Properties of the entropic risk measure EVaR in relation to selected distributions
Volume 11, Issue 4 (2024), pp. 373–394
Pub. online: 30 April 2024
Type: Research Article
Open Access
Received
3 March 2024
3 March 2024
Revised
15 April 2024
15 April 2024
Accepted
15 April 2024
15 April 2024
Published
30 April 2024
30 April 2024
Abstract
Entropic Value-at-Risk (EVaR) measure is a convenient coherent risk measure. Due to certain difficulties in finding its analytical representation, it was previously calculated explicitly only for the normal distribution. We succeeded to overcome these difficulties and to calculate Entropic Value-at-Risk (EVaR) measure for Poisson, compound Poisson, Gamma, Laplace, exponential, chi-squared, inverse Gaussian distribution and normal inverse Gaussian distribution with the help of Lambert function that is a special function, generally speaking, with two branches.
References
Ahmadi-Javid, A.: Entropic value-at-risk: A new coherent risk measure. J. Optim. Theory Appl. 155, 1105–1123 (2012). MR3000633. https://doi.org/10.1007/s10957-011-9968-2
Ahmadi-Javid, A., Fallah-Tafti, M.: Portfolio optimization with entropic value-at-risk. Eur. J. Oper. Res. 279(1), 225–241 (2019). MR3968161. https://doi.org/10.1016/j.ejor.2019.02.007
Ahmadi-Javid, A., Pichler, A.: An analytical study of norms and Banach spaces induced by the entropic value-at-risk. Math. Financ. Econ. 11, 1–24 (2017). MR3709386. https://doi.org/10.1007/s11579-017-0197-9
Artzner, P., Delbaen, F., Eber, J.-M., Heath, D.: Coherent measures of risk. Math. Finance 9(3), 203–228 (1999). MR1850791. https://doi.org/10.1111/1467-9965.00068
Axelrod, A., Chowdhary, G.: A dynamic risk form of entropic value at risk. Conference: AIAA Scitech 2019 Forum (2019). https://doi.org/10.2514/6.2019-0392
Barndorff-Nielsen, O.E.: Normal inverse Gaussian distributions and stochastic volatility modelling. Scand. J. Stat. 24(1), 1–13 (1997). MR1436619. https://doi.org/10.1111/1467-9469.00045
Chennaf, S., Ben Amor, J.: Entropic value at risk to find the optimal uncertain random portfolio. Soft Comput. 27, 1–13 (2023). https://doi.org/10.1007/s00500-023-08547-5
Corless, R., Gonnet, G., Hare, D., Jeffrey, D., Knuth, D.: On the Lambert W function. Adv. Comput. Math. 5, 329–359 (1996). MR1414285. https://doi.org/10.1007/BF02124750
Delbaen, F.: Remark on the paper “Entropic value-at-risk: A new coherent risk measure” by Amir Ahmadi-Javid, J. Optim. Theory Appl., 155(3) (2001) 1105–1123”. In: Barrieu, P. (ed.) Risk and Stochastics, pp. 151–158. World Scientific (2019). MR4404374. https://doi.org/10.1142/9781786341952_0009
Föllmer, H., Knispel, T.: Entropic risk measures: coherence vs. convexity, model ambiguity, and robust large deviations. Stoch. Dyn. 11(2–3), 333–351 (2011). MR2836530. https://doi.org/10.1142/S0219493711003334
Föllmer, H., Schied, A.: Convex measures of risk and trading constraints. Finance Stoch. 6(4), 429–447 (2002). MR1932379. https://doi.org/10.1007/s007800200072
Föllmer, H., Schied, A.: Stochastic Finance. An Introduction in Discrete Time. De Gruyter, Berlin (2016). MR3859905. https://doi.org/10.1515/9783110463453
Jeffrey, D.J., Hare, D.E.G., Corless, R.M.: Unwinding the branches of the Lambert W function. Math. Sci. 21(1), 1–7 (1996). MR1390696
Luxenberg, E., Boyd, S.: Portfolio construction with Gaussian mixture returns and exponential utility via convex optimization. Optim. Eng. (2023). MR4707664. https://doi.org/10.1007/s11081-023-09814-y
Pichler, A., Schlotter, R.: Entropy based risk measure. Eur. J. Oper. Res. 285(1), 223–236 (2020). MR4083060. https://doi.org/10.1016/j.ejor.2019.01.016
Pichler, A.: A quantitative comparison of risk measures. Annals of Operations Research, Springer 254(1), 251–275 (2017). MR3665746. https://doi.org/10.1007/s10479-017-2397-3