Transformation of Various Measures of Financial Risks with their Limitation on Outcomes Associated with Losses

In assessing the risk of investing in various financial assets, risk management focuses on the analysis of the worst possible losses (the right tail of the loss distribution). At the same time, most often, when speaking about losses, it is assumed that losses can, in principle, take on negative values (which corresponds to receiving positive profits). However, there are many theoretical studies suggesting that losses take only positive values. Many risk managers use only a portion of the sample of data that corresponds to positive losses when assessing the relevant risk measures using the statistical method or the Monte Carlo method. The purpose of this paper is to study the transformation of risk estimates of various levels of catastrophicity with such a change in the space of elementary events, and hence the law of loss distribution. The paper uses methods of analysis of financial risks of various levels of catastrophicity, including methods developed in the author’s previous papers. As a result of the study, it turned out that with such a transformation of the random value of losses, all the most important estimates are significantly transformed with the help of risk measures of various catastrophicity. The author concludes that the theoretical conclusions of the work will also contribute to a more conscious understanding of the theoretical results and the results of practical risk assessments, depending on the basis on which this assessment was made: allowing losses to accept negative values or focusing only on their positive values.

1. Borovkov A. A. Probability theory. Moscow: Nauka; 1986. 431 p. (In Russ.).

2. Santolino M., Belles-Sampera J., Sarabia J.M., Guillen M. An examination of the tail contribution to distortion risk measures. Journal of Risk. 2021;23(6):88–113. DOI: 10.21314/JOR.2021.014

3. Crouhy M., Galai D., Mark R. The essentials of risk management. New York, NY: McGraw-Hill Book Co.; 2005. 416 p. (Russ. ed.: Crouhy M., Galai D., Mark R. Osnovy risk-menedzhmenta. Moscow: Urait; 2018. 390 p.).

4. Hull J. C. Risk management and financial institutions. New York, NY: Pearson Education International; 2007. 576 p.

5. Jorion P. Value at risk: The new benchmark for managing financial risk. New York, NY: McGraw-Hill Education; 2007. 624 p.

6. Minasyan V.B. New ways to measure catastrophic financial risks: “VaR to the power of t” measures and how to calculate them. Finance: Theory and Practice. 2020;24(3):92–109. DOI: 10.26794/2587–5671–2020–24–3–92–109

7. Minasyan V. B. New risk measures “VaR to the power of t” and “ES to the power of t” and distortion risk measures. Finance: Theory and Practice. 2020;24(6):92–107. DOI: 10.26794/2587–5671–2020–24–6–92–107

8. Denuit M., Dhaene J., Goovaerts M., Kaas R. Actuarial theory for dependent risks: Measures, orders and models. Chichester: John Wiley & Sons, Ltd; 2005. 480 p. DOI: 10.1002/0470016450

9. Zhu L., Li H. Tail distortion risk and its asymptotic analysis. Insurance: Mathematics and Economics. 2012;51(1):115–121. DOI: 10.1016/j.insmatheco.2012.03.010

10. Yang F. First- and second-order asymptotics for the tail distortion risk measure of extreme risks. Communications in Statistics — Theory and Methods. 2015;44(3):520–532. DOI: 10.1080/03610926.2012.751116

11. Yin C., Zhu D. New class of distortion risk measures and their tail asymptotics with emphasis on Va R. Journal of Financial Risk Management. 2018;7(1):12–38. DOI: 10.4236/jfrm.2018.71002

12. Belles-Sampera J., Guillén M., Santolino M. Beyond value-at-risk: GlueVaR distortion risk measures. Risk Analysis. 2014;34(1):121–134. DOI: 10.1111/risa.12080

13. Belles-Sampera J., Guillén M., Santolino M. GlueVaR risk measures in capital allocation applications. Insurance: Mathematics and Economics. 2014;58:132–137. DOI: 10.1016/j.insmatheco.2014.06.014

14. Minasyan V. B. Risk assessment models of the companies implementing R&D projects. Finance: Theory and Practice. 2019;23(1):133–146. DOI: 10.26794/2587–5671–2019–23–1–133–146

15. Minasyan V. B. New risk measure VaR in the square and its calculation. Case of general law loss distributions, comparison with other risk measures. Upravlenie finansovymi riskami = Financial Risk Management Journal. 2019;(4):298–320. (In Russ.).

16. Minimum capital requirements for market risk. Basel Committee on Banking Supervision. Basel: Bank for International Settlements; 2019. 136 p. URL: https://www.bis.org/bcbs/publ/d457.pdf

17. Minasyan V.B. On comparison of certain measures of catastrophic risks. Upravlenie finansovymi riskami = Financial Risk Management Journal. 2022;(4):284–289. (In Russ.). DOI: 10.36627/2221–7541–2022–4–4–284–289