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Properties of the entropic risk measure EVaR in relation to selected distributions

Yuliya Mishura, Kostiantyn Ralchenko, Petro Zelenko, Volodymyr Zubchenko

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.

Properties of the entropic risk measure EVaR in relation to selected distributions

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.
Paper Structure (13 sections, 10 theorems, 83 equations, 7 figures)

This paper contains 13 sections, 10 theorems, 83 equations, 7 figures.

Table of Contents

  1. Introduction
  2. General properties of EVaR
  3. Calculation of EVaR for selected distributions
  4. Lambert $W$ function
  5. Poisson distribution
  6. Compound Poisson distribution
  7. Gamma distribution
  8. Laplace distribution
  9. Inverse Gaussian distribution
  10. Normal inverse Gaussian distribution
  11. Concluding Remarks
  12. EVaR for normal distribution
  13. List of related probability distributions with density functions

Key Result

Theorem 3.1

Let $X \sim \mathop{\mathrm{Pois}}\nolimits(\lambda)$, $\lambda>0$, then for all $\alpha \in [0,1)$ where $W_0$ stands for the principal branch of the Lambert function. The expression for $\mathop{\mathrm{EVaR}}\nolimits_\alpha(X)$ is continuous in $\alpha$, and for any fixed $\alpha$ is continuous in $\beta\in \mathbb R$.

Figures (7)

  • Figure 1: The two real branches of the Lambert $W$ function: $W_0(x)$ (solid line) and $W_{-1}(x)$ (dashed line).
  • Figure 2: EVaR for Poisson distribution with constant intensity $\lambda$.
  • Figure 3: EVaR for compound Poisson distribution with normal distribution of jumps ($\sigma=1$).
  • Figure 4: EVaR for Gamma distribution with parameter $\theta=1$.
  • Figure 5: EVaR for Laplace distribution.
  • ...and 2 more figures

Theorems & Definitions (24)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • ...and 14 more