these countries with similar historical and political development. We run only Dickey–Fuller test to detect the existence of significant change in the earnings management of Slovak and Bulgarian enterprises, not its modified versions. Further research may support these results by additional use of these tests. The standard normal homogeneity test was used to determine the year of a significant change in the earnings management of Slovak and Bulgarian enterprises, but this test is very sensitive when detecting the breaks near the beginning and the end of the series. Our results focus a priori on the parametric test. In the future investigations, the results of Dickey–Fuller tests may be compared with a nonparametric Phillips–Perron test, Kwiatkowski–Phillips–Schmidt–Shin test with the nonparametric test for stationarity in continuous-time Markov processes, von Neumann’s test and standard normal homogeneity test with nonparametric Pettitt’s test. Author Contributions: Conceptualization, P.D. and V.K.; methodology, P.D.; software, P.D.; validation, D.C., V.K., K.V. and I.A.; formal analysis, V.K., K.V. and I.A.; investigation, P.D. and K.V.; resources, D.C., K.V. and I.A.; data curation, K.V.; writing—original draft preparation, P.D.; writing—review and editing, K.V.; visualization, P.D.; supervision, D.C., V.K., K.V. and I.A.; project administration, K.V.; funding acquisition, K.V. All authors have read and agreed to the published version of the manuscript. 52 Risks 2020,8,57 Funding: This research received no external funding. Acknowledgments: This research was financially supported by the Slovak Research and Development Agency—Grant NO. APVV-17-0546: Variant Comprehensive Model of Earnings Management in Conditions of the Slovak Republic as an Essential Instrument of Market Uncertainty Reduction and VEGA 1/0210/19: Research of innovative attributes of quantitative and qualitative fundaments of the opportunistic earnings modeling. Conflicts of Interest: The authors declare no conflict of interest. References Agha, Omar M. A. Mahmood, Ça˘gatay S. Ba˘gçacı, and Nermin ¸Sarlak. 2017. Homogeneity analysis of precipitation series in North Iraq. IOSR Journal of Applied Geology and Geophysics 5: 57–63. [CrossRef] Alexandersson, Hans. 1986. A homogeneity test applied to precipitation data. Journal of Climatology 6: 661–75. [CrossRef] Andries, Katheleen, Martine Cools, and Steve Van Uytbergen. 2017. To shift or not to shift? Intertemporal income shifting as a response to the risk capital allowance introduction in Belgium. European Accounting Review 26: 531–59. [CrossRef] Archibald, T. Ross. 1967. The return to straight-line depreciation: An analysis of a change in accounting method. Journal of Accounting Research 5: 164–80. [CrossRef] Bai, Jushan, and Serena Ng. 2005. Tests for skewness, kurtosis, and normality for time series data. Journal of Business & Economic Statistics 23: 49–60. Balcerzak, Adam P., Tomas Kliestik, Dalia Streimikiene, and Lubos Smrcka. 2017. Non-parametric approach to measuring the efficiency of banking sectors in European union countries. Acta Polytechnica Hungarica 14: 51–70. Barefield, Russell M., and Eugene E. Comiskey. 1972. The smoothing hypothesis: An alternative test. The Accounting Review 47: 291–98. Ben Amar, Anis, and Salma Chakroun. 2018. Do dimensions of corporate social responsibility affect earnings managements? Evidence from France. Journal of Financial Reporting and Accounting 16: 348–70. [CrossRef] Beneish, Messod D. 1997. Detecting GAAP violation: Implications for assessing earnings management among firms with extreme financial performance. Journal of Accounting and Public Policy 16: 271–309. [CrossRef] Bera, Anil K., and Carlos M. Jarque. 1981. Efficient tests for normality, homoscedasticity and serial independence of regression residuals: Monte Carlo evidence. Economics Letters 7: 313–18. [CrossRef] Berti- É quille, Laure, Ji Meng Loh, and Tamraparni Dasu. 2015. A masking index for quantifying hidden glitches. Knowledge and Information Systems 44: 253–77. [CrossRef] Bowman, Kimiko O., and Lyndon R. Shenton. 1975. Omnibus test contours for departures from normality based on√b1andb2.Biometrika 62: 243–50. [CrossRef] Box, George E. P., and David A. Pierce. 1970. Distribution of residual autocorrelations in autoregressive-integrated moving average time series models. Journal of the American statistical Association 65: 1509–26. [CrossRef] Buishand, T. Adri. 1982. Some methods for testing the homogeneity of rainfall records. Journal of Hydrology 58: 11–27. [CrossRef] Burgstahler, David, and Illia Dichev. 1997. Earnings management to avoid earnings decreases and losses. Journal of Accounting and Economics 24: 99–126. [CrossRef] Bzeouich, Bilel, Faten Lakhal, and Neila Dammak. 2019. Earnings management and corporate investment efficiency: Does the board of directors mater? Journal of Financial Reporting and Accounting 17: 650–70. [CrossRef] Christensen, Hans B., Edward Lee, Martin Walker, and Cheng Zeng. 2015. Incentives or Standards: What determines accounting quality changes around IFRS adoption? European Accounting Review 24: 31–61. [CrossRef] Cimini, Riccardo. 2015. How has the financial crisis affected earnings management? A European study. Applied Economics 47: 302–17. [CrossRef] Copeland, Ronald M. 1968. Income smoothing. Journal of Accounting Research 6: 101–16. [CrossRef] 53 Risks 2020,8,57 Cugova, Aneta, Juraj Cug, and George Lazaroiu. 2019. Descriptive Analysis of Company’s Profit as a Basis for Earnings Management. Paper presented at 33rd International-Business-Information-Management- Association (IBIMA) Conference: Education Excellence and Innovation Management Through Vision 2020, Granada, Spain, April 10–11; pp. 5629–38. Dascher, Paul E., and Robert E. Malcom. 1970. A note on income smoothing in the chemical industry. Journal of Accounting Research 8: 253–59. [CrossRef] DeAngelo, Linda Elizabeth. 1986. Accounting numbers as market valuation substitutes: A study of management buyouts of public stockholders. Accounting Review 61: 400–20. Dechow, Patricia M., and Douglas J. Skinner. 2000. Earnings management: Reconciling the views of accounting academics, practitioners, and regulators. Accounting Horizons 14: 235–50. [CrossRef] Dechow, Patricia M., Richard G. Sloan, and Amy P. Sweeney. 1995. Detecting earnings management. Accounting Review 70: 193–225. Degeorge, Francois, Jayendu Patel, and Richard Zeckhauser. 1999. Earnings management to exceed thresholds. The Journal of Business 72: 1–33. [CrossRef] Dickey, David A., and Wayne A. Fuller. 1979. Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74: 427–31. Dopuch, Nicholas, and David D. Drake. 1966. The effect of alternative accounting rules for nonsubsidiary investments. Journal of Accounting Research 4: 192–219. [CrossRef] Dufour Jean-Marie, Abdeljelil Farhat, Lucien Gardiol, and Lynda Khalaf. 1998. Simulation-based finite sample normality tests in linear regressions. Econometrics Journal 1: 154–73. [CrossRef] Eldenburg, Leslie G., Katherine A. Gunny, Kevin W. Hee, and Naomi Soderstrom. 2011. Earnings management using real activities: Evidence from nonprofit hospitals. The Accounting Review 86: 1605–30. [CrossRef] Elder, John, and Peter E. Kennedy. 2001. Testing for unit roots: What should students be taught? The Journal of Economic Education 32: 137–46. [CrossRef] Ferrer Garcia, Cristina, and Jose Antonio Lainez Gadea. 2013. Detecting differences on the earnings quality measurement: Empirical evidence on Spanish firms. Revista de Metodes Cuantitativos para la Economia y la Empresa 16: 5–28. Garcia, Francisco Augusto Alcaraz. 2012. Tests to Identify Outliers in Data Series. Rio de Janeiro: Pontifical Catholic University of Rio de Janeiro, Industrial Engineering Department. Ghosh, Dhiren, and Andrew Vogt. 2012. Outliers: An evaluation of methodologies. Joint Statistical Meetings, 3455–60. Available online: https://ww2.amstat.org/meetings/jsm/2012/onlineprogram/AbstractDetails.cfm? abstractid=304068 (accessed on 30 May 2020). Gordon, Myron J. 1964. Postulates, principles and research in accounting. The Accounting Review 39: 251. Gordon, Myron J., Bertrand N. Horwitz, and Philip T. Meyers. 1966. Accounting measurements and normal growth of the firm. Research in Accounting Measurement 23: 223–41. Graham, John R., Campbell R. Harvey, and Shiva Rajgopal. 2005. The economic implications of corporate financial reporting. Journal of Accounting and Economics 40: 3–73. [CrossRef] Guay, Wayne R., S. P. Kothari, and Ross L. Watts. 1996. A market-based evaluation of discretionary accrual models. Journal of Accounting Research 34: 83–105. [CrossRef] Gunny, Katherine A. 2010. The relation between earnings management using real activities manipulation and future performance: Evidence from meeting earnings benchmarks. Contemporary Accounting Research 27: 855–88. [CrossRef] Hacker, Scott R., and Abdulnasser Hatemi-J. 2010. The Properties of Procedures Dealing with Uncertainty about Intercept and Deterministic Trend in unit Root Testing. CESIS Electronic Working Paper Series; Stockholm: Centre of Excellence for Science and Innovation Studies, The Royal Institute of Technology, pp. 1–19. Healy, Paul. 1985. The impact of bonus schemes on the selection of accounting principles. Journal of Accounting and Economics 7: 85–107. [CrossRef] Hepworth, Samuel R. 1953. Smoothing periodic income. The Accounting Review 28: 32–39. Hoang, Trang Cam, and Dempsey Michael Joseph. 2019. The effect of new corporate accounting regime on earnings management: Evidence from Vietnam. Journal of International Studies 12: 93–104. Hoglund, Henrik. 2012. Detecting earnings management with neural networks. Expert Systems with Applications 39: 9564–70. [CrossRef] 54 Risks 2020,8,57 Hudakova, Maria, Matej Masar, Maria Luskova, and Milan Robin Patak. 2018. The dependence of perceived business risks on the size of SMEs. Journal of Competitiveness 10: 54–69. [CrossRef] Iatridis, George, and George Kadorinis. 2009. Earnings management and firm financial motives: A financial investigation of UK listed firms. International Review of Financial Analysis 18: 164–73. [CrossRef] Istrate, Costel. 2019. Detecting earnings management using Benford’s Law: The case of Romanian listed companies. Journal of Accounting and Management Information Systems 18: 198–223. [CrossRef] Jarque, Carlos M., and Anil K. Bera. 1980. Efficient tests for normality, homoscedasticity and serial independence of regression residuals. Economics Letters 6: 255–59. [CrossRef] Jones, Jennifer J. 1991. Earnings management during import relief investigations. Journal of Accounting Research 29: 193–228. [CrossRef] Kang, Ho Ming, and Fadhilah Yusof. 2012. Homogeneity tests on daily rainfall series. International Journal of Contemporary Mathematical Sciences 7: 9–22. Kanovsky, Martin. 2018. The research effectivity of Slovak universities: Quantitative analysis of trends 2008–2017. Slovak Sociological Review 50: 429–47. [CrossRef] Kaplan, Robert S. 1985. Evidence on the effect of bonus schemes on accounting procedure and accrual decisions. Journal of Accounting and Economics 7: 109–13. [CrossRef] Kempen, R. 2010. Earnings Management in Public and Private Companies in the Netherlands. Master’s thesis, Accounting and Finance, Erasmus University Rotterdam, Rotterdam, The Netherlands; pp. 1–32. Khanh, Hoang Mai Thi, and Anh Phung Thu. 2019. The effect of financial leverage on real and accrual-based earnings management in Vietnamese firms. Economics and Sociology 12: 299–333. [CrossRef] Kral, Pavol, Viera Valjaskova, and Katarina Janoskova. 2019. Quantitative approach to project portfolio management: Proposal for Slovak companies. Oeconomia Copernicana 10: 797–814. [CrossRef] Kwiatkowski, Denis, Peter C. B. Phillips, Peter Schmidt, and Yongcheol Shin. 1992. Testing the null hypothesis of stationarity against the alternative of a unit root. Journal of Econometrics 54: 159–78. [CrossRef] Leybourne, Stephen. J., Terence C. Mills, and Paul Newbold. 1998. Spurious rejections by Dickey–Fuller tests in the presence of a break under the null. Journal of Econometrics 87: 191–203. [CrossRef] Lizinska, Joanna, and LeszekCzapiewski. 2018. Towardseconomic corporate sustainabilityin reporting: Whatdoes earnings management around equity offerings mean for long-term performance? Sustainability 10: 4349. [CrossRef] Ljung, Greta M., and George E. P. Box. 1978. On a measure of lack of fit in time series models. Biometrika 65: 297–303. [CrossRef] Madgerova, Raya, Vyara Kyurova, Anny Atanasova, and Blagovesta Koyundzhiyska-Davidkova. 2019. Study of the business environment for the development of youth entrepreneurship in Bulgaria. Theoretical and practical aspects. Revista Inclusiones 6: 90–104. Marin, Nikolay, and Preslav Dimitrov. 2018. Economic Effects of the membership of the Republic of Bulgaria in the European Union. Kutafin University Law Review 5: 209–21. McNichols, Maureen, and Peter G. Wilson. 1988. Evidence of earnings management from the provision for bad debts. Journal of Accounting Research 26: 1–31. [CrossRef] Meyers, Thomas D., Ladislav Vagner, Katarina Janoskova, Iulia Grecu, and Gheorghe Grecu. 2019. Big data-driven algorithmic decision-makingin selecting andmanaging employees: Advanced predictive analytics, workforce metrics, and digital innovations for enhancing organizational human capital. Psychosociological Issues in Human Resource Management 7: 49–54. Nagy, Ladislav. 2016. Ratio indicators database of HGN1 model for measuring enterprise performance. Journal of Knowledge Society 4: 1–11. Orazalin, Nurlan. 2019. Board gender diversity, corporate governance, and earnings management. Gender in Management 35: 37–60. [CrossRef] Orazalin, Nurlan, and Rassul Akhmetzhanov. 2019. Earnings management, audit quality, and cost of debt: Evidence from a Central Asian economy. Managerial Auditing Journal 34: 696–721. [CrossRef] Pavlovic, Vladan, Goranka Knezevic, and Radica Bojicic. 2018. Board gender diversity and earnings management in agricultural sector—Does it have any influence? Custos E Agronegocio On Line 14: 340–63. Pavlovic, Vladan, Goranka Knezevic, and Radica Bojicic. 2019. Does board of director’s age impact earnings management in agriculture? Case of Serbian agricultural sector. Custos E Agronegocio on Line 15: 254–72. 55 Risks 2020,8,57 Peasnell, Ken V., Peter F. Pope, and Steven Young. 2000. Detecting earnings management using cross-sectional abnormal accruals models. Accounting and Business Research 30: 313–26. [CrossRef] Penman, Stephen H., and Xiao-Jun Zhang. 2002. Accounting conservatism, the quality of earnings, and stock returns. The Accounting Review 77: 237–64. [CrossRef] Pina, Vicente, Maria Jose Arcas, and Caridad Marti. 2012. Accruals and accounting numbers management in UK executive agencies. Public Money & Management 32: 273–80. Piosik, Andrzej, and Ewa Genge. 2019. The influence of a company’s ownership structure on upward real earnings management. Sustainability 12: 152. [CrossRef] Pohlert, Thorsten. 2016. Non-Parametric Trend Tests and Change-Point Detection. Creative Commons License (CC BY-ND 4.0). Available online: https://mran.microsoft.com/snapshot/2016-06-30/web/packages/trend/ vignettes/trend.pdf (accessed on 30 March 2020). Rahman, Ashiqur, Jaroslav Belas, Tomas Kliestik, and Ladislav Tyll. 2017. Collateral requirements for SME loans: Empirical evidence from the Visegrad countries. Journal of Business Economics and Management 18: 650–75. [CrossRef] Rodriguez-Perez, Gonzalo, and Stefan van Hemmen. 2010. Debt, diversification and earnings management. Journal of Accounting and Public Policy 29: 138–59. [CrossRef] Roychowdhury, Sugata. 2006. Earnings management through real activities manipulation. Journal of Accounting and Economics 42: 335–70. [CrossRef] Sosnowski, Tomasz. 2018. Earnings management in the private equity divestment process on Warsaw Stock Exchange. Equilibrium. Quarterly Journal of Economics and Economic Policy 13: 689–705. [CrossRef] Sosnowski, Tomasz, and Anna Wawryszuk-Misztal. 2019. Diversity on management and supervisory board and accuracy of management earnings forecasts in IPO prospectuses. Ekonomia i Prawo. Economics and Law 18: 347–63. [CrossRef] Spuchlakova, Erika, and Juraj Cug. 2015. Credit risk and LGD modelling. Procedia Economics and Finance 23: 439–44. [CrossRef] Tanchev, Stoyan, and Ivan Todorov. 2019. Tax buoyancy and economic growth: Empirical evidence of Bulgaria. Journal of Tax Reform 5: 236–48. Susanto, Yulius Kurnia, Kashan Pirzada, and Adrianne Sheryl. 2019. Is tax aggressiveness an indicator of earnings management? Polish Journal of Management Studies 20: 516–27. Svabova, Lucia, and Marek Durica. 2019. Being an outlier: A company non-prosperity sign? Equilibrium. Quarterly Journal of Economics and Economic Policy 14: 359–75. [CrossRef] Svabova, Lucia, and Lucia Michalkova. 2018. Pre-processing of data in earnings management. Business Economics and Management 3: 65–72. Trading Economics. 2020. Available online: https://tradingeconomics.com/indicators (accessed on 20 April 2020). Turlea, Carmen, Laura Brad, Florin Dobre, and Ioan Codrut Turlea. 2019. The Influence of Financial Auditor on the Earnings Management Process: Mandatory Application of IFRS for the First Time in Romania. Paper presented at BASIQ International Conference on New Trends in Sustainable Business and Consumption, Bari, Italy, May 30–June 1. Vagner, Ladislav. 2017. Facility Management as a Partner of Cost Controlling at Costs Optimization in the Selected Enterprise. Paper presented at 17th Annual Conference on Finance and Accounting: New Trends in Finance and Accounting, Prague, Czech Republic, May 27; pp. 621–29. Valaskova, Katarina, Tomas Kliestik, and Maria Kovacova. 2019. Assessment of Selected Models of Earnings Management in Economic Conditions of Slovakia. Paper presented at 33rd International-Business- Information-Management-Association (IBIMA) Conference Education Excellence and Innovation Management Through Vision 2020, Granada, Spain, April 10–11; pp. 3922–31. Velte, Patrick P. 2019. The bidirectional relationship between ESG performance and earnings management—Empirical evidence from Germany. Journal of Global Responsibility 10: 322–38. [CrossRef] Von Neumann, John. 1941. Distribution of the ratio of the mean square successive difference to the variance. The Annals of Mathematical Statistics 12: 367–95. [CrossRef] 56 Risks 2020,8,57 White, Gary E. 1970. Discretionary accounting decisions and income normalization. Journal of Accounting Research 8: 260–73. [CrossRef] Young, Steven. 1999. Systematic measurement error in the estimation of discretionary accruals: An evaluation of alternative modelling procedures. Journal of Business Finance & Accounting 26: 833–62. © 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). 57 risks Article A Raroc Valuation Scheme for Loans and Its Application in Loan Origination Bernd Engelmann and Ha Pham * Faculty of Finance-Banking, Ho Chi Minh City Open University, 35-37 Ho Hao Hon, Dist 1, Ho Chi Minh City 700000, Vietnam; [email protected] *Correspondence: [email protected] Received: 27 April 2020; Accepted: 4 June 2020; Published: 10 June 2020 Abstract: In this article, a risk-adjusted return on capital (RAROC) valuation scheme for loans is derived. The critical assumption throughout the article is that no market information on a borrower’s credit quality like bond or CDS (Credit Default Swap) spreads is available. Therefore, market-based approaches are not applicable, and an alternative combining market and statistical information is needed. The valuation scheme aims to derive the individual cost components of a loan which facilitates the allocation to a bank’s operational units. After its introduction, a theoretical analysis of the scheme linking the level of interest rates and borrower default probabilities shows that a bank should only originate a loan, when the interest rate a borrower is willing to accept is inside the profitability range for this client. This range depends on a bank’s internal profitability target and is always a finite interval only or could even be empty if a borrower’s credit quality is too low. Aside from analyzing the theoretical properties of the scheme, we show how it can be directly applied in the daily loan origination process of a bank. Keywords: loan pricing; RAROC; loan origination JEL Classification: C69; C19 1. Introduction A loan is probably the most traditional banking product. However, when different people in different countries or even different people in the same country working in different customer segments speak about a loan, and they probably do not speak about the same product. The only common feature is that a lender gives money to a borrower and hopes to get back more than he has lent. Besides that, differences can be substantial. Critical drivers of product structure are the availability of funding and collateral. In some countries, only short-term funding is available to a bank. For this reason, interest rates of loans are rarely fixed over a long-term horizon but can be adjusted by a bank on short notice. In other countries, long-term funding is available, and loans are often fixed-rate or floating-rate loans, where the floating rate follows an objective rule like 6M Ibor plus a spread. 