# Conditional Tail Expectation and Premium Calculation under Asymmetric Loss

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Premium Calculation Minimizing the CTE under Asymmetric Loss

**Definition**

**1.**

**Definition**

**2.**

**Proposition**

**1.**

**Proof.**

**Remark**

**1.**

**Remark**

**2.**

**Proposition**

**2.**

- (i)
- Translativity: any increase in the liability by a deterministic amount c should result in the same increase in the capital. If the risk increases by a fixed amount c, then the premium also increases by that amount, i.e., ${\widehat{P}}_{X+c,\alpha}={\widehat{P}}_{X,\alpha}+c$ for all random variables and each constant c.
- (ii)
- Monotonicity: if $X{\le}_{st}Y$ then ${\widehat{P}}_{X,\alpha}\le {\widehat{P}}_{Y,\alpha}$, assuming that ${F}_{X}^{-1}(\xb7)$ and ${F}_{Y}^{-1}(\xb7)$ exist, where ${\le}_{st}$ is the usual stochastic order.
- (iii)
- Subadditivity: this reflects the idea that risk can be reduced by diversification, i.e., ${\widehat{P}}_{X+Y,\alpha}\le {\widehat{P}}_{X,\alpha}+{\widehat{P}}_{Y,\alpha}$.
- (iv)
- Positive homogeneity or scale invariance: independence with respect to the monetary units used, i.e., ${\widehat{P}}_{cX,\alpha}=c{\widehat{P}}_{X,\alpha}$ for all random variables and any constant c. As ${F}_{cX}\left(x\right)={F}_{X}(x/c)$, then using Proposition 1, $({\widehat{P}}_{cX,\alpha},{\delta}_{\alpha})$ is the optimal solution of the equations$$\begin{array}{ccc}\hfill {F}_{cX}(P-\delta )& =& \frac{1-\alpha}{2(1-\gamma )},\hfill \\ \hfill {F}_{cX}(P+\delta )& =& 1-\frac{1-\alpha}{2\gamma}.\hfill \end{array}$$
- (v)
- No-rip off: if $X\le c$, then ${\widehat{P}}_{\alpha}\le c$. ${\mathcal{P}}_{{X}_{\Theta ,\alpha}}\le min\left\{x\right|{\overline{F}}_{\Theta ,\alpha}\left(x\right)=0\}$, for all random variables. It is useless to keep more capital than the maximal loss value. If the random variable is unbounded then the premium is infinite.
- (vi)
- Constancy (or no unjustified loading): if $X\equiv c$, then ${\widehat{P}}_{X,\alpha}=c$. To deal with a loss of c, the insurer only needs to have a capital of the same amount at its disposal.

**Proof.**

- (i)
- As ${F}_{X+c}\left(x\right)={F}_{X}(x-c)$, then using Proposition 1, $({\widehat{P}}_{X+c,\alpha},{\delta}_{\alpha})$ is the optimal solution of the equations$$\begin{array}{ccc}\hfill {F}_{X+c}(P-\delta )& =& \frac{1-\alpha}{2(1-\gamma )},\hfill \\ \hfill {F}_{X+c}(P+\delta )& =& 1-\frac{1-\alpha}{2\gamma}.\hfill \end{array}$$
- (ii)
- If $X{\le}_{st}Y$, then ${\overline{F}}_{X}\left(x\right)\le {\overline{F}}_{Y}\left(x\right)$ for all $x\in {\mathbb{R}}^{+}$ and therefore ${F}_{X}^{-1}\left(u\right)\le {F}_{Y}^{-1}\left(u\right)$ for all $u\in (0,1)$. Thus ${\widehat{P}}_{X,\alpha}\le {\widehat{P}}_{Y,\alpha}$.
- (iii)
- It is direct.
- (iv)
- This property is satisfied since the CTE is a coherent risk measure. To be named coherent, a risk measure must be positive homogeneous, translative, and subadditive (see [20]).
- (v)
- The result follows since$${\widehat{P}}_{X,\alpha}=\frac{1}{2}\left[{F}_{X}^{-1}\left(\frac{1-\alpha}{2(1-\gamma )}\right)+{F}_{X}^{-1}\left(1-\frac{1-\alpha}{2\gamma}\right)\right]\le \frac{2c}{2}=c.$$
- (vi)
- The result is easily verified since$${\widehat{P}}_{c,\alpha}=\frac{1}{2}\left[{F}_{c}^{-1}\left(\frac{1-\alpha}{2(1-\gamma )}\right)+{F}_{c}^{-1}\left(1-\frac{1-\alpha}{2\gamma}\right)\right]=c.$$

