# Price and Profit Optimization for Financial Services

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Individual Profit as Influenced by Price Elasticity

#### 3.1. The Individual Profit Model

#### 3.2. Model Estimation

#### 3.3. Optimization of Expected Profit

**Proposition**

**1.**

## 4. Real Data Study

#### 4.1. Parameter Estimation

**Step**

**1.**

**Step**

**2.**

- Price change: The price change ${c}_{ik}$ is a continuous explanatory factor only present in the model for ${p}_{ik}^{\left(2\right)}$.
- Tariff premium: The tariff premium ${\widehat{\pi}}_{0ik}$ (in hundreds of euros) is a continuous explanatory factor with minimum, mean and maximum equal to 1.001, 4.038, and 15.000, respectively.
- Age: Age of the person contacted as a categorical factor; ≤40, 41–60, and >60. Sample frequencies are 27.6%, 53.8%, and 18.6%, respectively.
- House: Indicator variables describing whether or not the person owns their home (affirmative for 40.3% in the sample).
- Fuel: The type of fuel the car runs on; petrol or diesel. Sample frequencies are 81.7% and 18.3%, respectively.

#### 4.2. Profit Maximization and Calculation of Optimal Price Changes

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

**Proof**

**of (7).**

## References

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**Figure 1.**A comparison of the sales probability models ${\widehat{p}}_{ik}^{\left(1\right)}$ and ${\widehat{p}}_{ik}^{\left(2\right)}$ for different values of the price change ${c}_{ik}$. Note that ${\widehat{p}}_{ik}^{\left(1\right)}>{\widehat{p}}_{ik}^{\left(2\right)}$ for ${c}_{ik}<0$.

Price Change | Number of Offers | Average Sales Probability |
---|---|---|

−5% | 7898 | 10.90% |

0% | 65,861 | 9.21% |

+5% | 8095 | 7.96% |

**Table 2.**Estimated logistic regression parameters for the two alternative sales probability models. Standard error for the parameter estimates are in parentheses.

Factor | ${\mathit{p}}_{0\mathit{i}\mathit{k}}$ | ${\mathit{p}}_{\mathit{i}\mathit{k}}^{\left(2\right)}$ |
---|---|---|

intercept | −1.797 (0.055) | −1.806 (0.056) |

tariff premium | −0.202 (0.008) | −0.200 (0.008) |

Age (41–60) | −0.270 (0.028) | −0.270 (0.028) |

Age (61–) | −0.328 (0.037) | −0.327 (0.037) |

House (Y) | 0.0440 (0.025) | 0.440 (0.025) |

Fuel (petrol) | 0.321 (0.038) | 0.323 (0.038) |

price change | - | −2.877 (0.551) |

**Table 3.**Summary statistics of the parameter estimates, optimal price changes and corresponding expected profit. Min and max are the value (lowest and highest) of a single observation of the corresponding magnitude in the first column.

Definition | Parameter | Min | Mean | Max |
---|---|---|---|---|

Sales cost | ${\widehat{\omega}}_{k}$ | 11 | 11 | 11 |

Exp. claims cost | ${\widehat{s}}_{ik}$ | 22 | 413 | 1.507 |

Tariff price | ${\widehat{\pi}}_{0ik}$ | 110 | 454 | 1.646 |

Opt. price change | ${\widehat{c}}_{ik}^{\ast \left(1\right)}$ | −0.547 | −0.008 | 0.467 |

Opt. price change | ${\widehat{c}}_{ik}^{\ast \left(2\right)}$ | −0.329 | 0.050 | 0.350 |

Sales probability | ${\widehat{p}}_{0ik}$ | 0.0062 | 0.0874 | 0.225 |

Sales probability | ${\widehat{p}}_{ik}^{\left(1\right)}$ | 0.0026 | 0.107 | 1 |

Sales probability | ${\widehat{p}}_{ik}^{\left(2\right)}$ | 0.032 | 0.0788 | 0.321 |

Exp. Profit | ${\widehat{\mu}}_{ik}\left(0\right)$ | −10.8 | −0.229 | 16.1 |

Exp. Profit | ${\widehat{\mu}}_{ik}\left({\widehat{c}}_{ik}^{\ast \left(1\right)}\right)$ | −9.16 | 0.285 | 32.9 |

Exp. profit | ${\widehat{\mu}}_{ik}\left({\widehat{c}}_{ik}^{\ast \left(2\right)}\right)$ | −9.19 | 0.107 | 16.2 |

**Table 4.**Statistics for only the price offers associated with a positive expected profit, ${\widehat{\mu}}_{ik}\left(0\right)>0$, ${\widehat{\mu}}_{ik}\left({\widehat{c}}_{ik}^{\ast \left(1\right)}\right)>0$ and ${\widehat{\mu}}_{ik}\left({\widehat{c}}_{ik}^{\ast \left(2\right)}\right)>0$, respectively.

Description | c_{ik} = 0 | ${\mathit{c}}_{\mathit{i}\mathit{k}}={\widehat{\mathit{c}}}_{\mathit{i}\mathit{k}}^{\ast \left(1\right)}$ | ${\mathit{c}}_{\mathit{i}\mathit{k}}={\widehat{\mathit{c}}}_{\mathit{i}\mathit{k}}^{\ast \left(2\right)}$ |
---|---|---|---|

Number of offers with ${\widehat{\mu}}_{ik}>0$ | 31,383 | 35,692 | 33,807 |

Expected total profit (€) | 84,181 | 105,425 | 92,448 |

© 2018 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bolancé, C.; Guillen, M.; Nielsen, J.P.; Thuring, F.
Price and Profit Optimization for Financial Services. *Risks* **2018**, *6*, 9.
https://doi.org/10.3390/risks6010009

**AMA Style**

Bolancé C, Guillen M, Nielsen JP, Thuring F.
Price and Profit Optimization for Financial Services. *Risks*. 2018; 6(1):9.
https://doi.org/10.3390/risks6010009

**Chicago/Turabian Style**

Bolancé, Catalina, Montserrat Guillen, Jens Perch Nielsen, and Fredrik Thuring.
2018. "Price and Profit Optimization for Financial Services" *Risks* 6, no. 1: 9.
https://doi.org/10.3390/risks6010009