1 The most prominent type of loans that is linked to a particular collateral type is the mortgage. Here, often long maturities up to 30 years are observed. However, still, there are some differences between countries. For instance, in the US, a borrower can pass the key of a house to a bank when the house loses in value while in Germany defaulting on a mortgage is not that easy, and the borrower still is responsible for the residual amount between the loan balance and house value. There are a lot more differences. In some countries, borrowers have prepayment rights on their loans; in other countries, prepayment rights 1In this article, Ibor stands for all kinds of official floating rates like Libor, Euribor, etc. Risks 2020,8, 63; doi:10.3390/risks8020063 www.mdpi.com/journal/risks 59 Risks 2020,8,63 are less popular, but floating-rate loans are embedded with caps and floors. A lot more covenants can be included, like interest rates that increase with rating downgrades or minimum requirements on collateral value. In this article, we develop a loan pricing scheme based on risk-adjusted return on capital (RAROC) as a performance measure. The purpose of this article is twofold. First, we propose a scheme that is directly implementable in banking practice drawing on input data that is readily available in most banks. This data consists of quotes from the interbank market, like deposit and swap rates, internal costs for funding and operation, and credit risk parameters related to borrower and collateral, i.e., statistical default probabilities and loss rates. We explicitly assume that no market information related to borrower credit quality, like bond or CDS spreads, is observable. The proposed scheme is most valuable during loan origination where it provides a bank not only with the loan performance related to a particular offer of the loan’s interest rate but, in addition, with a decomposition of the interest rate into cost components associated with different bank operations related to lending. These are the funding of a bank loan by deposits, the management of loss risks, the hedging of interest rate risks, and the coverage of operational costs. Therefore, the scheme could be helpful in determining fund transfer prices between a bank’s operational units. In addition, the scheme delivers a loan valuation. This could be valuable in price negotiations when loan portfolios are sold to investors. In economies with negative interest rate like the European Union, life insurers and pension funds are increasingly attracted by residential mortgages (European Central Bank 2019) where they are still able to get a positive return in contrast to most government bonds. To determine a price during sales negotiations and for the investor’s ongoing reporting, our proposed model could be applied. The second contribution of this article is an analysis of the RAROC scheme’s theoretical properties. Roughly speaking, RAROC is defined as (interest income − costs) / capital. On first glance this means if a bank would raise interest rates, leaving all other quantities unchanged, profitability would increase. However, it is intuitively clear that raising interest payments beyond the income of a borrower or the profit of a firm will lead to a default. This means, an economically meaningful performance measure should reflect this by having a low value in this scenario. We will show that, when properly relating borrower default probability and loan interest rate, this is indeed reflected by RAROC, i.e., that there is only a finite interval of interest rates where it is economically sensible for a bank to originate a loan. The market power of banks in a particular loan segment will then determine whether a bank can originate a loan at the lower or the upper end of this interval. This interval might be empty in cases where a borrower’s credit quality is too low, meaning that a bank should not originate a loan to this borrower regardless of the interest rate. The general pricing framework presented in this article is not linked to a particular loan segment and is applicable both for corporate and retail lending. In the next Section, we will provide a literature review. In Section 3, the loan pricing formula and its parameterization will be explained. In Section 4, the RAROC pricing scheme will be developed, and the calculation of all its cost components will be derived. After that, in Section 5the theoretical properties of this scheme will be analyzed by linking the level of interest rates to the default rates of a borrower, and it will be shown how meaningful loan acceptance rules can be derived. In Section 6, numerical examples are presented for illustration. The final Section concludes. 2. Literature Review Our aim is developing a RAROC scheme that is applicable in banks world-wide drawing on inputs that are readily available in most banks. This is data related to internal costs and risk parameters measured statistically possible amended by expert judgment. As already outlined in the introduction, we explicitly assume that there is no market information of a borrower available, i.e., the borrower did not issue any bonds but all his debt consists exclusively of bank loans. This implies that the stream of literature using risk-neutral probabilities for asset pricing, e.g., as in Jarrow et al. (1997)or 60 Risks 2020,8,63 Choi et al. (2020) , which are based on trading strategies in arbitrage-free markets, is not applicable in this context. A recent article on fair value approaches for loans is Skoglund (2017) where the utilization of market data for loan valuation is discussed. In most loan markets world-wide this is not feasible since the required market data is available only in countries with developed capital markets. Even in these countries, market valuations can only be applied to a small segment of the loan market, typically large corporates. The RAROC concept dates back to the 70s, where it has been developed by practitioners. Often, it is applied in a one-period model analyzing one year only even if the loan maturity is longer (Crouhy et al. 1999) . The earliest extensions of RAROC to a multi-period framework are Aguais et al. (1998); Aguais and Forest (2000); Aguais and Santomero (1998) . However, the description is very sketchy and loan pricing aspects that became important after the financial crisis are, of course, not included since these articles were written well before. Besides that, there is no split of the interest rate into cost components related to different operational units of a bank provided. Closing this gap is one purpose of this article. When applying a RAROC scheme in practice, a number of aspects have to be considered that are not covered by our work but are required as inputs. These are the definition of capital for performance measurement, the allocation of total bank capital to sub-units, and the determination of target equity returns. As outlined in the introduction, RAROC is broadly defined as (interest income − costs)/capital. There are two definitions of capital, regulatory capital that is defined by supervisory rules Basel Committee on Banking Supervision (2006,2011), and economic capital. Economic capital is measured by a credit portfolio model that should reflect economic reality more closely than the regulatory rules, e.g., by quantifying concentration risks. Most economic capital models are based on Gupton et al. (1997) and CSFB (1997). Once the loss distribution of a credit portfolio is computed a risk measure is derived. Usually expected shortfall is used to compute portfolio risk and the risk is distributed among the single credits in a process called capital allocation. For details see Kalkbrener et al. (2004), Kalkbrener (2005) and Balog et al. (2017). Both academia and practice have used predominantly economic capital models for performance measurement. A recent study is Chun and Lejeune (2020) where a multi-period model for loan pricing and profit maximization in an economic capital framework is developed. Compared to our work they do not model the interaction of default probabilities and interest rates but link the interest rate to a probability that a borrower accepts a loan offer. Besides that, their framework cannot be used for fund transfer pricing as they do not provide a loan’s cost components. As pointed out by Klaassen and Van Eeghen (2018), economic capital usually exceeded regulatory capital in the past. For this reason, measuring loan performance by economic capital did not interfere with the regulatory rules since it was more conservative. However, this has changed with the recent Basel reforms resulting in higher minimum requirements of regulatory capital. When regulatory capital is higher than economic capital, performance measurement based on economic capital no longer works because it is overstating true performance. This motivated banks to move from economic to regulatory capital for performance measurement, as confirmed by a survey in (Ita 2016, p. 38), and recent empirical research (Akhtar et al. 2019;Oino 2018). The RAROC scheme in this article does not depend on a particular capital definition and will work with either economic or regulatory capital. We will use regulatory capital for illustration and concrete examples because it is more tractable and increasingly more common in practice. An important process where RAROC (or the related alternative RORAC) is applied, is the allocation of bank capital to business units. Here, a bank faces the problem that the managers of business units, who have only limited information about the total bank, should optimize the performance of their unit in a way leading to the maximum performance of the organization. The first article, demonstrating the usefulness of RAROC for this purpose is Stoughton and Zechner (2007). Extensions of this work are Buch et al. (2011), Baule (2014), Turnbull (2018) and Kang and Poshakwale (2019) where the latter even empirically demonstrate the usefulness of this approach. In this article, we do not analyze capital allocation on the business unit level but assume 61 Risks 2020,8,63 calculation of the capital buffer E is simple and E is independent of credit risk parameters like PD and LGD. Most of the internationally active banks, however, apply the Internal Ratings Based Approach which allows banks to compute minimum capital buffers from internal estimates of PD and LGD using the formula E=NLGDΦΦ−1(PD)+√ρΦ−1(0.999) 1−ρ−PD, (22) where PD = 1 −v( 1 ) is the one-year default probability of the borrower, LGD can be computed from the collateralization at the loan’s start, and ρ is the asset correlation which is defined in Basel Committee on Banking Supervision (2006) depending on the borrower segment. 3 As already outlined in Section 2, using regulatory capital for performance measurement becomes increasingly more popular in banking practice due to tightened capital requirements. However, the scheme would also work for E derived from a more complex credit portfolio model. The capital E is allocated to the loan and cannot be used for other investments. A bank defines a target return wt on its equity capital. As already outlined in Section 2this could be done by some modeling approach or by expert decision. The equity capital E is not lying in a safe but invested in assets like government bonds where it generates a return wr . The difference between wt and wr has to be generated by the interest income of the loan. This leads to an additional interest rate margin sUL , the unexpected loss margin, which is computed as sUL =(wt−wr)E ND. (23) Finally, the operating costs of a bank, like staff salaries or office costs, have to be covered by a loan’s interest income. These costs are summarized in an additional cost margin c . The calculation of c depends on the institutional details of a bank, and there is no general rule that is applicable to any bank. To include these costs into RAROC an adjustment is required reflecting the fact that only surviving borrowers can cover the costs. This leads to a cost margin scwhich is computed as sc=c∑t<TiNiτiδi(Ti) ∑t<TiNiτiδi(Ti)v(Ti)(24) Note, that this assumption is not required for economic capital. The reason is that by construction the expected loss margin sEL should be sufficient to cover expected loss and economic capital is only a buffer against unexpected events. Once a borrower defaults and a loss provision is built, the capital is freed and can be used for other investments of the bank. Putting all cost components together gives the hurdle rate zh of a loan, i.e., the interest rate that covers all costs and profitability targets of a bank. It is computed as zh=ys+sf+sb+sEL +sUL +sc. (25) Note, that this calculation is true only if the PD of a borrower does not depend on the interest rate z . If this is not the case (25) has to be replaced by a numerical algorithm as discussed in the following two sections. If the interest rate z is given, the return on equity capital, or, equivalently, a loan’s RAROC can be computed as RAROC =z−ys−sf−sb−sEL −sc E/ND+wr. (26) 3 Note that for calculating E under Basel II, a different set of PD and LGD is applied than for expected loss calculations. For regulatory purposes, PD is a long-term average reflecting average default risk over an economic cycle while LGD is computed under worst-case assumptions. We do not go into these details here since the precise calculation of E does not affect the structure of the RAROC scheme. 68 Risks 2020,8,63 This equation allows a bank to measure the impact of interest rates different from the hurdle rate zh on the return on economic capital. Furthermore, (26) can be used to measure the performance of already existing loans. 5. Properties of RAROC When looking at (26) it seems that if z becomes arbitrarily large, so does RAROC. This means that this performance measure suggests that banks should charge as high as possible interest rates to maximize profitability. Obviously, this reasoning is flawed since, at a certain interest rate level, a borrower is unable to service his debt and will default. In order to make RAROC realistic, we have to link default probabilities and interest rates. When building internal models for the default risk of a borrower, banks often include a variable known as the debt service ratio (DSR) into the list of explanatory risk factors. DSR computes the ratio of annual interest and amortization payments on all loans of a borrower and the available funds to pay interest. In the case of a company, these funds are net profit before interest and taxes. In the case of a retail client, it is net annual income. The interest rate z of a loan enters DSR linearly as DSR =β0+z·β1 . The coefficient β0 contains payments on other existing credit products a borrower might have, while β1 is the ratio of the loan’s balance divided by available funds. If β1 is small, then PD can be approximately considered as independent of z . The larger β1 , however, the more this approximation leads to wrong conclusions. To analyze the properties of RAROC when default risk and interest rates are coupled, we use a simplified setup to maintain analytical tractability. We assume a bullet loan with a balance of one that pays interest annually at times Ti= 1, ... , m and t= 0. Furthermore, we assume funding costs, hedging costs and operational costs of a bank are zero. In addition, we assume the interbank curve is flat, and all zero rates are zero, i.e., all discount factors are equal to one. Finally, we assume wr equals zero and Ri is a constant R . Concerning economic capital, we assume that a bank follows the Basel Standardized Approach, i.e., that economic capital E is independent of PD. This leads to a simplified RAROC formula RAROC =z−sEL E, (27) with a simplified sEL as sEL =(1−R)·(1−v(m)) ∑m i=1v(i). (28) For the latter equation, note that in (21) Ai is zero for all i<m for a bullet loan and one for i=m . The default part VD simplifies because Ri is constant and discount factors are one which results in a telescoping sum that can be simplified to 1 −v(m). To model survival probabilities, we assume DSR is part of the risk factors in (10) and we condense all other risk factors into the constant β0 . Furthermore, we assume a constant h . This leads to the survival probability v(i)at time iof v(i)=exp −exp(β0+z·β1)·i 0hds  =exp (−exp(β0+z·β1)·hi) =exp (−exp(β0+z·β1))hi =qhi, (29) q:=exp(−exp(β0+z·β1)). (30) The properties of RAROC under these assumptions are summarized in Theorem 1. Theorem 1. Under the assumptions of (27)–(29) where the constants β1 , h and R are required to fulfill β1> 0, h>0, and 0<R<1, RAROC(z) as a function of the loan interest rate z has the properties: 1. lim z→−∞RAROC(z)=−∞ 69 Risks 2020,8,63 2. lim z→+∞RAROC(z)=−∞ 3. There exists a unique interest rate zmax with RAROC(z)≤RAROC(zmax),∀z∈R The proof of Theorem 1is provided in the Appendix A. The economic interpretation of Theorem 1 is quite intuitive. The first relation means that if a bank pays huge interest until the verge of bankruptcy, RAROC becomes arbitrarily small. If, on the other hand, the borrower pays huge interest which brings him close to bankruptcy, RAROC becomes arbitrarily small, too. Therefore, if a loan brings either the borrower or the bank into trouble, this is adequately reflected by RAROC. Somewhere in between these extreme cases, there is an optimum from the perspective of the bank which allows the bank to generate high income while keeping default risk manageable if the client is creditworthy. Theorem 1has direct consequences on the loan origination process of a bank. There exists only a finite range of interest rates that should be considered as acceptable from a bank’s perspective. Given the profitability target of a bank wt only loans should be accepted with a RAROC greater or equal wt . This translates directly into a set of acceptable interest rates which we call the profitability range. Theorem 2. Define the profitability range P for a loan as the set of interest rates leading to a RAROC greater or equal wt P={z:RAROC(z)≥wt}. (31) Then exactly one of the three cases is true: 1. P is empty 2. P consists of one point zmax 3. P consist of an interval [zh,zmax] The proof of Theorem 2follows directly from Theorem 1. In the case of RAROC(zmax)<wt , P is empty and if RAROC(zmax)=wt then P={zmax} . Finally, if RAROC(zmax)>wt , there exists an interval [zl , zu] where RAROC(z)≥wt . The lowest interest rate of this interval is the hurdle rate zh which is the minimum interest rate that covers all costs and risks associated with the loan. Note, when PD is a function of z , the hurdle rate can no longer be determined by (25) but has to be computed by a numerical algorithm finding min zRAROC(z)≥wt . Although there exists interest rates z with z>zmax and RAROC(z)>wt it does not make sense for a bank to charge them. It can achieve the same profitability at a lower interest rate making it more likely that the client will accept the offer from the bank and not from a competitor. To conclude this Section, we remark that we suppose that Theorem 1holds in more general setup. In numerical examples, when using (22) for calculating economic capital E we still get a unique maximum RAROC in numerical examples. While it is quite easy to show that Parts 1 and 2 of Theorem 1still hold, the third part becomes rather complex since the most general Basel formula includes PD as a function of z and the asset correlation ρ as a function of PD and, therefore, as a function of z which makes the analytical treatment of RAROC in this case rather difficult. Yet from our numerical experiments, we suppose that Theorem 1holds in the more general setup. 