## 3. Analytical Expressions of the Premium for Composite Models

## 4. A Specific Model

## 5. Numerical Application

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Artzner, P.; Delbaen, F.; Eber, J.M.; Heath, D. Coherent measures of risk. Math. Financ.
**1999**, 9, 203–228. [Google Scholar] [CrossRef] - Klugman, S.A.; Panjer, H.H.; Willmot, G.E. Loss Models. From Data to Decisions, 3rd ed.; John Wiley: Hoboken, NJ, USA, 2008. [Google Scholar]
- Furman, E.; Zitikis, R. Weighted premium calculation principles. Insur. Math. Econ.
**2008**, 42, 459–465. [Google Scholar] [CrossRef] - Furman, E.; Zitikis, R. Weighted risk capital allocations. Insur. Math. Econ.
**2008**, 42, 263–269. [Google Scholar] [CrossRef] - Heilmann, W.R. Decision theoretic foundations of credibility theory. Insur. Math. Econ.
**1989**, 8, 77–95. [Google Scholar] [CrossRef] - Gómez-Déniz, E. A generalization of the credibility theory obtained by using the weighted balanced loss function. Insur. Math. Econ.
**2008**, 42, 850–854. [Google Scholar] [CrossRef] - Gómez-Déniz, E.; Calderín-Ojeda, E. Credibility premiums for natural exponential family and general 0–1 loss function. Chil. J. Stat.
**2015**, 6, 3–17. [Google Scholar] - Zellner, A. Bayesian and non-Bayesian estimation using balanced loss function. In Statistical Decision Theory and Related Topics; Gupta, S.S., Berger, J.O., Eds.; Springer: New York, NY, USA, 1994; pp. 371–390. [Google Scholar]
- Farsipour, N.S.; Asgharzadhe, A. Estimation of a normal mean relative to balanced loss functions. Stat. Pap.
**2004**, 45, 279–286. [Google Scholar] [CrossRef] - Jafari, M.; Marchand, E.; Parsian, A. On estimation with weighted balanced-type loss function. Stat. Probab. Lett.
**2006**, 76, 773–780. [Google Scholar] [CrossRef] - Hosomatsu, Y. Concepts, theory, and techniques. Asymmetric loss function and optimal policy under uncertainty: A simple proof. Manag. Sci.
**1980**, 26, 577–582. [Google Scholar] - Rockafellar, R.; Uryasev, S. Conditional value at risk for general loss distributions. J. Bank. Financ.
**2002**, 26, 1443–1471. [Google Scholar] [CrossRef] - Rockafellar, R.; Uryasev, S.; Zabaranking, M. Generalized deviations in risk analysis. Financ. Stoch.
**2006**, 10, 51–74. [Google Scholar] [CrossRef] - Wang, S. Premium calculation by transforming the premium layer density. ASTIN Bull.
**1996**, 26, 71–92. [Google Scholar] [CrossRef] - Heras, A.; Balbás, B.; Vilar, J.L. Conditional tail expectation and premium calculation. ASTIN Bull.
**2012**, 42, 325–342. [Google Scholar] - Grün, B.; Miljkovic, T. Extending composite loss models using a general framework of advanced computational tools. Scand. Actuar. J.
**2019**, 8, 642–660. [Google Scholar] [CrossRef] - Pigeon, M.; Denuit, M. Composite Lognormal–Pareto model with random threshold. Scand. Actuar. J.
**2011**, 3, 177–192. [Google Scholar] [CrossRef] - Scollnik, D.P.M. On composite Lognormal-Pareto models. Scand. Actuar. J.
**2007**, 1, 20–33. [Google Scholar] [CrossRef] - Sarabia, J.M.; Calderín-Ojeda, E. Analytical expressions of risk quantities for composite models. J. Risk Model Valid.
**2018**, 12, 75–101. [Google Scholar] [CrossRef] - Kaas, R.; Goovaertes, M.; Dhaene, J.; Denuit, M. Modern Actuarial Risk Theory; Springer; Heidelberg Gmbh & Co. Kg.: Berlin/Heidelberg, Germany, 2008. [Google Scholar]
- Cooray, K.; Ananda, M.M.A. Modeling actuarial data with a composite Lognormal–Pareto model. Scand. Actuar. J.
**2005**, 5, 321–334. [Google Scholar] [CrossRef] - Scollnik, D.P.M.; Sun, C. Modeling with Weibull-Pareto models. N. Am. Actuar. J.
**2012**, 16, 260–272. [Google Scholar] [CrossRef] - Calderín-Ojeda, E. On the Composite Weibull–Burr Model to describe claim data. Commun. Stat. Case Stud. Data Anal. Appl.
**2015**, 1, 59–69. [Google Scholar] [CrossRef] - Calderín-Ojeda, E.; Kwok, C.F.K. Modeling claims data with composite Stoppa models. Scand. Actuar. J.
**2016**, 9, 817–836. [Google Scholar] [CrossRef] - Acerbi, C.; Tasche, D. On the coherence of expected shortfall. J. Bank. Financ.
**2002**, 26, 1487–1503. [Google Scholar] [CrossRef]

