# Conditional Tail Expectation and Premium Calculation under Asymmetric Loss

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## 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$${\widehat{P}}_{X,\alpha}=\frac{1}{2}\left(\right)open="["\; close="]">{F}_{X}^{-1}\left(\right)open="("\; close=")">\frac{1-\alpha}{2(1-\gamma )}$$
- (vi)
- The result is easily verified since$${\widehat{P}}_{c,\alpha}=\frac{1}{2}\left(\right)open="["\; close="]">{F}_{c}^{-1}\left(\right)open="("\; close=")">\frac{1-\alpha}{2(1-\gamma )}$$

## 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

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**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 |

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**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