6. Numerical Example To illustrate the RAROC pricing scheme, we consider fixed-rate loans with or without amortization, and with or without collateralization. We consider a ten-year fixed-rate loan paying an interest rate of 4% with quarterly interest payments. The loan’s notional is N= 1,000, 000 which is paid in one tranche at the loan’s start date. We consider a bullet loan, i.e., a loan without amortization payments and an installment loan with an amortization rate of 5% annually. This means that in addition to the interest payment, the installment loan pays back 1.25% of the initial notional, i.e., Ai= 12,500 every quarter. Furthermore, the impact of collateral is illustrated. We assume that in this 70 Risks 2020,8,63 case, collateral with a cash equivalent value of C= 600,000 is available. For the unsecured parts of the loan, we assume a recovery rate Ru= 20%. This leads to a total of four different loans. For these loans RAROC is computed in the first part and hurdle rates and maximum RAROC in the second part. To carry out these calculations, information about interest rate markets and institutional details of the bank is required. In the first step, the information on funding and interest rate markets is collected. We assume that the funding of a bank is expressed as a spread over 12M Ibor rates, i.e., the bank funds itself by issuing bonds paying annual interest linked to a 12M Ibor rate. Furthermore, swap rates of fixed-to-floating swaps and basis swaps have to be included to account for the tenor mismatch in funding and lending. Assuming the European conventions, we have quotes for swaps exchanging a fixed-rate against a 6M Ibor rate. Furthermore, we need the spreads of basis swaps exchanging a 6 M Ibor rate against a 12M Ibor rate because of the funding tenor Λf= 12M, and we need the spreads of basis swaps exchanging a 3M Ibor rate against a 6M Ibor rate because of the loan’s tenor Λl= 3M. The data is summarized in Table 1. Table 1. Quotes of fixed-to-floating swaps, 3M Ibor against 6M Ibor basis swaps, 6M Ibor against 12M Ibor basis swaps, and funding spreads. All quotes are in percent. Tenor Swap Rate 3M→6M Spread 6M→12M Spread Funding Spread 3M 0.05 6M 0.15 1Y 0.22 0.10 0.08 0.10 2Y 0.45 0.10 0.08 0.12 3Y 0.58 0.10 0.08 0.14 4Y 0.75 0.10 0.08 0.17 5Y 0.95 0.10 0.08 0.20 6Y 1.13 0.10 0.08 0.22 7Y 1.30 0.10 0.08 0.25 8Y 1.47 0.10 0.08 0.28 9Y 1.62 0.10 0.08 0.30 10Y 1.76 0.10 0.08 0.33 12Y 1.96 0.10 0.08 0.40 15Y 2.12 0.10 0.08 0.50 The front part of the discount curves that is bootstrapped from fixed-to-floating swaps is built from deposit rates. In the example of Table 1the 3M and 6M deposit rate are used for computing the front part of δM,6M. The data in Table 1are not real market quotes but serves for illustration only. For the evaluation of default risk, we assume that a bank has established a rating system with six grades and uses a Cox proportional hazard model (10) to estimate term-structures of default probabilities. We assume that the loan’s interest rate is part of one risk factor, all other risk factors are summarized in the coefficient β0 and h is a constant as in (29). The parameters for each rating grade are summarized in Table 2while the default probabilities for each rating grade are illustrated in Figure 1using the interest rate of the example, 4%. Table 2. Parameters for the Cox proportional hazard model for each rating grade. Rating Grade β0β1h 1−6.0 10.0 1.0 2−5.5 10.0 1.0 3−5.0 10.0 1.0 4−4.0 10.0 1.0 5−3.5 10.0 1.0 6−2.5 10.0 1.0 71 Risks 2020,8,63 Figure 1. Term-structure of default probabilities for each rating grade. It remains to define economic capital and the operating costs of the bank. We assume an annual operating cost margin c= 0.50%. Economic capital is computed following the regulatory rules for corporate clients with an annual turnover above 50 million EUR where we use both the Standardized and the Internal Ratings Based Approach in our examples. In the case of the Standardized Approach, we assume that the company does not have an external rating. Finally, we assume a target RAROC wt of 10%. Cost components and RAROC are computed for the collateralized bullet loan (Loan I), the unsecured bullet loan (Loan II), the collateralized installment loan (Loan III), and the unsecured installment loan (Loan IV). The borrower rating is “3”, i.e., we assume a borrower with a one-year default probability of roughly 1%. The results are summarized in Table 3when E is computed as 0.08 ·NDand in Table 4when Eis computed by (22). Table 3. Cost components RAROC for the four example loans assuming rating grade 3 and using the Standardized Approach for computing E. All results are percentage values. Quantity Loan I Loan II Loan III Loan IV ys1.63 1.63 1.45 1.45 sf0.33 0.33 0.30 0.30 sb0.18 0.18 0.18 0.18 sEL 0.29 0.78 0.16 0.78 sc0.52 0.52 0.52 0.52 E8.00 8.00 8.00 8.00 RAROC 12.94 6.88 17.28 9.51 72 Risks 2020,8,63 Table 4. Cost components RAROC for the four example loans assuming rating grade 3 and using the Internal Ratings Based Approach for computing E. All results are percentage values. Quantity Loan I Loan II Loan III Loan IV ys1.63 1.63 1.45 1.45 sf0.33 0.33 0.30 0.30 sb0.18 0.18 0.18 0.18 sEL 0.29 0.78 0.16 0.78 sc0.52 0.52 0.52 0.52 E7.27 18.17 7.27 18.17 RAROC 13.83 2.94 18.48 4.07 The quantities ys and sf show the effect of the amortization rate. Both the swap curve and the funding spreads curve are steep. Since an amortization rate reduces the effective maturity of a loan both quantities are lower for amortizing loans. This effect is not seen in sb because both basis swap spread curves are flat. The expected loss margin sEL is considerably higher for the unsecured loan. For the amortizing collateralized loan, the expected loss margin is lowest because this loan becomes less risky when the outstanding balance is reduced due to the amortizations. This effect is not seen in E in Table 4because economic capital is based on a one-year horizon in the Basel II setup. We see that in both tables, only the collateralized loans pass the RAROC target of 10%. The unsecured loans show a RAROC below 10% and should be rejected if a bank strictly sticks to its profitability target. In the second example, we compute zh , zmax and maximum RAROC for Loan IV. Again we present the results for both regulatory regimes. The outcome for the Standardized Approach is displayed in Table 5while the numbers for the Internal Ratings Based Approach are shown in Table 6. Table 5. Hurdle rate, maximum RAROC and zmax for the unsecured installment loan Loan IV under the Standardized Approach. All results are percentage values. Rating Grade zhzmax RAROCmax 1 3.52 38.84 332.62 2 3.71 33.84 270.12 3 4.05 28.84 207.62 4 5.88 18.84 82.62 5 9.60 13.84 20.12 6 NA 3.84 -104.88 Table 6. Hurdle rate, maximum RAROC and zmax for the unsecured installment loan Loan IV under the Internal Ratings Based Approach. All results are percentage values. Rating Grade zhzmax RAROCmax 1 4.06 29.86 87.63 2 4.59 26.40 69.34 3 5.29 23.09 51.85 4 8.44 16.78 19.55 5 NA 13.39 4.66 6 NA 5.69 -23.49 We see that for the high-risk clients, no hurdle rate zh exists. This means that it is not possible for a bank to set an interest rate that makes the loan profitable. Therefore, a loan application of these clients should be rejected. We see that for Rating “3” in both cases, the hurdle rate is above 4%. This is consistent with the results in Tables 3and 4where RAROC was below the profitability target of 10% for Loan IV when an interest rate of 4% was used. Consistent with intuition, in both cases zh is increasing with borrower default risk while zmax and RAROCmax are decreasing. 73 Risks 2020,8,63 7. Discussion In this article, a loan pricing scheme is developed using the performance measure RAROC. Motivated by balance sheet considerations, i.e., the desire to match assets and liabilities, a calculation scheme is proposed which explicitly decomposes a loan’s interest rate into relevant cost components: Funding costs, costs for hedging interest rate risks, expected loss costs, target return on economic capital, and internal bank costs. For fixed-rate loans, a formula for the base swap rate was given in addition. These cost components are essential for internal fund transfer pricing processes between separate functions in a bank. The proposed pricing scheme is applicable for loans with the deterministic interest rate, i.e., fixed-rate loans and floating-rate loans linked to Ibor rates. We have analyzed the scheme mainly for the case where term-structures of default probabilities are estimated using a Cox proportional hazard model. This was motivated by the analytical tractability of this model. However, the scheme does not depend on this modeling assumption and could work with any term-structure of default probabilities regardless of its determination. In a theoretical analysis, it was shown in a slightly simplified setup that if a borrower’s default probability increases with a loan’s interest rate then RAROC becomes −∞ in the limiting cases of arbitrarily large negative and positive interest rates which means that both the cases of bank and borrower bankruptcy are treated within economic intuition by RAROC. It was further shown that RAROC has a unique maximum and that at most a finite interval of interest rates exists at which a bank should accept a loan application. In cases where interest rates a borrower is willing to accept are outside this interval or when the acceptance range is empty, a bank should reject a loan application. Numerical examples illustrated the application of this loan pricing framework. The examples suggest that the main results of the article hold in a more general setup than we were able to prove formally. The main challenge of applying this framework in practice is finding a link between a loan’s interest rate and borrower default rates empirically. In real data sets important information for determining this relationship like the total interest a borrower is paying on all his existing loan products or timely income information is often missing in retail data sets which makes the parameters β0and β1of our examples very hard to estimate. The benefits of implementing this approach in practice are threefold. First, the scheme delivers a split of a loan’s interest rate into cost components for internal fund transfer pricing. Second, when interest rate costs are properly included in a credit scorecard, the scheme allows the calculation of the profitability range for a loan. Only rates within this range a bank should offer when originating a loan. Finally, since the scheme delivers a loan valuation, it could, in addition, be valuable in the price determination of loan portfolio transactions when loans are sold to investors. One shortcoming of the profitability range is that this interest rate interval models only the perspective of the bank. These are the interest rates that ensure that the profitability requirements of the bank are met, but there is no view from the borrower’s perspective included. It would be very helpful if a bank would have, in addition to the profitability range, some information about the likelihood that a borrower will accept a loan offer and how this likelihood changes within the profitability range. Chun and Lejeune (2020) use the probability that a borrower accepts a loan offer in their model but merely on a theoretical basis using several distribution functions without giving any suggestion on empirical verification. Complementing the profitability range by including the borrower perspective would further increase the value of the RAROC scheme. We leave this challenging task for future research. Author Contributions: Conceptualization, B.E. and H.P.; methodology, B.E.; validation, B.E. and H.P.; writing—original draft preparation, B.E.; writing—review and editing, H.P.; supervision, H.P.; project administration, H.P.; funding acquisition, H.P. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the Ministry of Education and Training in Vietnam, Ho Chi Minh City Open University and Open Source Investor Services B.V. grant number B2019-MBS-03. 74 Risks 2020,8,63 Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results. Abbreviations The following abbreviations are used in this manuscript: CAPM capital asset pricing model CDS credit default swap LGD loss given default PD default probability RAROC risk-adjusted return on capital Appendix A. Proof of Theorem 1 Theorem 1consists of three parts which we each proof in a separate step. We start with the proof of 1. For the survival probabilities vi , it is easy to see that lim z→−∞v(i)= 1. This means that lim z→−∞zEL = 0 and lim z→−∞(z−zEL)/E=−∞. The proof of Part 2 is a bit more evolved. We have to show that lim z→+∞z−zEL =−∞ .From(29) we see that lim z→+∞v(i)= 0. Therefore, lim z→+∞zEL =+∞ and lim z→+∞z=+∞ which makes it not obvious to see how RAROC will behave in the limit. A reformulation of z−zEL leads to z−zEL =z−(1−R)(1−v(m)) ∑m i=1v(i)=z∑m i=1v(i)−(1−R)(1−v(m)) ∑m i=1v(i) For each product zv(i), we can show that lim z→+∞zv(i)=0: lim z→+∞zv(i)= lim z→+∞zqhi =lim z→+∞ z q−hi =lim z→+∞ 1 −hi ·q−hi−1·qlog(q)β1 =lim z→+∞−qhi hi ·log(q)β1=0. At the third equality sign we have used the rule of L’Hospital and the derivative of q with respect to z dq dz =d dz exp(−exp(β0+zβ1)) = −exp(−exp(β0+zβ1))exp(β0+zβ1)β1=qlog(q)β1. This results allows us to compute (we use a bit sloppy notation in the end) lim z→+∞z−zEL =lim z→+∞ z∑m i=1v(i)−(1−R)(1−v(m)) ∑m i=1v(i)=0−(1−R) 0=−∞. For the proof of Part 3, we will show that the first derivative of RAROC with respect to z is between 1 and −∞ and it is monotonically decreasing which implies that there exists exactly one root of dRAROC(z)/dz which proofs the Theorem. We start with dRAROC(z)/dz where we use the abbreviations L:=1−Rand D:=z−zEL: dD dz =1−d dz L(1−qhm) ∑m i=1qhi =1−L−∑m i=1qhi ·hm ·qhm−1dq(z) dz −(1−qhm)∑m i=1hi ·qhi−1dq(z) dz ∑m i=1qhi2 =1+Llog(q)β1∑m i=1qhihm ·qhm +(1−qhm)∑m i=1hi ·qhi ∑m i=1qhi2 75 Risks 2020,8,63 =1+Llog(q)β1∑m i=1hi ·qhi +h(m−i)qh(i+m) ∑m i=1qhi2 We have lim z→−∞q(z)= 1 and, therefore, lim z→−∞ d(z−zEL) dz = 1. On the other end, we have lim z→∞q(z)= 0 which leads to lim z→∞ d(z−zEL) dz =−∞. The reason for the latter is that lim z→∞log(q)=−∞and lim q→0 ∑m i=1hi ·qhi +h(m−i)qh(i+m) ∑m i=1qhi2=+∞, which can be seen after applying the rule of L’Hospital 2 m− 1 times. Note that the highest exponent of q in the numerator is 2 m− 1 because the coefficient of q2m is zero and the highest exponent of the denominator is 2 m . This leaves one q in the denominator after 2 m− 1 times applying L’Hospital’s rule while there is none in the numerator. Since d(z−zEL) dz is continuous, it must have at least one root. To show that this root is unique, we show that d(z−zEL) dz is decreasing monotonically by proving that d2D dz2=d2(z−zEL) dz2is negative for all z. d2D dz2=Lβ11 q dq dz ∑m i=1hi ·qhi +h(m−i)qh(i+m) ∑m i=1qhi2 +Llog(q)β1∑m i=1qhi2∑m i=1h2i2·qhi−1+h2(m2−i2)qh(i+m)−1 ∑m i=1qhi4 dq dz −Llog(q)β1 2∑m i=1qhi·∑m i=1hi ·qhi−1·∑m i=1hi ·qhi +h(m−i)qh(i+m) ∑m i=1qhi4 dq dz =Lβ2 1log(q)∑m i=1hi ·qhi +h(m−i)qh(i+m) ∑m i=1qhi2 +Llog(q)2β2 1∑m i=1qhi∑m i=1h2i2·qhi +h2(m2−i2)qh(i+m) ∑m i=1qhi3 −Llog(q)2β2 1 2·∑m i=1hi ·qhi·∑m i=1hi ·qhi +h(m−i)qh(i+m) ∑m i=1qhi3 =Lβ2 1log(q)∑m i=1hi ·qhi +h(m−i)·qh(i+m) ∑m i=1qhi2 +Llog(q)2β2 1 ∑m i,j=1qhj h2i2·qhi +h2(m2−i2)·qh(i+m) ∑m i=1qhi3 −Llog(q)2β2 1 2·∑m i,j=1hj ·qhj ·hi ·qhi +h(m−i)·qh(i+m) ∑m i=1qhi3 =Lβ2 1log(q)∑m i=1hi ·qhi +h(m−i)·qh(i+m) ∑m i=1qhi2 −Llog(q)2β2 1 ∑m i,j=1h2(2ij −i2)·qh(i+j)+h2(2j(m−i)−m2+i2)·qh(i+j+m) ∑m i=1qhi3 Since log(q)< 0 ) the first term of this expression is negative. To prove that the full expression is negative, it is sufficient to prove that each coefficient of qhi in the numerator of the second term is non-negative and at least one is strictly positive. In total there are 3 m− 1 terms qhk with k= 2, ... ,3 m . 76 Risks 2020,8,63 For k= 2, ... , m+ 1 only the first part of the double sum is relevant, from k=m+ 2, ... ,2 m both parts contribute, while from k=2m+1,...,3monly the last part counts. We start with k= 2, ... , m+ 1. For fixed k , the index i can run from 1 to k− 1 while j is set to k−i . Then i+j=k and all possible combinations leading to i+j=k are covered. Summing over the coefficients that contribute to qhk yields k−1 ∑ i=1 2(i(k−i)) −i2= k−1 ∑ i=1 2ki −3i2 Using the relations ∑k i=1i=k(k+1) 2and ∑k i=1i2=k(k+1)(2k+1) 6leads to k−1 ∑ i=1 2ki −3i2=2k(k−1)k 2−3(k−1)k(2k−1) 6=k(k−1) 2>0. Next, we look at k=m+ 2, ... ,2 m . We parametrize this as m+k , k= 2, ... , m . In this case we have contributions from both terms of the double sum and the coefficient of qm+kis computed as m ∑ i=k2i(m+k−i)−i2+ k−1 ∑ i=12(k−i)(m−i)−m2+i2=1 2(m2−m)+k(m−k+1)>0. The derivation of the above result is a bit lengthier as in the first case but uses the same reasoning. Finally, we report the result for k= 2 m+ 1, ... ,3 m . Similar as before, we run k from 1 to m and ensure that i+j= 2 m+k . Summing over all combinations of i and j fulfilling this condition leads to m ∑ i=k2(m−i+k)(m−i)−m2−i2=1 2m2+k2−km +1 2(m−k)≥0. Here, the sum can become zero in the case of k=m . In all other cases, it is strictly positive. This concludes the proof that the second derivative of RAROC is always strictly negative. References Aguais, Scott D., Larry Forest, Suresh Krishnamoorthy, and Tim Mueller. 1998. Creating value from both loan structure and price. Commercial Lending Review 13: 13–25. Aguais, Scott D., and Lawrence R. Forest. 2000. The future of risk-adjusted credit pricing in financial institutions. RMA Journal 83: 26–31. Aguais, Scott D., and Anthony M. Santomero. 1998. Incorporating new fixed income approaches into commericial loan valuation. Journal of Lending & Credit Risk Management 58–65. [CrossRef] Akhtar, Yasmeen, Kayani Mujtaba, and Tahir Yousaf. 2019. The effects of regulatory capital requirements and ownership structure on bank lending in emerging asian markets. Journal of Risk and Financial Management 12: 142. doi:10.3390/jrfm12030142. [CrossRef] Balog, Dora, Tamas Batyi, Peter Csoka, and Miklos Pinter. 2017. Properties and comparison of risk capital allocation methods. European Journal of Operational Research 259: 614–25. [CrossRef] Banasik, John, Jonathan N. Crook, and Lyn C. Thomas. 1999. Not if but when will borrowers default. Journal of the Operational Research Society 50: 1185–190. [CrossRef] Basel Committee on Banking Supervision. 2006. International Convergence of Capital Measurement and Capital Standards: A Revised Framework. Available online: http://www.bis.org/publ/bcbsca.htm (accessed on 30 March 2019). Basel Committee on Banking Supervision. 2011. Basel III: A Global Regulatory Framework for More Resilient Banks And Banking Systems. Available online: http://www.bis.org/publ/bcbs189.htm (accessed on 30 March 2019). Baule, Rainer. 2014. Allocation of risk capital on an internal market. European Journal of Operational Research 234: 186–96. [CrossRef] 77 Risks 2020,8,89 frequencies corresponding to periods of 5 up to 40,960 minutes, that is, up to 28.5 days, or approximately one month. Table 1. Frequency bands of the first 13 decomposition level. Decomposition Level (j) Frequency Band (1 2j+1;1 2j) Period Band In minutes (5×2j;5×2j+1) In days (5×2j)/1,440 (1) (2) (3) (4) 0 (original) 0 −1/2 - - 11/4−1/2 5 to 10 0.017 21/8−1/4 10 to 20 0.014 31/16−1/8 20 to 40 0.03 41/32−1/16 40 to 80 0.06 51/64−1/32 80 to 160 0.11 6 1/128 −1/64 160 to 320 0.22 7 1/256 −1/128 320 to 640 0.44 8 1/512 −1/256 640 to 1280 0.89 9 1/1024 −1/512 1280 to 2560 1.78 10 1/2048 −1/1024 2560 to 5120 3.56 11 1/4096 −1/2048 5120 to 10,240 7.11 12 1/8192 −1/4096 10,240 to 20,480 14.22 13 1/16, 384 −1/8192 20,480 to 40,960 28.44 Source: Authors’ computations. We can examine our data by means of the DWT described above. The multi-horizon nature of this approach gives it the name “multi-resolution analyses” (MRA). In the empirical application presented in Section 5, we employed several well-established long-memory parameter estimators on the exchange-rate data decomposed using MRA. These include the R/S estimator ( Mandelbrot and Van Ness 1968 ), aggregated variance estimator (Dieker and Mandjes 2003), differenced variance and absolute moment estimators (Teverovsky and Taqqu 1997) and Higuchi estimator (Higuchi 1981). See (Vo and Vo 2019b) for discussions regarding these estimators. Over the last decade, the wavelet-based methodology has gained considerably more attention in the financial volatility modelling literature thanks to its ability to offer powerful insights with respect to horizon-specific dynamics of data generating processes. Recently, (Boubaker 2020) carried out Carlo simulations to compare several wavelet-based estimators and concluded that the Wavelet Exact Local Whittle estimator outperforms the Wavelet OLS and Wavelet Geweke–Porter-Hudak estimators and generates more accurate results to identify the fractional integration parameter for symmetric heavy-tailed distributions. One of the most important applications of this novel line of research is the analyses of co-movement patterns among asset classes at different investment frequencies that could potentially offer hedging strategies to mitigate market-wise and sector downside risks. Recent related studies include (Ghosh et al. 2020), who adopted a wavelet-based time-varying dynamic approach for estimating the medium- and long-range conditional correlation among various financial and energy assets to determine their hedge ratios. Along a similar vein, (Kang et al. 2019) found strong evidence of volatility persistence, causality and phase differences between Bitcoin and gold futures prices. In addition, wavelet-filtered data are used to capture movements of Bitcoin returns at various investment horizons, and form the basis for the examination of Bitcoin’s ability to hedge global uncertainty (Bouri et al. 2017). To the best of our knowledge, applications of wavelet-based methodology to the currency markets are much more limited compared to other financial markets, a fact we seek to change with the contributions of this paper. 84 Risks 2020,8,89 3.2. Testing for Structural Breaks in the Presence of Long Memory In this section; we describe a procedure with which we can test for the existence of possible multiple structural breaks; which is a source of long memory in volatility. Previous research has documented that structural breaks in the mean can partly explain the persistence of realised volatility ( Choi et al. 2010 ), but the effect of structural breaks could also mask that of true long memory and thus lead to misspecifications (Sibbertsen 2004). Therefore; we need to account for the possibility of structural breaks in our data. Specifically; we were firstly interested in fitting a univariate GARCH model to our data using several alternative specifications that are prominent in the literature. In GARCH models, the normalised density function is often written in terms of the location and scale parameters as: αt=(μt,σt,ω), where the conditional mean and variance are given by: μt=μ(θ,rt)=E(vt|rt);σ2 t=σ2(θ,rt)=E[(vt−μt)2rt], with rt and vt denoting returns and volatility, and ω=ω(θ,rt) is the remaining parameters of the distribution. Here, for simplicity, we assumed an ARIMA (1,1) model for the mean equation, normal distribution of the error terms and different GARCH (1,1) specifications for the variance equation. These include the standard GARCH (Bollerslev 1986), the exponential GARCH/eGARCH (Nelson 1991 ) and the GJRGARCH (Glosten et al. 1993), which account for asymmetric volatilities and the fractionally-integrated GARCH/fiGARCH (Baillie et al. 1996), which accounts for fractionally integrated (long-memory) processes. 3 For example, the specific equations for the standard GARCH (1,1) models are: Φ(L)(1−L)(vt−μt)=Θ(L)εt;σ2 t=ω+α1ε2 t−1+β1σ2 t−1, with L denoting the lag operator and εt the residual from the mean filtration process. The asymmetric GARCH models (eGARCH and GJRGARCH) have an additional parameter γ capturing the degree of asymmetry. In contrast, the fiGARCH model has an additional parameter d that captures the degree of long memory. After fitting these GARCH models, we performed tests for structural breaks by applying a Change Point Model (CPM) on the corresponding model residuals. We aimed to detect multiple change points in a sequence of observations of the volatility process, with different CPMs such as the t-tests proposed by (Hawkins et al. 2003); the Bartlett test (Hawkins and Zamba 2005a); the Generalised Likelihood Ratio test (Hawkins and Zamba, Statistical process control for shifts in mean or variance using a changepoint formulation (Hawkins and Zamba 2005b); the Mann–Whitney test (Ross, Tasoulis, & Adams, Nonparametric monitoring of data streams for changes in location and scale, (Ross et al. 2011); the Mood test (Ross et al. 2011) and the Kolmogorov–Smirnov test (Ross and Adams 2012). While the first three methods are designed to capture change points in Gaussian processes, the others are for non-Gaussian processes. 