**Figure 1.**Plot of the pdf given in (11) for special values of its parameters.

**Left panel**, $(\theta ,\lambda ,\mu ,\sigma )=(2,2,1,1)$ and

**right panel**$(\theta ,\lambda ,\mu ,\sigma )=(3,3,0,0.5)$.

**Figure 2.**QQ plots of the Lomax (

**top left**), lognormal (

**top right**)and composite lognormal-Lomax (CLL) distribution for the Danish fire claims dataset.

**Table 1.**Optimal values of ${\widehat{P}}_{\alpha}$, ${\widehat{\delta}}_{\alpha}$ and ${\mathrm{CTE}}_{\alpha}\left({\widehat{P}}_{\alpha}\right)$ for different risk levels $\alpha $ and $\gamma $. Empirical VaR and TVaR and VaR and TVaR of X are also given for comparison purposes.

Risk Level $\mathit{\alpha}$ | ||||||
---|---|---|---|---|---|---|

Risk | Measure | 0.90 | 0.925 | 0.95 | 0.975 | 0.99 |

Empirical | VaR | 5.080 | 5.989 | 8.454 | 14.395 | 24.970 |

TVaR | 14.271 | 17.172 | 22.222 | 33.450 | 55.587 | |

Model | ${\pi}_{X}^{\alpha}$ | 5.165 | 6.281 | 8.248 | 13.052 | 23.741 |

${\mathrm{CTE}}_{\alpha}\left(X\right)$ | 14.971 | 18.068 | 23.524 | 36.848 | 66.492 | |

$\gamma =0.6$ | ${\widehat{P}}_{\alpha}$ | 3.333 | 3.951 | 5.043 | 7.723 | 13.707 |

${\widehat{\delta}}_{\alpha}$ | 2.515 | 3.152 | 4.270 | 6.988 | 13.015 | |

${\mathrm{CTE}}_{\alpha}\left({\widehat{P}}_{\alpha}\right)$ | 3.478 | 4.290 | 5.718 | 9.198 | 16.927 | |

$\gamma =0.8$ | ${\widehat{P}}_{\alpha}$ | 3.938 | 4.684 | 6.001 | 9.229 | 16.428 |

${\widehat{\delta}}_{\alpha}$ | 3.165 | 3.928 | 5.267 | 8.527 | 15.765 | |

${\mathrm{CTE}}_{\alpha}\left({\widehat{P}}_{\alpha}\right)$ | 7.210 | 8.759 | 11.485 | 18.136 | 32.922 | |

$\gamma =0.9$ | ${\widehat{P}}_{\alpha}$ | 4.211 | 5.017 | 6.440 | 9.923 | 17.688 |

${\widehat{\delta}}_{\alpha}$ | 3.476 | 4.297 | 5.738 | 9.251 | 17.051 | |

${\mathrm{CTE}}_{\alpha}\left({\widehat{P}}_{\alpha}\right)$ | 9.150 | 11.092 | 14.510 | 22.851 | 41.398 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Calderín-Ojeda, E.; Gómez-Déniz, E.; Vázquez-Polo, F.J.
Conditional Tail Expectation and Premium Calculation under Asymmetric Loss. *Axioms* **2023**, *12*, 496.
https://doi.org/10.3390/axioms12050496

**AMA Style**

Calderín-Ojeda E, Gómez-Déniz E, Vázquez-Polo FJ.
Conditional Tail Expectation and Premium Calculation under Asymmetric Loss. *Axioms*. 2023; 12(5):496.
https://doi.org/10.3390/axioms12050496

**Chicago/Turabian Style**

Calderín-Ojeda, Enrique, Emilio Gómez-Déniz, and Francisco J. Vázquez-Polo.
2023. "Conditional Tail Expectation and Premium Calculation under Asymmetric Loss" *Axioms* 12, no. 5: 496.
https://doi.org/10.3390/axioms12050496