# Optimal Premium as a Function of the Deductible: Customer Analysis and Portfolio Characteristics

## Abstract

**:**

## 1. Introduction

## 2. Customer’s Problem

**Corollary**

**1.**

**Remark**

**1.**

**Corollary**

**2.**

**Remark**

**2.**

**Remark**

**3.**

## 3. Portfolio Characteristics

#### 3.1. Stochastic Claim Frequencies

#### Exponentially-Distributed Claim Frequencies

**Remark**

**4.**

#### 3.2. Stochastic Risk Aversions

#### Gamma Distributed Risk Aversions

## 4. Ruin Probability

- (i)
- If $\mu (\tilde{p};K)>0$, then $\phi \left({r}_{0}\right)<1$ for all ${r}_{0}>0$, $K>0$ and p in some bounded open interval $I\subset (0,\infty )$ containing $\tilde{p}$.
- (ii)
- If $\mu (\tilde{p};K)\le 0$, then $\phi \left({r}_{0}\right)=1$ for all ${r}_{0}>0$ and $p>0$.

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## 5. Examples and Illustrations

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

## Acknowledgments

## Conflicts of Interest

## Appendix A. Calculations of the Present Value in Section 2

## Appendix B. Calculations of the Variance in Section 2

## Appendix C. Illustration of the Complexity of the Exponential Premium in Section 2

## References

- H. Schmidli. Stochastic Control in Insurance. Berlin, Germany: Springer Science & Business Media, 2007. [Google Scholar]
- D.L. Iglehart. “Diffusion approximations in collective risk theory.” J. Appl. Prob. 6 (1969): 285–292. [Google Scholar] [CrossRef]
- S. Asmussen, and H. Albrecher. Ruin Probabilities. Singapore: World Scientific, 2010, Volume 14. [Google Scholar]
- R. Rees, and A. Wambach. “The Microeconomics of Insurance.” Found. Trends Microecon. 4 (2008): 15–163. [Google Scholar] [CrossRef]
- S. Asmussen, B.J. Christensen, and M. Taksar. “Portfolio size as function of the premium: Modelling and optimization.” Stochastics 85 (2013): 575–588. [Google Scholar] [CrossRef]
- K. Burnecki, J. Nowicka-Zagrajek, and A. Weron. Pure Risk Premiums under Deductibles. A Quantitative Management in Actuarial Practice. Research Report; Wroclaw, Poland: Hugo Steinhaus Center, Wroclaw University of Technology, 2004. [Google Scholar]
- B. Højgaard. “Optimal dynamic premium control in non-life insurance. Maximizing dividend pay-outs.” Scand. Actuar. J. 2002 (2002): 225–245. [Google Scholar] [CrossRef]
- S. Asmussen. “Modeling and performance of bonus-malus systems: Stationarity versus age-correction.” Risks 2 (2014): 49–73. [Google Scholar] [CrossRef] [Green Version]
- M. Van Lieshout. Markov Point Processes and Their Applications. Singapore: World Scientific, 2000. [Google Scholar]
- H.U. Gerber, and G. Pafum. “Utility functions: From risk theory to finance.” N. Am. Actuar. J. 2 (1998): 74–91. [Google Scholar] [CrossRef]
- H.U. Gerber. “On additive premium calculation principles.” Astin Bull. 7 (1974): 215–222. [Google Scholar] [CrossRef]
- S. Pitrebois, M. Denuit, and J. Walhin. “Fitting the belgian bonus-malus system.” Belg. Actuar. Bull. 3 (2003): 58–62. [Google Scholar]
- F. Bichsel. “Erfahrungstarifierung in der Motorfahrzeug-Haftpflichtversicherung.” Mitt. Verein. Schweiz. Versich. Math. 64 (1964): 119–130. [Google Scholar]
- C. Hipp, and M. Taksar. “Optimal non-proportional reinsurance control.” Insur. Math. Econ. 47 (2010): 246–254. [Google Scholar] [CrossRef]
- L.G. Benckert, and J. Jung. “Statistical models of claim distributions in fire insurance.” Astin Bull. 8 (1974): 1–25. [Google Scholar] [CrossRef]
- R.M. Corless, G.H. Gonnet, D.E. Hare, D.J. Jeffrey, and D.E. Knuth. “On the Lambert W function.” Adv. Comput. Math. 5 (1996): 329–359. [Google Scholar] [CrossRef]

**Figure 1.**The premium (14) as function of K for different values of customer characteristics. The interest rate is chosen to be $2\%$, and the estimates $\widehat{\mu}=1.6$ and $\widehat{\sigma}=1.99$ are used.

**Figure 3.**A contour of the relation $\mu (p;K)/\sigma {(p;K)}^{2}$ for deductibles in the range $[0,2000]$ and premiums in $[1210,4750]$.

**Figure 4.**The premiums ${p}^{*}\left(K\right)$ and $\tilde{p}\left(K\right)$ as functions of the deductible K.

**Figure 5.**Mesh of the optimal premium $max\{{p}^{*}\left(1000\right),\tilde{p}\left(1000\right)\}$ for different values of the parameters. If a parameter does not vary, then it has the same value as in previous graphs. (

**a**) Mesh of the optimal premium for $L,N\in [500,20500]$; (

**b**) mesh of the optimal premium for $\beta ,b\in [0.5,6.5]$.

© 2016 by the author; 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 (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Thøgersen, J.
Optimal Premium as a Function of the Deductible: Customer Analysis and Portfolio Characteristics. *Risks* **2016**, *4*, 42.
https://doi.org/10.3390/risks4040042

**AMA Style**

Thøgersen J.
Optimal Premium as a Function of the Deductible: Customer Analysis and Portfolio Characteristics. *Risks*. 2016; 4(4):42.
https://doi.org/10.3390/risks4040042

**Chicago/Turabian Style**

Thøgersen, Julie.
2016. "Optimal Premium as a Function of the Deductible: Customer Analysis and Portfolio Characteristics" *Risks* 4, no. 4: 42.
https://doi.org/10.3390/risks4040042