4. Results Our five-minute USD/AUD nominal exchange rates (measured as the AUD cost of 1 USD, instead of the default AUD/USD rate) are provided by the commercial data vendor Bloomberg. The data coverage period is from 18:05, 7th August 2019 to 9:25, 16th September 2019—a total of T= 8481 intervals/observations. This period was selected when the analysis was conducted. In addition, we 3 Note that when d= 0, the FIGARCH (1,d,1) model collapses to the standard GARCH(1,1), while when d= 1, it collapses to iGARCH(1,1). 85 Risks 2020,8,89 considered that a total of 8481 observations is sufficient for the analysis using our selected technique. We selected the closing ask USD/AUD rate as our subject of study. Figure 2presents MRA plots for this time series, starting with the original level in the top-left plot and ends with the coarsest smoothed series in the bottom-right plot. In between these cases are detailed series corresponding to the 13 decomposition levels presented in Table 1. Note that the original data can be reconstructed by the direct summation of all the components. It can be seen clearly that noisy fluctuations are captured by higher-frequency detail series ( D1 to D8 ) while these noises can be filtered out in lower-frequency details (D9to D13) and in the smooth component (S13). Figure 2. Multi-resolution plots of USD/AUD time series. Notes: This figure presents the MRA for the full data range from 7th August 2019, to 9:25, 16th September 2019. In each panel, the horizontal axis indicates the corresponding days of the two months. The continuously compounded exchange rate returns are computed as the differences of logarithmic five-minute exchange rates: rit =logpit −log pi,t−1(t=1, ...,T) . Exchange rate volatility is proxied by the 288-interval (or one-day) rolling standard deviations of returns, that is, σit =(1/288)289 j=2logpi,t+j−log pi,t+j−1−rit2 , where rit =(1/288)289 j=2logpi,t+j−log pi,t+j−1 is the rolling average return. This means the first 288 return observations are set aside for the computation of the first realized volatility value, leaving us with 8193 observations. 86 Risks 2020,8,89 Summary statistics of the volatility and return series are presented in Table 2. Overall, the return series distribution resembles normality, albeit having high kurtosis. On the other hand, the volatility is left-skewed, as, by construction, it only contains positive values. To examine the long-range dependence patter of our data, Figure 3illustrates the corresponding five-minute auto-correlograms, or visualised autocorrelation function (ACF), for the two series. As can be seen, the return series exhibit no significant autocorrelation pattern after the first lag, while the volatility series clearly demonstrates long-range dependence. Table 2. Summary statistic of five-minute USD/AUD returns and volatilities. Mean Median Variance Skewness Kurtosis JB LB(21) Returns 0.000001 0.00 0.00 0.72 422.12 3881823130.38 (0.00) 9022.43 (0.00) Volatilities 0.000412 0.000393 0.00 3.70 33.17 25166002.26 (0.00) 10405779.70 (0.00) Notes: Returns of the USD/AUD exchange rate are computed as the log-change of the corresponding five-minute spot USD/AUD: rit =logpit −log pi,t−1(t=1,...,T) , where pit denotes the nominal exchange rate. T denotes the number of observations (8193 five-minute intervals). Volatilities are defined as the one-day rolling standard deviations of rit . JB and LB denote the Jarque–Bera and the Ljung–Box statistics, respectively. p-values are in parentheses. Source: Authors’ computations. Figure 3. Auto-correlogramsofUSD/AUDhigh-frequencyreturnsandvolatilities. Notes: Exchange-rate returns are computed as the log-change of the corresponding five-minute closing spot rates: rit =logpit −logpi,t−1 (t=1, ... , 8193). Volatilities are defined as the one-day rolling standard deviations of rit. Source: Authors’ computations. 4.1. A Multi-Resolution Analysis Our main question of interest is “At which particular horizons does the long-memory behaviour of the USD/AUD exchange rate persist?” To answer this, Table 3presents estimates of the Hurst index as applied to the original series as well as the 12 levels of smooth components which capture all activities at frequencies lower than 1 2j+1 and thus preserve the underlying trends of the volatility. 4 As can be seen from this table, the volatility measures of five-minute AUD/USD returns exhibit a very persistent pattern of long memory, at all levels of decomposition, where the Hurst index estimated using all methods is significantly larger than 0.5. This means that there are no frequencies that are solely responsible for the long-range dependence characteristic of the Australian dollar. This could be explained by the fact that the trader base of this open-economy currency is quite diverse and active, who tend to switch trading 4 On the other hand, “detail” components reflect the fluctuations (or differences) of volatility series and thus are not representative of the long-range dependent behaviour. 87 Risks 2020,8,89 horizons frequently via diversification/rebalancing operations. This makes disentangling the impacts of activities at a particular frequency from those at other frequencies difficult.5 Table 3. Long-memory parameter estimates of exchange-rate volatility at different horizons. Smooth Level R/S aggVar diffVar absVal Higuchi Original 1.101 0.966 1.620 0.980 0.966 (0.062) (0.064) (0.179) (0.047) (0.030) 11.146 0.966 1.618 0.980 0.966 (0.085) (0.064) (0.166) (0.047) (0.030) 21.056 0.966 1.747 0.980 0.966 (0.063) (0.064) (0.184) (0.047) (0.030) 30.994 0.966 1.663 0.980 0.966 (0.044) (0.064) (0.142) (0.047) (0.030) 40.988 0.967 1.833 0.980 0.966 (0.033) (0.064) (0.164) (0.047) (0.030) 51.005 0.968 1.965 0.981 0.966 (0.034) (0.064) (0.187) (0.047) (0.030) 61.008 0.972 2.174 0.984 0.966 (0.034) (0.065) (0.236) (0.047) (0.030) 71.003 0.978 1.756 0.986 0.965 (0.027) (0.066) (0.196) (0.048) (0.030) 80.990 0.988 1.803 0.992 0.965 (0.017) (0.063) (0.168) (0.047) (0.030) 90.993 0.997 1.932 0.999 0.966 (0.014) (0.050) (0.127) (0.037) (0.030) 10 1.000 1.003 1.669 1.007 0.966 (0.008) (0.027) (0.154) (0.019) (0.030) 11 0.999 1.001 1.364 1.005 0.966 (0.007) (0.003) (0.240) (0.010) (0.030) 12 0.999 1.002 1.468 1.005 0.966 (0.006) (0.004) (0.152) (0.008) (0.030) Notes: Nomenclatures: (1) R/S: Rescaled range; (2) aggVar: Aggregated variance; (3) diffVar: Differenced variance; (4) AbsVar: Absolute moments; (5) Higuchi: Higuchi’s method. Heteroskedasticity-robust standard errors are in parentheses. Refer to Table 2for interpretation of the decomposition levels/time-scales. Source: Authors’ computations. 4.2. Horizon-Based Power Decomposition Given the highly persistent pattern of long memory observed in AUD/USD volatility, it is now fruitful to analyse the multi-scale composition of power (or variations) of the original nominal five-minute exchange rate. To do this, in Figure 4we present a “heat map” representation of the MRA as proposed by (Torrence and Compo 1998), which illustrates the power scale of the original series through both time and frequencies. Stronger colours (i.e., red or orange) at any frequency and time represent higher power scales and stronger cyclical behaviour. 6 We performed wavelet decomposition only up to the horizon corresponding to 512 five-minute intervals. This design allowed us to investigate 5 Additionally, the behaviour of exchange rates can be related to the dynamic long-memory properties of other economic variables, such as the aggregated price levels (via the purchasing power parity relationship) or the interest rates (via the uncovered interest parity relationship). 6 The computation is done this time with a continuous Morlet wavelet transform, rather than a DWT. Due to some issues with this operator, certain information outside the region outlined by the parabolic curve (the “cone of influence”) should be ignored. (e.g., (Daubechies 1992) for details.) 88 Risks 2020,8,89 the interaction dynamics of the intraday volatility process. The map reveals features that are in close conjunction with the cyclical behaviour of the series, which is not easy to discern without the map. Specifically, frequencies corresponding to the periods of 256 to 512 five-minute intervals are observed to exhibit the highest power while no strong cyclical pattern can be observed at higher frequencies. These results corroborate those of (Caporale et al. 2019). In agreement with Figure 2, though at shorter horizons there are only small intraday noises, there exist large (but infrequent) movements at the longer horizons in the AUD/USD exchange-rate dynamics. Interestingly, it can also be seen that there are two episodes of volatility spillover between the low frequencies and the high frequencies in this sample: The August 13th and the September 10th. These days are also associated with episodes of relatively high volatility. The former effect is tied in with the release of the statement on monetary policy by the Reserve Bank of Australia on August 9th, while the more prominent effect on the second date could be a result of market anticipation during the week leading to the meetings of the Federal Open Market Committee (US) and Reserve Bank Board (Australia) meetings, both of which are on September 17th. In the next subsection, we investigated possible breaks in the volatility process in more details. Figure 4. Wavelet heat map of five-minute nominal USD/AUD exchange rate. Notes: Horizontal axis ranges from 0 to 8193, the number of five-minute intervals in our sample. The vertical axis indicates the horizons (in five-minute) corresponding to the frequencies at which the underlying time series fluctuates. The power meter is located beneath the graph. The area within the parabolic region indicates the “zone of influence”. The plots were drawn using functions provided in the R package dplR (Bunn 2008). Source: Authors’ computations. 4.3. Sources of Long Memory: Structural Breaks Table 4presents the estimated results for the alternative GARCH (1,1) models discussed in Section 3. The likelihood value and information criteria are in agreement that the most appropriate model is the eGARCH, followed by the standard GARCH, then the fiGARCH and GJRGARCH. Interestingly, the long-memory parameter estimates implied by fiGARCH (d=0.85) again confirm the long-memory characteristic of the volatility process. 89 Risks 2020,8,89 Table 4. GARCH (1,1) models estimates. GARCH eGARCH GJRGARCH fiGARCH μ0.00 (0.00) 0.00 (0.33) 0.00 (0.00) 0.00 (0.00) AR1−0.07 (0.07) 0.13 (0.02) −0.06 (0.70) −0.10 (0.05) MA1−0.15 (0.07) −0.35 (0.00) −0.13 (0.40) −0.16 (0.00) ω0.00 (0.00) −2.85 (0.00) 0.00 (1.00) 0.00 (0.98) α10.07 (0.03) 0.03 (0.00) 0.05 (0.00) 0.06 (0.00) β10.90 (0.02) 0.82 (0.00) 0.90 (0.00) 0.86 (0.00) γor d- - 0.32 (0.00) 0.05 (0.00) 0.85 (0.00) Log-likelihood 54,649.75 54,767.86 54,228.44 54,358.09 Information Criteria Akaike −13.34 −13.37 −13.24 −13.27 Bayes −13.33 −13.36 −13.23 −13.26 Shibata −13.34 −13.37 −13.24 −13.27 Hannan–Quinn −13.34 −13.37 −13.23 −13.27 Notes: This table presents estimation results for different GARCH (1,1) specifications described in Section 4. The last row of parameter estimates refers to the asymmetric parameter γ for the eGARCH and GJRGARCH models, while for the fiGARCH model this refers to the fractional differential parameter d .p-Values based on robust standard errors in parentheses. Based on these estimates, we were able to extract the residuals of these models and apply the CPMs described in Section 4to test for breakpoints of the (conditional) volatility process. Test results are presented in Table 5. We can see that the number of structural breaks detected is substantial, given the high frequency of our data. 7 Importantly, the breaks exist for models exclusively designed to capture long memory, such as fiGARCH, regardless of the assumption of the underlying distribution. Table 5. Number of breakpoints detected using different test statistics. Test Type Test For GARCH eGARCH GJRGARCH fiGARCH Student Mean changes 102 113 93 112 Bartlett Variance changes 181 177 191 181 GLR Mean and variance changes 155 154 163 146 B. Tests in a (possibly unknown) non-Gaussian process MW Location shifts 115 115 112 113 M Scale shifts 48 54 44 57 KS Arbitrary changes 61 71 53 65 Notes: The tests listed are applied to residuals from the GARCH models described in Table 5. Nomenclatures: GLR (Generalised Likelihood Ratio), MW (Mann–Whitney), M (Mood) and KS (Kolmogorov  Smirnov). Sources: Authors’ examinations and computations. 5. Discussions, Conclusions and Implications 5.1. Discussions We contribute to the existing literature by providing a careful examination of the time series characteristics of the AUD/USD exchange rate at a very high frequency, which has important economic implications. As a final note, we conjecture that long memory is observed for this series and is persistent throughout the trading horizons, which implies that investors should be wary of such changes when managing their portfolios. Secondly, rather than focusing on short-term fluctuations and gains, an optimal trading horizon would preferably be longer than half-day, as this could capture more fundamental trend information of the exchange-rate return processes. 7 The exact time periods when the breaks are detected are not presented here to conserve space but are available upon request. 90 Risks 2020,8,89 Our study complements the findings of (Caporale et al. 2019), who documented the persistence of both returns and volatility processes of the EUR/USD and USD/JPY exchange rates at lower trading frequencies. In agreement with this paper, we concur that such evidence against random-walk behaviour implies predictability and is inconsistent with the Efficient Market Hypothesis since abnormal profits can be made using trading strategies based on trend analysis. We also extended this research by introducing the wavelet-based long-memory estimator, as opposed to relying on conventional tools such as the R/S statistic or the fractional integration analysis. Another recent study related to ours is (Boubaker 2020) whose Monte Carlo simulation results suggest that when it comes to estimating the long-memory parameter in stationary time series, the Wavelet Exact Local Whittle estimator outperforms the Wavelet OLS and Wavelet Geweke–Porter-Hudak estimators in terms of smaller bias. It would be interesting to extend this comparison exercise to include our Wavelet MLE approach and apply these estimators on actual data (rather than on simulations). This study is subject to two qualifications. First of all, due to our limited access to high-frequency exchange-rate data, we were unable to examine further the implication of our results for the Australian dollar for other (commodity) currencies. A possibly more general conclusion can be drawn when more of these valuable data are available to us. 8 Secondly, our research is limited to the currency market. Applying the same approach to other financial markets such as stocks, bonds or commodity futures to examine their volatility persistence and the workings of the EMH offers an interesting future research venue. 5.2. Conclusions and Implications Existing literature indicates that the choice of an appropriate statistical tool for analysing exchange-rate dynamics should ultimately be made based on the long-memory properties of the underlying data generating process, which varies across different trading horizons. The Australian dollar is generally considered as a representative commodity currency given the performance of the Australian economy is mainly driven by commodities and the Australian dollar is one of the top 10 most frequently traded currencies in the world. As such, this study was conducted to examine the Australian dollar  US dollar exchange rates—one of the most popular and frequently traded pairs of currencies. This study covers the period from 18:05, 7th August 2019 to 9:25, 16th September 2019 with a total of 8481 observations—a sufficient number of observations required for our analysis. In this paper, we used a wavelet-based approach that allows for modelling long-memory characteristics of this important currency pair at different trading horizons. The high-frequency behaviour of exchange rates observed from our study would be valuable for designing and evaluating exchange-rate models and/or forecasts. More generally, these insights can potentially be used to evaluate the currency risks related to the Australian trade balance, trade flows, terms-of-trade, prices of foreign-exchange futures (or options) and/or international asset portfolio formation. Author Contributions: Theoretical frameworks surveyed conducted are done by L.H.V. Both authors conduct reviews of empirical analyses. The original draft is prepared by D.H.V. Reviewing and editing are done by both authors. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by Ho Chi Minh City Open University, Vietnam [E2020.14.1]. 8 Nevertheless, in a recent study (Vo and Vo 2019b) have applied wavelet-based estimators on daily data of six heavily traded currencies, including AUD, and have shown cross-currency results that are similar to ours. 91 Risks 2020,8,89 Acknowledgments: Part of this research was completed when Long Vo was a Master student at the School of Economics and Finance, Victoria University of Wellington, where he received financial support from a New Zealand-ASEAN Scholar Award provided by the New Zealand Ministry of Foreign and Trade. Duc Vo acknowledges the financial assistance from Ho Chi Minh City Open University. We thank the editor and three anonymous referees whose comments helped improve the paper. All remaining errors are our own. Conflicts of Interest: The authors declare no conflict of interest. References Akgül, Isil, and Hulya Sayyan. 2008. Modelling and forecasting long memory in exchange rate volatility vs. stable and integrated GARCH models. Applied Financial Economics 18: 463–83. [CrossRef] Aloy, Marcel, Mohamed Boutahar, Karine Gente, and Anne Peguin-Feissolle. 2011. Purchasing power parity and the long memory properties of real exchange rates: Does one size fit all? Economic Modelling 28: 1279–90. [CrossRef] Baillie, Richard, Tim Bollerslev, and Hans Ole Mikkelsen. 1996. Fractionally integrated generalised autoregressive conditional heteroskedasticity. Journal of Econometrics 74: 3–30. [CrossRef] Baqaee, David. 2010. Using wavelets to measure core inflation: The case of New Zealand. North American Journal of Economics and Finance 21: 241–55. [CrossRef] Bollerslev, Tim. 1986. Generalised autoregressive conditional heteroskedasticity. Journal of Econometrics 31: 307–27. [CrossRef] Booth, Geoffrey, Fred Kaen, and Peter Koveos. 1982. R/S analysis of foreign exchange rates under two international monetary regimes. Journal of Monetary Economics 10: 407–15. [CrossRef] Boubaker, Heni. 2020. Wavelet estimation performance of fractional integrated processes with heavy-tails. Computational Economics 55: 473–98. [CrossRef] Bouri, Elie, Rangan Gupta, Aviral Kumar Tiwari, and David Roubaud. 2017. Does bitcoin hedge global uncertainty? Evidence from wavelet-based quantile-in-quantile regressions. Finance Research Letter 23: 87–95. [CrossRef] Bunn, Andrew. 2008. A dendrochronology program library in R (dplR). Dendrochronologia 26: 115–124. [CrossRef] Caporale, Guglielmo Maria, Luis Gil-Alana, and Alex Plastun. 2019. Long memory and data frequency in financial markets. Journal of Statistical Computation and Simulation 89: 1763–79. [CrossRef] Cashin, Paul, Luis Cespedes, and Ratna Sahay. 2004. Commodity currencies and the real exchange rate. Journal of Development Economics 75: 239–68. [CrossRef] Chen, Yu-chin, and Kenneth Rogoff. 2003. Commodity currencies. Journal of International Economics 60: 133–60. [CrossRef] Choi, Kyongwook, Wei-Choun Yu, and Eric Zivot. 2010. Long memory versus structural breaks in modeling and forecasting realised volatility. Journal of International Money and Finance 29: 857–75. [CrossRef] Coakley, Jerry, Jian Dollery, and Neil Kellard. 2008. The role of long memory in hedging effectiveness. Computational Statistics and Data Analysis 52: 3075–82. [CrossRef] Daubechies, Ingrid. 1992. Ten Lectures on Wavelets. Philadelphia: Society for Industrial and Applied Mathematics. Dieker, Antonius, and Michael Mandjes. 2003. On spectral simulation of fractional Brownian motion. Probability in the Engineering and Informational Sciences 17: 417–34. [CrossRef] Downes, Peter, Kevin Hanslow, and Peter Tulip. 2014. The Effect of the Mining Boom on the Australian Economy. (RDP 2014–08). Research Discussion Paper. Sydney: Reserve Bank of Australia. Drine, Imed, and Christophe Rault. 2005. Can the Balassa-Samuelson theory explain long-run real exchange rate movements in OECD countries? Applied Financial Economics 15: 519–30. [CrossRef] Friedman, Milton. 1953. The case for flexible exchange rates. In Essays in Positive Economics. Chicago: University of Chicago Press, pp. 157–203. Gencay, Ramazan, Faruk Selcuk, and Brandon Whitcher. 2002. An Introduction to Wavelets and Other Filtering Methods in Finance and Economics. San Diego: Academic Press. Gencay, Ramazan, Nikola Gradojevic, Faruk Selcuk, and Brandon Whitcher. 2010. Asymmetry of information flow between volatilities across time scales. Quantitative Finance 10: 895–915. [CrossRef] Ghosh, Indranil, Manas Sanyal, and Rabin Jana. 2020. Co-movement and dynamic correlation of financial and energy markets: An integrated framework of nonlinear dynamics, wavelet analysis and DCC-GARCH. Computational Economics.[CrossRef] 92 Risks 2020,8,89 Gil-Alana, Luis, and Hector Carcel. 2020. A fractional cointegration var analysis of exchange rate dynamics. The North American Journal of Economics and Finance 51: 100848. [CrossRef] Glosten, Lawrence, Ravi Jagannathan, and David Runkle. 1993. On the relation between the expected value and the volatility of the nominal excess return on stocks. Journal of Finance 48: 1779–1801. [CrossRef] Hawkins, Douglas, and Gideon K. D. Zamba. 2005a. A change-point model for a shift in variance. Journal of Quality Technology 37: 21–31. [CrossRef] Hawkins, Douglas, and Gideon K. D. Zamba. 2005b. Statistical process control for shifts in mean or variance using a changepoint formulation. Technometrics 47: 164–73. [CrossRef] Hawkins, Douglas, Peihua Qiu, and Chang Wook Kang. 2003. The changepoint model for statistical process control. Journal of Quality Technology 35: 355–66. [CrossRef] Higuchi, Tomoyuki. 1981. Approach to an irregular time series on the basis of the fractal theory. Physica D: Nonlinear Phenomena 31: 277–83. [CrossRef] Hurst, Harold. 1951. Long term storage capacity of reservoirs. Transaction of the American Society of Civil Engineer 116L: 770–99. In, Francis, and Sangbae Kim. 2013. An Introduction to Wavelet Theory in Finance: A Wavelet Multiscale Approach. Singapore: World Scientific. International Monetary Fund. 2019. International Financial Statistics. Available online: https://data.imf.org/?sk= 4C514D48-B6BA-49ED-8AB9--52B0C1A0179B (accessed on 5 April 2019). James, Jessica, Ian Marsh, and Lucio Sarno. 2012. Handbook of Exchange Rates. Hoboken: Wiley. Jiang, Yongchong, He Nie, and Weihua Ruan. 2018. Time-varying long-term memory in Bitcoin market. Finance Research Letters 25: 280–84. [CrossRef] Jitmaneeroj, Boonlert. 2018. The effect of the re-balancing horizon on the tradeoffbetween hedging effectiveness and transaction costs. International Review of Economics and Finance 58: 282–98. [CrossRef] Kang, Sang Hoon, Ron McIver, and Jose Arreola Hernandez. 2019. Co-movements between Bitcoin and gold: A wavelet coherence analysis. Physica A: Statistical Mechanics and Its Applications 536: 120888. [CrossRef] Lien, Donald, and Yiu Kuen Tse. 1999. Fractional cointegration and futures hedging. Journal of Futures Markets 19: 457–74. [CrossRef] Lo, Andrew. 1991. Long-term memory in stock market prices. Econometrica 59: 1279–313. [CrossRef] Lothian, James. 2016. Purchasing power parity and the behavior of prices and nominal exchange rates across exchange-rate regimes. Journal of International Money and Finance 69: 5–21. [CrossRef] Mallat, Stéphane. 2009. A Wavelet Tour of Signal Processing: The Sparse Way. Amsterdam: Academic Press. Mandelbrot, Benoit, and John Van Ness. 1968. Fractional Brownian motions, fractional noises and applications. SIAM Review 10: 422–37. [CrossRef] Maples, William, Wade Brorsen, and Xiaoli Etienne. 2019. Hedging effectiveness of fertiliser swaps. Applied Economics 51: 5793–801. [CrossRef] Marsh, Ian, Evgenia Passari, and Lucio Sarno. 2012. Purchasing power parity in tradable goods. In J Handbook of Exchange Rates. Edited by Jessica James, Ian Marsh and Lucio Sarno. Hoboken: Wiley, vol. 2, pp. 189–220. Mensia, Walid, Shawkat Hammoudeh, and Seong-Min Yoon. 2014. Structural breaks and long memory in modeling and forecasting volatility of foreign exchange markets of oil exporters: The importance of scheduled and unscheduled news announcements. International Review of Economics and Finance 30: 101–19. [CrossRef] Moosa, Imad. 2003. The sensitivity of the optimal hedging ratio to model specification. Finance Letter 1: 15–20. Nason, Guy. 2008. Wavelet Methods in Statistics with R. Berlin: Springer. Nelson, Daniel. 1991. Conditional heteroskedasticity in asset returns: A new approach. Econometrica 59: 347–70. [CrossRef] Ouyang, Alice, Ramkishen Rajan, and Jie Li. 2016. Exchange rate regimes and real exchange rate volatility: Does inflation targeting help or hurt? Japan and the World Economy 39: 62–72. [CrossRef] Peng, Lu, Shan Liu, Rui Liu, and Lin Wang. 2018. Effective long short-term memory with differential evolution algorithm for electricity price prediction. Energy 162: 1301–14. [CrossRef] Ross, Gordon, and Niall Adams. 2012. Two nonparametric control charts for detecting arbitary distribution changes. Journal of Quality Technology 44: 102–16. [CrossRef] Ross, Gordon, Dimitris Tasoulis, and Niall Adams. 2011. Nonparametric monitoring of data streams for changes in location and scale. Technometrics 53: 379–89. [CrossRef] 93 Risks 2020,8,90 and the expected value of holding the option from ti to ti+1 , given asset’s value  Stl1,...,li i at time ti defined via gti( Stl1,...,li i )=Ee−r nfti+1( Stl1,...,li+1 i+1 ) Stl1,...,li i, where fti( Stl1,...,li i )=max{hti( Stl1,...,li i ),gti( Stl1,...,li i )} is the option value at time tiin state  Stl1,...,li i . Please note that ftn( Stl1,...,ln n )=fT( STl1,...,ln)=hT( STl1,...,ln)=(K−Sa Tl1,...,lnMb Tl1,...,ln)+. 2.2. Estimators We will now give the formulas for the estimators Θ and Φ which overestimate and underestimate the true price of the option, respectively. Then, we will state the main theorem showing that both estimators are asymptotically unbiased and that they converge to the theoretical price of the π -option. We also provide a detailed explanation of the estimation procedure based on the exemplary price tree. In all calculations we consider a π -put option with parameters a=− 1, b= 1 and K= 1. Additionally, we assume that the risk-free rate used for discounting the payoffs equals 5%. 2.2.1. The ΘEstimator The formula for the estimator is recursive and given by: Θti=max hti( Stl1,...,li i ),e−r n1 l l ∑ j=1 Θtl1,...,li,j i+1,i=0, . . . , n−1. At the option’s maturity, T, the value of the estimator is given by ΘT=fT( ST). The Θ estimator, at each node of the price tree, chooses the maximum of the payoff of the option’s early exercise at time ti,hti( Stl1,...,li i ), and the expected continuation value, i.e., the discounted average payoff of successor nodes. Figure 4shows how the value of Θestimator is obtained given the certain realization of a price tree. All calculations are also shown below: •a ⎧ ⎨ ⎩ Holding value: 0+10 115 +0 3e−0.05 ≈0.028 Early exercise: 0 •b ⎧ ⎨ ⎩ Holding value: 5 110 +0+13 110 3e−0.05 ≈0.052 Early exercise: 10 110 ≈0.091 •c ⎧ ⎨ ⎩ Holding value: 10 110 +15 110 +5 110 3e−0.05 ≈0.086 Early exercise: 20 110 ≈0.182 •d Holding value: 0.028+0.091+0.182 3e−0.05 ≈0.095 Early exercise: 10 110 ≈0.091 100 Risks 2020,8,90 100 115 100 90 115 110 110 110 120 120 105 115 125 125 105 110 110 110 97 110 100 110 95 110 105 110 t0t1t2 (a)Price tree 0.095 d 0.028 a 0.091 b 0.182 c 0 0.087 0 0.045 0 0.118 0.091 0.136 0.045 t0t1t2 (b)Evaluation of Θestimator Figure 4. Explanation of Θestimator. 2.2.2. The ΦEstimator The Φestimator is also defined recursively. Before we give the formula we need to introduce an auxiliary function ξby ξj tl1,...,li i = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ hti( Stl1,...,li i ),ifhti( Stl1,...,li i )≥e−r n1 l−1 l ∑ k=1 k=j Φtl1,...,li,k i+1 e−r nΦtl1,...,li,j i+1 ,ifhti( Stl1,...,li i )<e−r n1 l−1 l ∑ k=1 k=j Φtl1,...,li,k i+1 (4) for j=1, . . . , l. Now we can define the Φestimator in the following way: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ Φtl1,...,li i =1 l l ∑ j=1 ξj tl1,...,li i ΦT=fT( ST). (5) The formula for this estimator is more complicated. Therefore, we provide a detailed explanation of the mechanism behind the algorithm in the following part of this section. In our explanation we refer 101 Risks 2020,8,90 to Figure 5. Please note that in the following example, underlined numbers correspond to the final values associated with the specific branches of the tree. 100 115 100 90 115 110 110 110 120 120 105 115 125 125 105 110 110 110 97 110 100 110 95 110 105 110 t0t1t2 (a)Price tree 0.061 d 0 a 0.091 b 0.182 c 0 0.087 0 0.045 0 0.118 0.091 0.136 0.045 t0t1t2 (b)Evaluation of Φestimator Figure 5. Explanation of Φestimator. •a ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Early exercise: 0 Holding value for branch j=1: 0.087+0 2e−0.05 ≈0.041 >0→0 Holding value for branch j=2: 0+0 2e−0.05 =0≤0=hi( Sti)→0 Holding value for branch j=3: 0+0.087 2e−0.05 ≈0.041 >0→0 For the branch j= 1 we look at the two remaining ones to determine whether early exercising (payoff = 0) or holding the option (payoff = 0.087+0 2e−0.05 ) is more profitable. Obviously, early exercise is not optimal, so we hold the option and thus, as the value of ξ1 t1 1 we take the payoff of the branch j=1 which is 0. For the branch j= 2 both early exercise value and holding value from two other branches equals 0. Thus, from (4) the value of ξ2 t1 1equals the payoff of early exercise, which is 0. For the third branch, again holding the option is a more profitable decision (based on the payoffs of the two remaining branches). Thus, ξ3 t1 1 takes the value corresponding to the branch j= 3 and it is 0. 102 Risks 2020,8,90 Now the value of the estimator for node a is the sum of ξj t1 1across all branches: Φt1 1=1 3 3 ∑ j=1 ξj t1 1=0. Similarly, we have the following values of our estimator. •b ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Early exercise: 10 110 ≈0.091 Holding value for branch j=1: 0+0.118 2e−0.05 ≈0.056 <0.091 →0.091 Holding value for branch j=2: 0.045+0.118 2e−0.05 ≈0.078 <0.091 →0.091 Holding value for branch j=3: 0.045+0 2e−0.05 ≈0.021 <0.091 →0.091 In this case, the value of the estimator for the b node equals 0.091. •c ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Early exercise: 20 110 ≈0.182 Holding value for branch j=1: 0.045+0.136 2e−0.05 ≈0.086 <0.182 →0.182 Holding value for branch j=2: 0.091+0.045 2e−0.05 ≈0.065 <0.182 →0.182 Holding value for branch j=3: 0.091+0.136 2e−0.05 ≈0.108 <0.182 →0.182 For node c the value of the estimator is 0.182. •d ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Early exercise: 10 110 ≈0.091 Holding value for branch j=1: 0.091+0.182 2e−0.05 ≈0.13 >0.091 →0 Holding value for branch j=2: 0.0+0.182 2e−0.05 ≈0.087 <0.091 →0.091 Holding value for branch j=3: 0+0.091 2e−0.05 ≈0.043 <0.091 →0.091 The value of the estimator for this node equals 0+0.091+0.091 3= 0.061. This is also the (under)estimated value of the option. Following arguments of Broadie and Glasserman (1997), one can easily prove the following crucial fact. Theorem 1. Both Θ and Φ are consistent and asymptotically unbiased estimators of the option value. They both converge to the true price of the option as the number of price tree branches, l , increases to infinity. For a finite l : •The bias of the Θestimator is always positive, i.e., E[Θ0(l)] ≥f0( S0). •The bias of the Φestimator is always negative, i.e., E[Φ0(l)] ≤f0( S0). On every realization of the price tree, the low estimator Φ is always less than or equal to the high estimator Θ , i.e., P(Φtl1,...,li i≤Θtl1,...,li i )=1. 3. Numerical Analysis In this section, we will present results of the numerical analysis. First, we use the algorithm described above to price the American option with arbitrary parameters. This will allow us to confirm that our Monte Carlo algorithm produces precise estimates of options’ prices. We focus on options related to Microsoft Corporation stock. Next, we price π -options for several combinations of parameters. We also consider π -option on drawdown using the real market data and we compare 103 Risks 2020,8,90 it with an American put, which is one of the most popular tool for protecting our portfolio against price drops. 3.1. American Options First of all, we decided to check the robustness of the Monte Carlo pricing algorithm. We estimate prices of the American call options with different strike prices. In the example, the uderlying asset price S0 equals 100, σ= 20%, risk-free rate r= 5% and the maturity is 30 days. In Table 1we present the results of the estimation. Please note that when using the Broadie–Glasserman algorithm, we obtain the upper and the lower boundaries of the option price. To obtain the American option price estimate we average both values. Table 1. Comparison of the estimated and ’real’ American option prices with different strikes. Absolute percentage errors are also included. Strike Low Est. High Est. Estimated Price Real Price Abs. Perc. Err. $80 $20.16 $20.55 $20.36 $20.33 0.14% $85 $15.10 $15.54 $15.32 $15.35 0.19% $90 $10.28 $10.62 $10.45 $10.43 0.19% $95 $5.84 $6.02 $5.93 $5.89 0.68% $100 $2.54 $2.60 $2.57 $2.51 2.36% $105 $0.76 $0.77 $0.77 $0.73 5.33% $110 $0.16 $0.16 $0.16 $0.14 13.35% 3.2. π-Options We will analyze put π -option for various combinations of parameters a and b . We assume that parameter a is varying from − 1.1 to − 0.9 and b parameter between 0.9 and 1.1. The ranges of these parameters have been chosen arbitrarily for illustrative purposes. All input parameters for options pricing, S0 , M0 , volatility and interest rate are taken from the real market data for the Microsoft Corporation stock (MSFT) and are given in Table 2. The numerical results are presented in Figure 6. Table 2. Input parameters for pricing π-put option on the Microsoft Corporation stock. Parameter Value S0106.08 M0110.83 σ17.03% r1.5% l65 K1 104 Risks 2020,8,90 -0.9 -0.94 a parameter -0.98 -1.02 -1.06 -1.1 0.9 0.94 0.98 bparameter 1.02 1.06 1.1 0.4 0.3 0.2 0.1 0.5 0 0.6 0.7 Option value Figure 6. π-option price estimations for varying aand b—Microsoft Corporation stock. 3.3. π-Options on Relative Drawdown Recall that for a=−1 and b=1 the payoff of the π-option equals K−St Mt+ , (6) where St/Mt is the current value of the relative drawdown of the underlying asset. We believe that such contracts could be very efficiently used for hedging and managing portfolio risk against the volatile drops in underlying’s price (see Section 3.4). One can adjust the payoff function (6) by the appropriate choice of the strike K . The choice is arbitrary and solely dependent on the risk management goals of the option’s buyer. It allows the setting of the minimal size of drawdown we would like to protect against and let the buyer adjust and full control of the level of our exposure at risk associated with unexpected price drops. For example by setting K=9 10 , the payoff of our option becomes greater than zero only if the drop in the price of the underlying from its maximum exceeds 10%. Of course, the bigger the value of K, the more expensive the option is. We take a closer look at the impact of Mt and K on the price of this special case of π -option. Here, we assume that the maximum price Mt is between 100 and 120 and K ranges between 0.8 and 1. This time, the remaining parameters, namely S0 , r and σ , have been arbitrarily chosen for illustrative purposes and are given in Table 3. The results are shown in Figure 7. Table 3. Input parameters for pricing π-option on relative drawdown. Parameter Value S0100 σ20% r5% l100 105 Risks 2020,8,90 1 0.96 0.92 Strike 0.88 0.84 0.8 100 104 Maximum 108 112 116 0.15 0.2 0.1 0.05 0 120 Option value Figure 7. π-option on relative drawdown price estimations for varying Kand M0parameters. 3.4. π-Options on Relative Drawdown - Application We now focus on the potential application of π -options and compare the prices of American put and π -option on relative drawdown. We compare these particular instruments due to the fact that their values increase with the decrease of the underlying asset’s price. As an exemplary environment for the options comparison we choose two time series containing daily closing prices of the Microsoft Corporation’s stock (see Figure 8) as well as daily closing prices of the West Texas Intermediate (WTI) crude oil futures (see Figure 9). Both datasets are taken from www.finance.yahoo.com and span approximately one year, from 6 November 2017 to 9 November 2018. We use the first 9 months (from 6 November 2017 to 3 August 2018) to calibrate the historical volatility for both assets, which is one of the input parameters in our pricing algorithm. Then, using the historical volatility, we compute prices of π and American options (using assets’ prices from 3 August 2018), both expiring 3 months after the end of calibration period. Please note that the parameters for the π -option on a relative drawdown are a=− 1, b= 1 and K= 1. Input parameters for calculation and estimated options prices for both assets are given in Tables 4and 5. 106 Risks 2020,8,90 Figure 8. Daily closing prices of the Microsoft Corporation’s stock. Data spans from 6.11.2017 to 9.11.2018. Vertical dashed line indicates the end of volatility calibration period. Option prices are calculated based on the volatility and the stock’s price on 3.08.2018. Figure 9. Daily closing prices of the West Texas Intermediate crude oil futures contracts. Data spans from 6.11.2017 to 9.11.2018. Vertical dashed line indicates the end of volatility calibration period. Option prices are calculated based on the volatility and the asset’s price on 3.08.2018. Since the payoff of π -option on relative drawdown with K= 1 is always less than 1, to compensate against the drop in underlying’s price, we need a certain number of these contracts per each unit of stock in our portfolio. This number must be equal to M0 . Please note that in Tables 4and 5 , the real price of the single π-option on relative drawdown contract should be 0.0735 for MSFT and 0.0949 for WTI. However, in order to be able to compare the results to the American put values, we initially need to make the instruments pay the same amount in case of a price drop, therefore we multiply the price of single π -option on drawdown by M0 (110 and 74 for MSFT and WTI respectively). That is why in 107 Risks 2020,8,90 Tables 4and 5the price of π -option equals 0.0735 · 110 = 8.09 for the stock and 0.094 · 74 = 6.95 for the oil futures contract. Table 4. Input parameters for computation and estimated options’ prices for the MSFT dataset. MSFT American Put πon Drawdown Parameter Value Parameter Value K108.13 M0110.83 S0108.13 σ24.06% r2.25% l100 T3 Option price $5.47 $8.09 Table 5. Input parameters for computation and estimated options’ prices for the WTI dataset. WTI American Put πon Drawdown Parameter Value Parameter Value K68.49 M074.15 S068.49 σ23.97% r2.25% l100 T3 Option price $3.07 $6.95 It turns out that π -option is more expensive than vanilla put in case of both assets, which is not a surprise as it initially pays the amount equivalent to the present maximum drawdown. However, since the difference in price between these instruments is rather significant, a question emerges whether there exists a situation in which purchasing π -option on relative drawdown is more profitable than buying a simple vanilla put. To answer this question, let us focus on the dashed part of the Microsoft Corporation and WTI futures data from the beginning of this section. In Figures 10 and 11 we show the amount each instrument would pay (on each day) throughout the whole 3-month period until options’ maturity. 108 Risks 2020,8,90 Figure 10. Microsoft Corporation stock closing prices ( top ) and the corresponding payoffs of π -option on relative drawdown and American put (bottom) with the parameters from Table 4. Figure 11. WTI crude oil futures contract closing prices ( top ) and the corresponding payoffs of π -option on relative drawdown and American put (bottom) with the parameters from Table 5. 109 Risks 2020,8, 110 At present, there are many different approaches that are based on the assessment of individual components of the security of a company’s activities, and there is no structured methodology that includes intangible assets and intellectual property. The purpose of the study was to develop an algorithm to determine the economic security of businesses based on valuation of intangible assets in accordance with the IVS. In order to accomplish the purpose of the study, the following objectives were set: - Based on literature review, to define the position of intangible assets in the activities of economic entities and the availability of approaches that can be used to assess the impact of intangible assets and intellectual property on economic security of companies; - To consider the impact produced by intangible assets on the value and economic security of business entities; - To develop methodology to determine the economic security of a business based on the valuation of intangible assets using the IVS. The paper is structured as follows: literature review, description of the applied models and methods, substantiation of the applied data, specification of the research results and reliability analysis of the calculated research results. The paper also details an algorithm for determining the economic security of businesses based on the valuation of intangible assets according to the IVS, along with discussion of the results and conclusion. 2. Literature Review A universal algorithm for determining the economic security of enterprises based on intellectual property has not been developed so far. Starting from the 1980s, foreign scholars conducted major scientific research on the whole range of questions related to the role of intellectual capital for business development. They note a considerable effect produced by intangible assets on company security (Barth et al. 2001). American economists V. Andonova and Ru í z-Pava highlight that companies are highly dependent on their intangible assets. The authors also conclude that intangible assets are a major factor in the productivity of enterprises and determine their competitive advantages in the external environment (Andonova and Guillermo 2016). According to the International Valuation Standards 2020, section IVS 210, an intangible asset is defined as: “a non-monetary asset that manifests itself by its economic properties. It does not have physical substance but grants rights and/or economic benefits to its owner”. Tsai et al. (2016) present a study that is based on comparison of various types of machine learning for intangible assets. Clausen and Hirth (2016) in their work introduce a profit indicator, related to the value of intangible assets based on the productivity of intangible assets. Gu and Li (2015) in their study investigate the matters related to investing in companies based on intangible assets. Vasconcelos et al. (2019) presented work aimed at studying the relationship between the intangible assets, macroeconomic environment and market value of public companies in Germany, the UK and Portugal. They also investigated the impact of intangible assets on the market value of companies using sensitivity tests. In their research, the authors Montresor and Vezzani (2016) highlight the innovative impact of intangible investments and claim that via intangible investments companies acquire knowledge assets that increase their innovativeness. Matos et al. (2018) formulated a hypothesis that future results of many companies will depend on intangible assets. They carried out analysis of intangible assets for a number of European Union countries. The research by Basso et al. (2015) shows the contribution of intangible assets in the creation of the value of companies using the methodology suggested by Gu and Li. In his research, Nejati (2016) explains the main components of intangible assets, namely human capital, structural capital and relational capital. Russell (2016) considers the intangible assets of pharmaceutical companies and compares the value of these assets in terms of their significance. 116 Risks 2020,8, 110 In their research, Pastor et al. (2017), Bontis (2001), Bouteiller and Karyotis (2010) and Pastor et al. (2017) carried out analysis and review of the literature dedicated to intangible assets and their valuation as well as the examples of methods that can be used to evaluate individual intangible assets. The work by Plaskova et al. (2019) carried out analysis, based on which a clear definition of an innovative asset as an element of an organization’s intangible assets was given. Proposals were made to create a solid business image and investment attractiveness of an organization. Authors Boj et al. (2014) look at intangible assets and intellectual capital as the key drivers creating value and competitive advantages for organizations (Rodionov et al. 2018a) They suggest methodology for defining, measuring and managing the relevance of intangible assets in achieving the strategic goals of an organization (Bouteiller and Karyotis 2010). In the method described by Kaplan and Norton (2004), a firm initiates the most important processes and determines human, information and organizational capital necessary for these processes (Rodionov et al. 2018b). Del Giudice and Paola (2017) consider intangible assets and intellectual property from the perspective of the fact that they ensure competitiveness, prosperity and growth of the enterprise. Based on the literature review it can be concluded that the issues concerning the impact created by intangible assets on economic security of companies have not been extensively studied to date. Moreover, not enough attention is paid to economic security on the basis of intangible assets. According to the data presented in the survey conducted by Brand Finance GIFT, the Top 100 Companies by Total Intangible Value, among 100 large companies, more than 50 have intangible assets exceeding 90% of the value of the entire business. Examples of these companies are Johnson & Johnson, Visa Inc., The Procter & Gamble Co., Anheuser-Busch InBev., Comcast Corp., Mastercard Inc., Novartis AG, Amazon.com Inc., and Microsoft Corp. (Brand Finance 2019). Thus, many business entities carry out their activities only because they have trademarks, patents, new technologies, intangible assets and intellectual property (Chernogorsky 2018). Accordingly, new R&D, advanced technologies and know-how are becoming more and more actively involved in business processes, which increases the importance of intellectual property and intangible assets, so determining economic security in this field is becoming increasingly important. At the same time, it was observed that to date no algorithm has been developed for determining the economic security of a business based on valuation of intangible assets in accordance with the IVS. At present, there are many different approaches that are based on the assessment of individual components of the security of a company’s activities. In addition, it should be noted that the presented approaches do not have a structured methodology. Some proposed methods have the following disadvantages: - There is no possibility of practical implementation due to the absence of assessment criteria or an established scale of values; - There is no approach to assessing business security that would take into account all the components of the economic security of an enterprise; - The assessment of the company’s security is based only on the threats or risks of its implementation; - Underestimation of the impact of intangible assets on the economic security of companies. Accordingly, a distinctive feature of this study is the special attention paid to intangible assets and intellectual property and their impact on the economic security of companies. 3. Models and Methods In the course of the study, we identified the parameters that could be used to judge the factors that affect the value and economic security of business entities. The indicators of companies that are not interdependent act as factor characteristics, X. These characteristics include revenue, intangible assets, intellectual property, fixed assets, assets under construction, financial investments, current assets and long-term and short-term liabilities. 117 Risks 2020,8, 110 Capitalization or the value of business entities (the resulting characteristic, Y) is understood as the product of the market value of one company’s share (share price) and the number of shares in circulation. The imbedded Excel package “Data Analysis” and statistics data analysis package “Stata” were used for modeling. In the course of the study it was established that there is a ratio between the resulting indicator and variables. Its direction was defined, as well as the correlation ratio and adequacy of the model obtained, which implies the degree to which the theoretical model that was built to describe the relationship between the characteristics reflects the actual dependence between these characteristics, i.e., whether the model is practically admissible. In order to check the presence of heteroscedasticity of random errors in the regression model obtained according to the initial values of the characteristics in logarithmic form, the White test was used. The test is based on checking a time series for heteroscedasticity. An important accompanying problem is to verify the causal link between the time series of the factor and resulting characteristic, which was settled using the Granger causality test. The test can be used to answer the question: Is it true that change in the value of intangible assets and intellectual property (X) will entail change in the company capitalization (Y)? It was checked using a linear regression model of Yvalues on previous Xand Yvalues. In other words, Yvalues are presented in the following form: Yi=ui+akyi−k+bkxi−k+Ei(1) Yiis the value of variable Yat time i; Xiis the value of variable Xat time i; kis the time delay (in our case, a lag). If in the regression obtained coefficients kof the formula can be neglected, it is believed that the previous Xvalues do not help to predict Yand, consequently, Xis not the cause of Yaccording to the Granger causality test. Based on the model obtained, an algorithm determining the economic security of businesses was suggested. The uniqueness of the algorithm is in the fact that it unites all the basic functions of intangible assets and intellectual property that provide economic security. In addition, the algorithm is versatile and can be used by companies operating in different industries. 4. Data The largest companies were chosen as objects of the research, since they are clearly indicative representatives of the oil industry among Russian companies whose shares are listed on the stock market and that represent 90% of the market in the sector. The analytical data posted on the official websites of Russian organizations formed the information basis for the research. The financial and economic indicators of the Russian companies whose shares are listed on the OJSC Moscow Stock Exchange and the Russian Trading System (RTS) were the empirical basis of the research. In addition, the totality of indicators was determined from the existing indicators of the financial statements of the business entities over the last seven years for each company operating in the oil sector. 5. Results The results of the calculations and the regression model built for the oil industry are presented below. 118 Risks 2020,8, 110 5.1. Adequacy Analysis of the Calculated Research Results The indicators of companies, which are not interdependent among themselves, are used as factor signs “x”. These features are revenue, intangible assets, intellectual property, fixed assets, construction in progress, financial investments, current assets and long-term and short-term liabilities. By capitalization or the value of business entities (resultant attribute “Y”) we mean the market value of one share of the company (share price) per the number of shares in circulation. Data from financial statements for the last five years were used for calculations. For this study, annual data were used. The tightness of the relationship between linearly dependent features was determined using a linear correlation coefficient (r), the calculation of which is automated using statistical data analysis packages. The linear model of pair regression between the value of intangible assets and company capitalization has the following form: Y=−1.946987x1+307.4673x2−0.4629237x3−2.445406x4−0.4796452x5 +6.288961x6−0.1429317x7−103000000 (2) The regression coefficient under xshows that if the value of intangible assets and intellectual property increases, the market capitalization of the company increases too. Input data for the computational model are presented in Appendix A. The model was checked for adequacy. The results presented in the Tables 1–4. Table 1. Regression analysis data of the relationship between the value of intangible assets (x 2 ) and the capitalization of Russian companies. Source Coefficient Std. Error t-Ratio p-Value (p>t) const 307.4673 52.09164 5.90 0.004 l_lnt_rus − 1.03000000 5.4900000 −1.88 0.134 Thus, the result of 0.004 means that the hypothesis is confirmed as the result was less than 0.134. Table 2. Regression analysis data of the relationship between the value of intangible assets and the capitalization of Russian companies. Sum Squared Resid 43,566,000,000,000,000 R-squared 0.9971 Adjusted R-squared 0.9922 F(7, 6) 113.93 p-value (F) 0.0001 Since the significance level ap (p-value), calculated for coefficients a0 and a1 is lower than the set significance level a =0.01, both these coefficients are recognized as non-random (i.e., typical for the general population). The value of the determination index R2 (R-squared in the table) is equally 0.9971. This value is over 0.5, which is evidence of the good approximation of the source (actual) data using the built linear function of relation. Table 3. The regression output p-value of each variable. Source p>|t| x1 0.007 x2 0.004 x3 0.073 x4 0.095 x5 0.436 x6 0.001 x7 0.828 _cons 0.134 119 Risks 2020,8, 110 The adequacy of the regression model to the actual data was also established by Fisher’s ratio test, which evaluates the statistical significance (non-randomness) of the determination index as typical, so the linear model of relation between characteristics X and Y is to a greater degree applicable to the general population of enterprises as a whole. Then, a heteroscedasticity test was used. The presence of heteroscedasticity leads to the following negative effects: the estimations of the standard errors of regression coefficients are displaced, the estimations of regression coefficients using the method of least squares are ineffective and t-statistics of regression coefficients are inadequate. Table 4. The results of the heteroscedasticity test. Source chi2 df p Heteroscedasticity 12.00 11 0.3636 Skewness 7 Kurtosis 1 Total 19 As a result of the test, it was revealed that in the majority of cases heteroscedasticity is satisfactory, so general statistical methods can be used. According to the results of the test, it was concluded that the p-value is higher than the significance level chosen as 5% (0.3336 >0.05), so hypothesis zero about the lack of heteroscedasticity was not rejected, i.e., the random disturbance dispersion does not depend on X and the regression model (3) detailed above is homoscedastic. This proves the adequacy of the statistical valuations of the quality of the linear regression model. Calculations were made according to the Granger test for the period from 2013 to 2019 with the time lag being 1. To study the directions of the causal relationships between the intangible assets and capitalization, the Granger test was used, where x1 is the intangible assets of the company. If it is >0.05, it cannot be claimed that the hypothesis “A is NOT the Granger cause of B” is true. Thus, capitalization is dependent on intangible assets, since the coefficient is 0.224 and 0.997. Typically, the Granger test tests two null hypotheses: “xis not the cause of y by Granger” and “(Y is not the cause of Xby Granger”. The p-values are small, so we accept the hypothesis that X1 is the Granger cause of Y1. Further, when the situation is reversed, p-values are greater than 0.05; therefore, we reject the hypothesis that Y1 is the Granger cause for X1. According to the above information it can be concluded that despite industry specific features, which affect the quantitative values of intangible assets and intellectual property, the value of business entities and their level of economic security are affected. 5.2. Algorithm to Determine Economic Security of a Business Based on Valuation of Intangible Assets According to the IVS According to the results of the study, an algorithm was developed to determine the economic security of businesses. This algorithm is based on a multi-stage comprehensive analysis of intangible assets and intellectual property. As an example, one of the large oil companies represented on the Russian market was considered. At the first stage the company performance was preliminarily analyzed considering the specifics of the sector where it operates. The performance analysis was carried out on the example of the Neft Y company, for which indicators for the period 2017–2019 were analyzed. The main indicators of the financial status and performance of Neft Y were selected and grouped according to the qualitative characteristics in the period analyzed. The company performance is defined by the following indicators: • The net assets exceed the equity capital, and an increase in the net assets was observed during the analyzed period; 120 Risks 2020,8, 110 • A positive change in the organization’s own capital in relation to the total change in the organization’s assets; •A growth in revenue by 94.4% was observed during the analyzed period; •The share of self-cost in revenue was 91.08–93.49 % during the analyzed period; • Profits grew by 49% during the analyzed period, and net profit was obtained (2,366,408 thousand rubles); •A growth in fixed assets and intangible assets was observed; •Borrowed money is actively used in the company’s operations. Based on the above analysis, it can be concluded that positive dynamics of the main indicators (revenue, net profit) are observed in the performance of Neft Y. The company actively involves intangible assets in its operations. At the second stage, more profound analysis of the indicators was carried out. Firstly, in Block 1 we analyzed the existing intangible assets, including the rights for the results of intellectual activity that are not accounted for in books, as well as the efficiency of the intangible assets management system of the business entity. According to the conducted analysis, Neft Y has the following intangible assets: a license for exploration and production of raw hydrocarbons and a patent. Thus, intangible assets are applied in the operations of the company, which allows it to use new technologies and explore deposits for producing raw hydrocarbons. In Block 2, investments were calculated. In this block, investments in intangible assets and intellectual property were calculated. The value of intangible assets and intellectual property was calculated according to the IVS to achieve a high quality of calculations along with transparency and reliability. Since Neft Y acquired a new license for exploration and production of raw hydrocarbons, the value of the required investments was estimated, as detailed in Section 5. In Block 3 the sources of the effect were analyzed. In order to determine the source of the effect (benefits, profits) from using intangible assets and intellectual property, it is important to carry out a comprehensive study, which represents a legal and engineering study. The legal study includes defining the title documents based on which the rights for intellectual property are vested. In the engineering study, the quantitative and qualitative technological and engineering characteristics and parameters of the goods produced due to the presence of intellectual property are established. When the sources of the effect were analyzed, the following intangible assets were identified for Neft Y: a license for exploration and production of raw hydrocarbons and a patent for a gravel filter. The patent is a title document. The invention is specific to the oil and gas industry and can be used to install gravel filters and to overhaul boreholes. The validity period of the patent is 20 years. Neft Y has a registered trademark, which is not accounted for in books. Thus, the trademark of Neft Y can be accounted for in books according to the market value. Stage 3.1. Using intangible assets in business activities (calculating the annual income from using them). In this case it is assumed that the business entity is the holder of exclusive rights due to which the business entity has a right to produce unique goods and services. Stage 3.2. Using intangible assets in commercial turnover, license for intangible assets. According to the license contract, the holder of the exclusive right (licensor) grants the other party (licensee) the right to use the intellectual property. The transfer of non-exclusive rights is another source of income from applying intellectual property. Neft Y has not made license contracts so far but plans to consider the possibility of granting non-exclusive rights for the use of the patent for the gravel filter. Stage 3.3. Using intangible assets when exclusive rights belong to three parties. In this case the business entity uses intangible assets in its activities that belong to the right of use of a non-exclusive 121 Risks 2020,8, 110 license. Prior to making a license contract, a feasibility study has to be conducted to make sure it is reasonable to conclude this contract and to adequately calculate the price of the right of use. Neft Y lacks such contracts, so no analysis was performed at this stage. Stage 3.4. Using intangible assets as a collateral for attracting investments. A mandatory condition for collateral is the state registration of the above list of assets. In order to obtain the collateral, the market value of the asset has to be defined. In this case, special attention must be paid to the quality of the valuation report, which will be used as a basis for taking a decision about the collateral. It is the IVS that ensure the quality, transparency, fairness and reliability of the valuation. This is extremely important for taking investment decisions and for the purposes of collateral. The registered trademark and the patent for the gravel filter can be the subject of collateral for the Neft Y company. Stage 3.5. Using intangible assets to make a payment in the business entity’s equity capital. Exclusive rights for intangible assets can be introduced into the company’s equity capital. All intangible assets are introduced into the equity capital of the business entity at market value calculated in the valuation report that is prepared according to the IVS. Increasing the equity capital helps to attract investments for the activities of the company. In Block 4 the current expenses of the business entity were analyzed. Stage 4.1. Analysis and calculation of patent taxes to maintain the patent in force. Stage 4.2. Tax analysis and calculation. Periodic (current) payments for the use of rights for the results of intellectual activity and rights for individualization means (in particular, the rights emerging from patents for inventions, useful models, industrial samples) are included in the composition of the company’s expenses. Thus, due to an increase in expenses, the size of the profit tax goes down. In addition to the income obtained by business entities due to intangible assets, it is reasonable to account for the tax benefits for the rights holder. Tax benefits include reduction in the amounts of taxes and an effective increase in the cash flow of the business entity. For some objectives of valuation, such as financial statements, the tax benefit from depreciation should be included in the valuation when applying the income approach to intangible assets (International Valuation Standards 2020). Thus, intangible assets give the company real tax benefits due to depreciation, which is in many tax jurisdictions. The calculation of results are presented in the Table 5. Table 5. The profit tax calculated prior to and after the intangible assets were accounted for and depreciated. Item 2019 (without Accounting for the License for Intangible Assets) 2019 (Accounting for the License for Intangible Assets) Revenue from selling goods, products, work and services (in current prices), thousand rubles 41,785,958 41,785,958 Full self-cost of sold goods, work, services, thousand rubles 38,666,550 38,761,413 Depreciation of fixed assets 118,266 118,266 Depreciation of intangible assets 50 94,913 Earnings before interest and tax (EBIT) 2,958,010 2,863,147 Profit tax, thousand rubles 591,602 572,629 Difference in profit tax for one year including and excluding the depreciation of intangible assets, thousand rubles 18,973 Compiled by the authors. 122 Risks 2020,8, 110 Thus, the profit tax due to the depreciation of the business entity’s intangible assets can be 18,973 thousand rubles lower per year. Stage 4.3. Analyzing and calculating royalty fees. The company paying royalty fees to the authors for using intellectual property is one of the most important issues. Royalty fees were not calculated in this study because the company lacks patents wherein the authors have the right to receive royalty fees. Stage 4.4. Analyzing and calculating payments under license contracts. Payments under license contracts can be defined by one of the following options: royalties (payments represent a percentage of the licensee’s revenue from the products sold), a lump sum payment (a single payment, which represents a fixed amount) and a combined payment (part of the amount is paid in one installment, and the second part represents payments in form of royalties). Stage 4.5. Expenses related to risks in the sphere of intellectual rights (legal expenses). Legal expenses in the sphere of patent disputes can amount to substantial costs that business entities bear in case of litigation. These expenses arise if legal disputes are dealt with. Stage 4.6. Expenses related to loan payments in case intangible assets are used as collateral. In this case, expenses related to payment interest on loans arise only if the business entity has a loan and occur according to the terms of the contract. In Block 5 the value of the effect was calculated. The effect from intangible assets and intellectual property can be expressed in the ways described below. Stage 5.1. Calculating the market value of intangible assets. The market value of the asset is determined according to the IVS. The calculation of the market value of the license for production of raw hydrocarbons is presented as an example in Section 6and Table 7. Stage 5.2. Calculating the profits from using intangible assets and intellectual property. Earnings from the use of intangible assets and intellectual property can be formed by regular royalty payments, depreciation deductions of intangible assets, tax benefits and collateral benefits. Receiving regular royalty fees is possible in case a license contract is made to transfer non-exclusive right of use of the patent for the gravel filter. In case the license contract is concluded, Neft Y can receive annual income amounting to, on average, 1.91% from the earnings formed with the application of the above patent. Thus, if a medium company in the oil and gas sector applies the patent, it can bring the holder of the exclusive ownership rights 612,350 thousand rubles, on average, with the average earnings being 32,103,215 thousand rubles and the average value of the royalty rate being 1.91%. Below is given the calculation of the amount of license fee for use of the patent for one year Table 6. Table 6. The calculated royalty rate (annual payment). Name 2017 2018 2019 Revenue 21,495,399 33,028,289 41,785,958 Gross profit 1,917,073 2,148,678 3,119,408 Pe =(Gross profit/Revenue), % 8.92% 6.51% 7.47% The average value of Pe 7.63% Licensor’s share in the licensee’s profit 25% Royalty rate R=D×Pe 1+Pe,1.91% Compiled by the authors. 6. Valuation of Intangible Assets According to the IVS In order to implement the algorithm determining the economic security of a business at the investment stage, it is necessary to appraise the investments required for acquiring or creating intellectual property and intangible assets. The market value of the intellectual property and intangible assets is determined according to the IVS. The IVS are key guidelines for carrying out qualitative valuation all over the world. Applying the IVS gives us a high quality, reliable assessment which is internationally recognized (IVSC 2020). 123 Risks 2020,8, 110 The market value of the license for the right to produce raw hydrocarbons for Neft Y based on IVS 210 Intangible Assets (IVS 210 Intangible Assets) was calculated using a comparative approach. According to the IVS, corrections were introduced into the calculations to reflect the specific features of the intangible assets that were evaluated. The method of comparative transactions was used in terms of the comparative approach according to IVS 210 (IVS 210). In order to estimate the interest discount of Urals oil price to Brent oil price on the markets of Western Europe and the USA, the average level of oil prices for the period 2012 through 2018 was used. The average value according to agency Platts was 1.2%. The average oil prices were according to the source https://ru.investing.com. The average value of the specific indicator of the resource value (price of the license/recoverable resources) was calculated based on the results of the tenders and auctions for obtaining licenses for exploration and production of raw hydrocarbons (www.torgi.gov.ru). The average Urals oil price on the world market as of the tender/auction date was used in the calculations. After the calculations were made, the corrected value of the stock was 85.957 rub./t. The calculation of results are presented in the Table 7. Table 7. The results of the calculated market value of the license. Deposit The Quantity of Resources by Category C1 as of 30 March 2020, thousand t The Quantity of Resources by Category C2 as of 30 March 2019, thousand t Data on the Extracted Oil Since 30 March 2020 till the Valuation Date, thousand t Extracted Oil Resources as of the Valuation Date, Reduced to Category C1, thousand t Corrected Value of Resources, rub./t Market Value of Resources as of the Valuation Date, thousand rub. Deposits (investments of Neft Y) 54,717 986 0 55,210.0 85.957 4,745,665 Total: 54,717 986 0 55,210.0 4,745,665 Source: data of the customer, authors’ own calculations. An algorithm for determining the value of intangible assets according to the IVS is presented above. It was considered on the example of Neft Y and represents a sequence of actions to be taken to determine the value of the license for production of raw hydrocarbons. This structure is part of the algorithm for determining economic security of a business, because intangible assets are one of the major components that provide economic security of economic entities. 7. Discussion and Conclusions This study analyzed the impact that the value of intangible assets and intellectual property has on capitalization of companies and their level of economic security, based on calculated values. The study relies on pair correlation relationships between the factor and performance characteristics. The impact of revenue, intangible assets, intellectual property, fixed assets, assets under construction, financial investments, current assets and long-term and short-term liabilities was analyzed. The calculated results of the study are presented for the example of the oil and gas sector. The effect of intangible assets and intellectual property on the company value and economic security were determined. An algorithm was developed to determine economic security based on valuation of intangible assets according to the IVS. It includes the entire cycle of the enterprise’s use of intangible assets and intellectual property to calculate economic security. The algorithm includes analysis of the business entity’s activities, which consists of two stages (preliminary analysis and in-depth analysis of indicators). Five interrelated blocks are presented: 1—analysis of the intangible assets and intellectual property existing in the enterprise; 2—calculation of the investments necessary for intangible assets and intellectual property; 3—analysis of the sources of the effect (possible earnings from the intangible assets and intellectual property are identified as well as the ways they can be used to attract investments in the company and increase the value of the company’s assets); 4—possible expenses of the business 124 Risks 2020,8, 110 entity, as well as the possible options for reducing them. This section presents possible benefits in terms of profit tax due to depreciation deductions on the company’s intangible assets. Thus, in the presented algorithm, qualitative assessment of the value of intangible assets and intellectual property according to the IVS is the major component revealing the economic security of business entities. The multiple stages of the suggested algorithm make it versatile and suitable for application by companies that use intangible assets and intellectual property to different extents. The uniqueness of the presented algorithm is due to the fact that it contains a full set of stages to manage intangible assets and intellectual property within which the values of the assets are defined in accordance with the International Valuation Standards. This is an essential component of the algorithm that determines the economic security of businesses. The algorithm can be used to evaluate the company’s activities in a new way, to prevent risks and use new possibilities related to the application and valuation of intangible assets and intellectual property according to the IVS. The practical significance of the research is that the results of the study may be used in the operations of modern companies that apply intangible assets and intellectual property in their activities to determine sustainable development and form an effective system for economic security management due to the use of intangible assets. Further research will involve goodwill accounting and valuation according to the IVS aimed at determining the economic security of companies. Author Contributions: Conceptualization, D.R., O.P.; methodology, D.R., O.P.; software O.P.; validation, O.N., formal analysis, D.R., O.P. and O.N.; investigation, D.R., O.P. and O.N.; data curation and writing–original draft preparation, D.R., O.P. and O.N.; visualization, D.R., O.P.; supervision, D.R., O.P. and O.N.; projectadministration, Dmitrii Rodionov, O.P.; funding acquisition D.R., O.P. All authors have read and agreed to the published version of the manuscript. Funding: This research was supported by the Academic Excellence Project 5-100 proposed by Peter the Great St. Petersburg Polytechnic University. Acknowledgments: The authors thank everyone who helped to make the research happen. Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results. Appendix A Table A1. Research results for the oil and gas industry (annual data, yearly data). Name Average Data Values for the Period 2012–2019 (Annual Data) X1 X2 X3 X4 Revenue, Thousand Rubles Intangible Assets and Intellectual Property Assets (Results of Research and Development, Unfinished R&D), Thousand Rubles Fixed Assets, Thousand Rubles Unfinished Construction Objects, Thousand Rubles PJSC “NK” LUKOIL “ 289,492,597 949,960 13,404,602 1,828,806 PJSC “GAZPROM” 4,262,855,065 15,753,276.38 7,054,326,754 704,993,665.6 PJSC TATNEFT 527,673,919.3 1,573,121 182,396,433.4 69,466,968.75 PJSC ANK “Bashneft” 576,570,420.9 2,046,907.625 133,885,790 14,934,543.88 OJSC “Surgutneftegas” 1,081,006,001 663,165.625 760,351,258.4 701,595,646.4 PJSC “Varyeganneftegaz” 30,764,814.13 416,054.25 28,743,199.25 1,814,514.375 PJSC “Gazprom Neft” 1,386,778,553 2,602,433.25 4,521,554.875 1,424,129.875 PJSC “Saratov Oil Refinery” 13,138,640.13 48,927.125 14,435,942 2,895,163.625 JSC “Slavneft-YANOS” 26,025,383.38 113,698.5 38,464,365.5 4,252,928.75 JSC “YATEK” 4,832,172.25 547,987.125 7,218,803.375 3,093,899.375 PJSC “Transneft” 800,467,224.3 5,555,425.875 58,474,417.5 0 OJSC “Slavneft-Megionneftegaz” 148,234,764.8 564,772.75 79,140,407.13 8,773,602.125 125 Risks 2021,9,1 case was GARCH, which had the smallest mean squared error (MSE). Ortiz Arango (2017), in turn, used GARCH models and neural network differentials (RND) to predict the future prices of financial assets, specifically the future development of the price of a barrel of oil. He found that neural networks produce better results than the basic GARCH model and are therefore a reliable alternative method for time series analysis. Lu et al. (2016), who predicted the volatility of log-returns in the Chinese energy market using the GARCH model and neural networks, also confirm the better predictive ability of neural networks. Similarly, Arneric et al. (2014), who examined the development of the Croatia stock market (CROBEX) or Mohamed (2013) index price by comparing GARCH models and neural networks for modeling financial returns in the market in the Arab Republic of Egypt, confirmed the better predictive power of neural networks in relation to GARCH models. Due to the findings from these studies, neural networks will be applied in this contribution for exchange rate prediction. The application of NN (neural networks) for predicting and trading the EUR/USD exchange rate is described by Dunis et al. (2011). Dhamija and Bhalla (2011) found that NNs can be effectively used for forecasting exchange rates and therefore also for business strategy proposals. Guresen et al. (2011) also argued that exchange rate forecasting is an important financial issue, one that is receiving an ever-increasing amount of attention. Over the last few years, a number of neural network models and hybrid models have been put forward to exceed traditional prediction results in an effort to surpass traditional linear and non-linear approaches. Guresen et al. (2011) assessed the effectiveness of neural network models, which are known to be dynamic and effective in financial market forecasting. The analyzed models were multi-layer perceptrons (MLP), dynamic artificial neural networks (DAN2), and hybrid neural networks that use generalized autoregressive conditional heteroscedasticity (GARCH) to extract new input variables. Sindelarova (2012) also dealt with the application of artificial neural networks (ANN) for the prediction of economic time series. First, she focused on the revision of the basic existing ANN architectures for predicting time series and described their application in predicting the CZK/EUR exchange rate. She also presented a hybrid version of ANN, as did Bielecki et al. (2008), which was based on the same network strategy, but tried to increase the prediction accuracy. The results of the studies are comparisons of the hybrid approach and the accuracy of traditional ANN settings for the CZK/EUR or USD/PLN exchange rates. However, due to the many parameters to be empirically assessed, it is not easy to choose a suitable NN architecture for the prediction of the exchange rate. Researchers frequently do not consider the influence of the neural network parameters on its performance. Zhang and Hu (1998) examined the effect of the number of the input and hidden nodes and the size of the training sample on the performance in and outside the sample. For a detailed examination, the GBP/USD exchange rate (prediction) was used. It was discovered that NNs outclass linear models, especially in the case of a short prediction horizon. Yin and Chen (2016) suggested a method for the application of the exponential generalized autoregressive conditional heteroscedasticity-M (EGARCH-M) model in connection with the Elman NN for predicting the return rate of the USD/CNY exchange rate. The EGARCH-M model captured the volatility asymmetry, plus the correlation between the return, and this one was past volatility; Elman’s NN was used so that it corresponded with the non-linear character of the return rate. GBP/CNY and USD/CNY exchange rate predictions were carried out by Liu et al. (2011) using predictions by RBF neural networks and GARCH models. CNY rates can be considered as a financial TS (time series) characterized by a high non-linearity and a change of behavior over time (Cai et al. 2012). CNY has grown from a trading currency to an investment currency and currently has the potential to be a worldwide reserve currency. The development of CNY as an international currency might balance the USD dominated system and add to regional and international financial stability (Ma and Mccauley 2011;Zhang and Sato 2012). Interestingly, the correlation between the exchange rate and stock market performance was approached by Tian and Ma (2010), who used the autoregressive distributed lag model—ARDL’s cointegration approach to examine the impact of financial liberalization on the relationship between 132 Risks 2021,9,1 the exchange rate and stock market performance in China. They found that there was a cointegration between the Shanghai stock index and the renminbi (RMB) against the US dollar and the Hong Kong dollar from 2005, the year in which the Chinese exchange regime became a flexible, managed floating system. The authors found that the exchange rate and the money supply affected the share price with a positive correlation. They also showed that the increase in the money supply had been largely due to the huge influx of “hot money” from other countries in recent years. The prediction of exchange rate changes, their link to other macroeconomic phenomena and possible geopolitical impacts have been the subject of an extremely extensive volume of research. For example, Ilzetzki et al. (2019) dealt with exchange rate arrangements and restrictive measures in 194 countries. Ho and Karim (2012) examined the significant relationship between exchange rates, macroeconomic fundamentals, and international trade in a group of Asian countries from 1980 to 2009. According to them, international trade is essential for developing countries for investment purposes and to attract foreign exchange in this liberalized and globalized world. Regression analyses show that market size and the exchange rate play a very important role in promoting international trade. Population growth has significant negative effects on developed countries like Japan and Singapore, but has positive effects on the Philippines. In addition, inflation rates have a negative impact on the Philippines and India, while financial market developments are only marginally significant in overall trade between Singapore and India. The results of the study represent the strategic policy implications for developing and developed Asian countries with regard to the facilitation of international trade and boosting growth. In this specific field, the first area of research was concerned with the correlation of exchange rates and inflation or business cycles (De Boer et al. 2020). Forbes et al. (2018) used vector autoregressive modelling to reveal the links between the exchange rate and inflation, and Nguyen and Sato (2020) used the same method to detect asymmetries in the Japanese yen. Using an autoregressive approach, Grabowski and Welfe (2020) identified four main determinants of the currency market: inflation, terms of trade, country-specific risks, and the state of the currency market. The correlation of exchange rates and consumer prices with a vector autoregressive model was then examined by Ha et al. (2020). Thee VAR and the ARDL (autoregressive distributed lag) models were used by Chiappini and Lahet (2020) to find the key factors for 24 emerging economies, thereby demonstrating China’s fundamental influence on the exchange rates of other Asian countries. The same method was used by Dogru et al. (2019) to analyze the effect of exchange rates on bilateral trade between the United States, Mexico, Canada, and the United Kingdom. Ponomareva et al. (2019) used time series regression for predicting the exchange rates of the US dollar, Japanese yen, British pound, and euro as well as the Australian and Canadian dollars. When using the baltic dry index to predict the exchange rates, Han et al. (2020) employed the method of time series. The use of other analytical methods is rather an exception. Behavioral equilibrium models used by Kharrat et al. (2020) are also relatively common as part of optimizing monetary investment strategies. It is mainly these investment strategies and optimal security that represent the second important area of research. Maggiori et al. (2020) focused on global portfolios and pointed out the difference between companies in the United States and other countries where securities were usually subscribed in foreign currency. Opie and Riddiough (2020) presented a new method for dynamically hedging currency exposure in international equity and bond portfolios using time series. The time series prediction test was also the basis of the spot exchange rate model for 16 currencies according to Narayan et al. (2020)Narayan et al. Bahmani-Oskooee and Hegerty (2007) provided an insight into history and stated that the increase in exchange rate volatility since 1973 has had indeterminate effects on international export and import flows. Although it can be assumed that an increase in risk may lead to a decrease in economic activity, the theoretical literature provides justification for positive or insignificant effects. Similar results were found in empirical tests. While modeling techniques have evolved over time to incorporate new developments into econometric analysis, no single degree of exchange rate volatility has dominated in the literature. 133 Risks 2021,9,1 New patterns in intraday currency trading were revealed by Khademalomoom and Narayan (2020); and a currency trading strategy that took into account the predictive power of currency implied volatility was presented by Ornelas and Mauad (2019) and Accominotti et al. (2019). Bulut (2018) successfully used Google Trends to predict exchange rates. Amo Baffour et al. (2019) dealt with the integration of an asymmetric model into an artificial neural network for the prediction of the exchange rates of five currencies. According to them, this hybrid solution dramatically increased the quality of the model. The significant risk of generalization in the search for suitable predictive models was pointed out by Cheung et al. (2019). According to their research, the performance of models varied fundamentally, depending on the length of the prediction. The effect of influencing the exchange rate in relation to the return on equity within the optimization models was revealed by Turkington and Yazdani (2020). An important topic is also investment in so-called safe-haven currencies, where, for example, Cho et al. (2020) are reducing the importance of the euro, which, according to them, is still one of the currencies that moves in opposition to global stock markets. The treasury-EuroDollar (TED) spread, and country-specific volatility and low liquidity factors were revealed by Maurer et al. (2019) as the two key sources of risk in foreign exchange (FX) markets. Another area is represented by the use of exchange rates as an indicator of the state of an economy. Augustin et al. (2020) used currency swap spreads for this purpose, and Dahlquist and Hasseltoft (2020) stated the need to include inflation and economic stability in monetary trading strategies. Another important topic is the interconnectedness of exchange rates and commodity prices (Liu et al. 2020). The link between the type of commodities and the exchange rate, or their collapse, was revealed by Bodart and Carpantier (2020), according to whom the impact on agricultural exports was significantly greater compared to the relatively small impact on energy and/or mineral exports. The impact of oil price shocks, especially in the long run, on exchange rates was identified by Huang et al. (2020). Chernov et al. (2018) dealt with the quantification of the risk of currency shocks through an empirical model of bilateral exchange rates. Colacito et al. (2018) also focused on the 10 most traded currencies in the world. They stated their heterogeneity of exposure to trade and currency shocks. A separate chapter of the research concerns the assessment of the effectiveness of monetary unions (Groll and Monacelli 2020), very often with an overlap to crises such as that in Greece (Kriwoluzky et al. 2019). Chari et al. (2020) addressed the performance of economies and the benefits of a single currency. Bonadio et al. (2020) focused on the speed of the impact of an exchange rate shock in Switzerland. Furthermore, the topic of interventions in exchange rates in order to support export potential due to events in global markets is also current. However, as Rajkovi´c et al. (2020) showed in the example of the currencies of the Balkans and Central and Eastern Europe, currency depreciation did not have a significant effect on the trade deficit. Interestingly, Xing (2018) found the complete opposite to be true, with rising wages and the cumulative appreciation of the RMB undermining China’s comparative advantage. This was also confirmed by Choi and Choi (2018), who found that the devaluation of the RMB had a direct effect on reducing unemployment. Min and Yang (2019) looked at the problem of debt risks in a currency other than the domestic currency in South Korean companies. With regard to the RMB, attention must also be paid to the impact of exchange rate changes on economic growth and income distribution. Ribeiro et al. (2020) stated that although low exchange rates lead to increased exports, they have a negative impact on the income of selected groups of the population. In a sample of 2500 pairs, Gopinath et al. (2020) primarily assessed the effect of exchange rates on business elasticity and determined the monetary paradigm. Research on the RMB is extensive, partly as a result of a series of analyses of the impact of the reform of the People’s Bank of China, which, according to Wen and Wang (2020), has led to reduced exchange rate volatility. This significant change was also addressed by Smallwood (2019), according to whom, exchange rate uncertainty has no effect on trade with the United 134 Risks 2021,9,1 States, or Cheung et al. (2018), who dealt with the impact of these changes on central parity. Liu and Woo (2018) also extensively analyzed the effects of the so-called trade war between these great powers, drawing attention to the rather vague term “equilibrium exchange rate” used by many politicians and economists. Ho (2020) pointed out the strong effects of virtual currencies and their exchange rates, even in terms of inflation and economic growth for Taiwan and China. 3. Materials and Methods The data for the analysis are accessible on the World Bank (2020) website. The information on the mutual exchange rates of the Czech crown (hereinafter referred to as “CZK”) and the Chinese yuan (hereinafter referred to as “RMB”) were used for the purpose of the analysis (i.e., the daily exchange rate records of these currencies). The time period began on 6 October 2009 and closed on 21 October 2018, which was the equivalent of 3303 data inputs. The unit was several CZK to one RMB. The descriptive characteristics of the dataset are presented in Table 1. Table 1. Characteristics of the dataset. Statistics Date–Input Variable RMB to CZK–Output (Aim) Minimum (training) 40,092.00 2.485800 Maximum (training) 43,394.00 4.163000 Diameter (training) 41,734.79 3.265645 Standard deviation (training) 939.26 0.392383 Minimum (testing) 40,102.00 2.496100 Maximum (testing) 43,393.00 4.155700 Diameter (testing) 41,755.71 3.272882 Standard deviation (testing) 957.97 0.394309 Minimum (validation) 40,111.00 2.498900 Maximum (validation) 43,388.00 4.152900 Diameter (validation) 41,768.68 3.246446 Standard deviation (validation) 1,438.25 0.499822 Minimum (overall) 40,092.00 2.485800 Maximum (overall) 43,394.00 4.163000 Diameter (overall) 41,743.00 3.263852 Standard deviation (overall) 953.64 0.392668 Source: Own research. Statistica software, version 12, by Dell Inc. was used for the data processing. Data mining, neural networks (i.e., automated neural networks (ANS)) were utilized for the computation of the neural structures. A regression was performed using neural structures. Multi-layer perceptron networks (MLP) and radial basis function (RBF) NNs were then generated. The MLP network has one or more hidden layers between the input and output layers, with the neurons arranged in layers, the connections always routed from the lower to higher layers, and with no interconnection between neurons in the same layer (see Figure 1)(Ramchoun et al. 2017). The RBF network in its simplest form is a three-layer forward neural network. The first layer corresponds to the inputs to the network, the second layer is a hidden layer consisting of a series of non-linear activation RBF units, and the last layer corresponds to the final output of the network. Activation functions in RBF are conventionally implemented as Gaussian functions (see Figure 2). Two sets of new neural networks were generated: 1. The self-sufficient variable was time and the dependent variable was defined as the CZK/RMB exchange rate. 135 Risks 2021,9,1 2. Time was an independent variable. The seasonal variable was characterized by a categorical variable represented by year, month, day of month, and day of week, in which the value was measured for each variable independently. The purpose was to work with the potential daily, monthly, and annual seasonal fluctuations in time series. The dependent variable was the CZK/RMB exchange rate. What follows next is the analogical work with the datasets. The time series was divided into three datasets (i.e., training, testing, and validation). The first dataset included 70% of the input data. The neural structures were created on the basis of the training set. Each of the two remaining datasets included 15% of the input data, respectively. Both of these datasets served to verify the reliability of the discovered neural structure (i.e., the discovered model). The time series delay was 1. In total, 100,000 neural networks were created, of which the five with the best traits were retained. The hidden layer contained at least two neurons and at most 50 neurons. For the radial basis function, the hidden layer contained at least 21 neurons and at most 30 neurons. The following distribution functions were considered for a multiple perceptron network in the hidden and output layers: Atanh, exponential, linear, logistic, and sinus. The performance of the individual datasets was defined in the form of a correlation coefficient. There were, of course, other performance measures such as root mean square error (RMSE), the mean absolute percentage error (MAPE), mean absolute bias error (MABE), and coefficient of determination (R2). The root mean square error (RMSE) is the square root of the mean square error (MSE). RMSE measures the differences between the values predicted by the hypothetical model and the observed values. In other words, it measures the quality of the fit between the actual data and the predicted model. Similarly, MAPE is a simple average of absolute percentage errors, a formula used to calculate an error in a statistical forecast that measures the magnitude of a predicted error. The coefficient of determination, R2, is a useful measure of the total value of the predictor variable(s) when predicting the resulting variable in a linear regression setting (Salkind 2010). Figure 1. MLP network structure (Source: Khalafi and Mirvakili 2011). 136 Risks 2021,9,1 Figure 2. RBF network structure (Source: Faris et al. 2017). The other settings remained in the default (as for ANS—automated neural networks). Finally, the results of both retained sets of neural networks were compared. 4. Results 4.1. Neural Structure A A total of 100,000 NNs were generated in the course of the above-defined procedure. The five that displayed the best parameters were retained and are presented in Table 2. Table 2. Retained neural networks. Network Training Perform. Testing Perform. Validation Perform. Training Error Testing Error Validation Error Training Algorithm Error Function Activ. of Hidden Layer Output Activ. Function 1RBF 1-30-1 0.983490 0.983020 0.984843 0.002516 0.002616 0.002319 RBFT Sum.quar. Gauss Identity 2RBF 1-26-1 0.984841 0.985412 0.984883 0.002312 0.002255 0.002309 RBFT Sum.quar. Gauss Identity 3RBF 1-25-1 0.986071 0.986443 0.985769 0.002126 0.002109 0.002179 RBFT Sum.quar. Gauss Identity 4RBF 1-26-1 0.985491 0.985337 0.984503 0.002213 0.002262 0.002367 RBFT Sum.quar. Gauss Identity 5RBF 1-30-1 0.984297 0.983784 0.984732 0.002394 0.002499 0.002339 RBFT Sum.quar. Gauss Identity Source: Own research; according to Machova and Marecek (2019). All were radial basis function NNs with only one variable in the input layer (i.e., time). The NNs contained from 25 to 30 neurons in the hidden layer. There was a solo neuron and a solo output variable (i.e., the CZK/RMB exchange rate) in the output layer. The RBFT (redundant byzantine fault tolerance) training algorithm was applied to all the networks. The hidden layer of neurons of all the neural networks was activated by the same function (i.e., the Gaussian curve). Likewise, the external layers of neurons used the same function for the purpose of activation (see Table 2). The search was for a network that performed equally well across all the datasets (note: the data distribution across the datasets took place randomly), while the error should be the smallest possible. The performance of the individual datasets was represented by a correlation coefficient. The values for the individual datasets for the retained NNs are presented in Table 2. 137 Risks 2021,9,1 The figures revealed that the performance of all the retained neural networks reached approximately the same results. The unimportant differences had no impact on the performance of the respective networks. The values of the correlation coefficients for all the training datasets was below 0.983. The values of the correlation coefficients for the testing datasets were very similar to the training datasets (i.e., always above 0.983) and was above 0.984 for the validation datasets. Note that the error for all the datasets was slightly above 0.002. The error differences for the equalized time series were almost insignificant for the datasets. A more detailed analysis is required to determine the most appropriate neural network. Table 3provides an overview of the basic statistical characteristics of the individual datasets for the five retained neural networks. Under ideal circumstances, the statistical characteristics of the neural networks should comply, in an interspace manner, in all the sets of a certain neural structure (i.e., minima, maxima, residuals, etc.). In the case of the retained neural networks, the differences between the equalized time series were minimal, both in terms of absolute values and residuals. It is therefore not clear which of the retained NNs generated the most suitable results. Therefore, all the neural networks seem to be applicable in practice. Table 3. Statistical characteristics of the individual datasets according to the retained neural network. Statistics 1.RBF1-30-1 2.RBF1-26-1 3.RBF1-25-1 4.RBF1-26-1 5.RBF1-30-1 Minimal prediction (training) 2.58183 2.55340 2.52919 2.52734 2.62556 Maximal prediction (training) 4.04950 4.09743 4.00225 4.00540 3.95151 Minimal prediction (testing) 2.58355 2.55342 2.52917 2.52741 2.62557 Maximal prediction (testing) 4.04944 4.09741 4.00223 4.00544 3.95152 Minimal prediction (validation) 2.58184 2.55531 2.53062 2.52749 2.62600 Maximal prediction (validation) 4.04951 4.09682 4.00226 4.00505 3.95129 Minimal residuals (training) −0.22414 −0.21614 −0.30694 −0.24314 −0.28141 Maximal residuals (training) 0.37317 0.23107 0.22900 0.21521 0.29266 Minimal residuals (testing) −0.21388 −0.18746 −0.28546 −0.22842 −0.26051 Maximal residuals (testing) 0.37307 0.23378 0.22341 0.20519 0.29323 Minimal residuals (validation) −0.21094 −0.17494 −0.17479 −0.20650 −0.23232 Maximal residuals (validation) 0.26023 0.22773 0.18504 0.21784 0.21065 Minimal standard residuals (training) −4.46833 −4.49505 −6.65757 −5.16815 −5.75120 Maximal standard residuals (training) 7.43936 4.80567 4.96689 4.57450 5.98108 Minimal standard residuals (testing) −4.18178 −3.94719 −6.21611 −4.80292 −5.21090 Maximal standard residuals (testing) 7.29438 4.92251 4.86501 4.31438 5.86540 Minimal standard residuals (validation) −4.38041 −3.64037 −3.74445 −4.24472 −4.80316 Maximal standard residuals (validation) 5.40392 4.73891 3.96396 4.47779 4.35511 Source: Machova and Marecek (2019). Figure 3is a line graph that shows the actual development of the CZK/RMB exchange rate at the individual intervals in a slightly different manner. The x-axis (case number) shows information about the input data (i.e., about the time series (marked by numbers due to the software settings)), whilst the y-axis shows the value of the CZK/RMB exchange rate. The blue line indicates the actual development of the exchange rate, and the other colors show the predictions according to the individually generated and retained networks (as presented in Table 2). The close similarity of the predictions of the individual networks is not important, but rather the extent of compliance to the actual development of the exchange rate. Within this context, it can be concluded that all the undistributed neural networks are seemingly very interesting. On the face of it, the basic directions of the lines, which assess the course of the CZK/RMB exchange rate, display the extremes in the development of the actual exchange rate. 138 Risks 2021,9,1 VWHSXVHGDVLQSXWVWHSSUHGLFWHGDKHDG 6DPSOHV7UDLQLQJ7HVWLQJ9DOLGDWLRQ &KLQHVH<XDQWR&]HFK.RUXQD >5%)@ >5%)@ >5%)@ >5%)@ >5%)@                       &DVHQXPEHU             &KLQHVH<XDQWR&]HFK.RUXQD2XWSXWYDULDEOH Figure 3. Actual and predicted (according to retained neural networks) development of CZK/RMB exchange rate during the monitored period (Source: Own research; according to Machova and Marecek 2019). Given that the network structure (as depicted in Figure 1) contains 3303 items of data on the CZK/RMB exchange rate, this may seem unclear. It is therefore appropriate to present the situation for a selected data interval. Therefore, the line graph in Figure 4compares the actual development of the CZK/RMB exchange rate for the final 100 days of the monitored period (i.e., from 14 July to 21 October 2018.) ϯϬϱϬϬϬϬ ϯϭϬϬϬϬϬ ϯϭϱϬϬϬϬ ϯϮϬϬϬϬϬ ϯϮϱϬϬϬϬ ϯϯϬϬϬϬϬ ϯϯϱϬϬϬϬ ZD<ĞdžĐŚĂŶŐĞƌĂƚĞ ϭZ&ϭͲϯϬͲϭ ϮZ&ϭͲϮϲͲϭ ϯZ&ϭͲϮϱͲϭ ϰZ&ϭͲϮϲͲϭ ϱZ&ϭͲϯϬͲϭ Figure 4. Actual and predicted (according to retained neural networks) development of the CZK/RMB exchange rate for the period from 14 July to 21 October 2018 (Source: Own research; according to Machova and Marecek 2019). The graph shows that none of the retained neural networks were completely and accurately able to trace the actual course of the CZK/RMB exchange rate during the monitored period. However, it was clear that the 3.RBF 1-25-1 and 5.RBF 1-30-1 networks came the closest to reality. Their predicted values were almost identical to the actual exchange rate at the beginning of the monitored period, with more significant differences showing at the end of the monitored period. The difference in both cases was about CZK 0.08 to one RMB. Even the least accurate network, namely 2.RBF 1-26-1, differed from the actual figures for the exchange rate by less than CZK 0.011. An examination of the residuals therefore seems appropriate. The development of the residuals during the period from 14 July to 21 October 2018 is presented in Figure 5. 139 Risks 2021,9,1 ͲϬϮϬϬϬϬϬ ͲϬϭϱϬϬϬϬ ͲϬϭϬϬϬϬϬ ͲϬϬϱϬϬϬϬ ϬϬϬϬϬϬϬ ϬϬϱϬϬϬϬ ϬϭϬϬϬϬϬ ϭZ&ϭͲϯϬͲϭ ϮZ&ϭͲϮϲͲϭ ϯZ&ϭͲϮϱͲϭ ϰZ&ϭͲϮϲͲϭ ϱZ&ϭͲϯϬͲϭ Figure 5. Development of residuals for the equalized time series during the period from 14 July to 21 October (Source: Own research; according to Machova and Marecek 2019). The graph shows that, with exception of the 5.RBF 1-30-1 network, the aggregate of the residuals for all the neural networks during the monitored period was almost zero. The residuals achieved quite high positive values in this period. To illustrate this, Table 4shows the aggregate of the residuals for the equalized time series. Table 4. Aggregate of the residuals for the individual equalized time series. Characteristics 1.RBF 1-24-1 2.RBF 1-29-1 3.RBF 1-30-1 4.RBF 1-28-1 5.RBF 1-26-1 Aggregate of residuals 0.150758 −1.025922566 −3.350398611 −1.785245346 −3.244516106 Source: Own research; according to Machova and Marecek (2019). Under ideal circumstances, if we ignore the residual fluctuations for the individual cases during the monitored period, the absolute value of the aggregates of the residuals will total zero. The absolute value of the aggregate of the residuals of the second neural network (2.RBF 1-29-1), which was nearly − 1.026, was the closest to zero. In contrast, the 3.RBF 1-30-1 and 5.RBF 1-26-1 networks produced the highest aggregate of residuals in absolute terms, with values above 3. However, it is necessary to point out that this value is minimal in relation to the 3303 measurements. It is therefore possible to state that the most accomplished neural networks were 3.RBF 1-25-1 and 5.RBF 1-30-1. 4.2. Neural Structure B A total of 100,000 NNs were generated on the basis of the defined procedure. The five that displayed the best parameters were retained and are presented in Table 5. 140 Risks 2021,9,1 Table 5. Retained neural networks. Network Training Perform. Testing Perform. Validation Perform. Training Error Testing Error Validation Error Training Algorithm Error Function Activ. of Hidden Layer Output Activ. Function 1MLP 61-11-1 0.998718 0.996990 0.997563 0.000197 0.000468 0.000374 BFGS (Quasi-Newton) 392 Sum quart. Tanh Identity 2MLP 61-11-1 0.998927 0.997313 0.997517 0.000165 0.000417 0.000382 BFGS (Quasi-Newton) 461 Sum quart. Logistic Identity 3MLP 61-11-1 0.998919 0.997606 0.997632 0.000166 0.000377 0.000364 BFGS (Quasi-Newton) 569 Sum quart. Tanh Identity 4MLP 61-11-1 0.998791 0.997572 0.997594 0.000186 0.000377 0.000372 BFGS (Quasi-Newton) 558 Sum quart. Tanh Exponential 5MLP 61-10-1 0.998640 0.997059 0.997641 0.000209 0.000457 0.000363 BFGS (Quasi-Newton) 436 Sum quart. Tanh Tanh Source: Own research. All were multi-layer perceptron neural networks. There were four variables (i.e., time, year, day of month, day of week, in the input layer). Time was represented by one neuron in the input layer, a year by 10 neurons, a month by 12 neurons, a weekday by 7 neurons, and a day of the month by 31 neurons, respectively. The total (i.e., 61 neurons) formed the input layer of the generated and retained neural networks. The neural networks contained either 10 or 11 neurons in the hidden layer. Consequently, there was a single neuron and a single output variable, which was the CZK/RMB exchange rate, in the output layer. The Quasi-Newton training algorithm was applied to all the networks. All the neural networks used either the hyperbolic tangent or logistic functions for the purpose of the activation of the neural hidden layer. For the activation of the neural output layer, the retained neural networks used the hyperbolic tangent, exponential, and identity functions (see Table 5). The search was for a network that performed equally well across all the datasets (note: the data distribution across the datasets took place randomly), while the error should be the smallest possible. The performance of the individual sets was represented by a correlation coefficient. The values for the individual datasets for the retained NNs are presented in Table 5. The table shows that the performance of all the retained neural networks was approximately the same. The insignificant differences bear no influence on the performance of the individual networks. The values of the correlation coefficients for all the training datasets significantly exceeded 0.998. The values of the correlation coefficients for the testing datasets exceeded 0.997, and for the validation datasets, they significantly exceeded 0.997. Note that the error for all the datasets fell within the interval >0.0001 to <0.0005. The error differences for the equalized time series were completely insignificant for the individual datasets. A more detailed analysis is required to determine the most appropriate neural network. Table 6 provides an overview of the basic statistical characteristics of the individual datasets for the five retained neural networks. 141