Credibility Theory for Variance Premium Principle

Abstract

The method of credibility theory plays a critical role in various research areas of actuarial science. Among others, the hypothetical mean and process variance are two quantities that convey crucial information to insurance companies when determining premiums for the insureds. The classical credibility model charges premiums by applying a linear combination of the mean of past claims and the population mean, and the corresponding estimator is proved to be the best estimator of the hypothetical mean under the mean squared loss criterion. Enlightened by the prestigious variance premium principle, we propose a credibility approach to estimate the linear combination of hypothetical mean and process variance under the quadratic loss function. Our proposed estimator consists of the linear form of observations and their quadratic terms, as well as some quantities representing population information. Meanwhile, a spin-off result is found and utilized to compare with the classical credibility model and the q-credibility model. The nonparametric estimators of structural quantities are also provided for ease of practical usage. Several numerical illustrations are carried out to demonstrate the performance of the estimator. A real dataset from a Swiss insurance company is also analyzed for the practical application of our results, where a data-driven procedure is proposed for determining the safety loading.

ACKNOWLEDGMENTS

This work was presented at the 5th International Conference on Econometrics and Statistics in June 2022 (EcoSta 2022) and the 25th International Congress on Insurance: Mathematics and Economics in July 2022. We thank the audience and sincerely appreciate their inspiring comments and useful suggestions. The authors are grateful for the comments and suggestions from two anonymous reviewers, which significantly improved the quality of this article.

DISCLOSURE STATEMENT

No potential conflict of interest was reported by the author(s).

Notes

1 Such requirement ensures that the kurtosis of claim severity exists. It is worth mentioning that it is not a strong condition in our setting because we usually adopt bounded random variables to model claim severity in many insurance scenarios, which naturally satisfies this assumption.

2 It is also known as the (individual) actuarially fair premium.

3 Usually, a loading factor, say $\kappa >0,$ is possibly attached to the hypothetical mean. In light of this, expression (1)(1) $$\mu (\Theta )+r{\sigma }^2(\Theta ),\hspace{1em}r\in {\mathbb{R}}_{+},$$(1) may be modified to admit the form $(1+\kappa )\mu (\Theta )+r{\sigma }^2(\Theta ).$ It corresponds to the mean–variance premium principle, combining the expected value and variance premium principle. The current study still works after some modifications to normal equations.

4 See the definition of “rate” from Statement of Principles Regarding Property and Casualty Insurance Ratemaking published by the Casualty Actuarial Society.

5 We use the following probability density function for the Pareto distribution:

$f_{\Theta }(x)=\frac{\eta {\chi }^{\eta }}{x^{\eta +1}}{\mathbf{1}}_{(x>\chi )},\hspace{1em}x>0,$

where η and x_m denote the shape and scale parameters, respectively.

6 The Gamma distribution has the following probability density function:

$f_{\Theta }(x)=\frac{{\lambda }^{\omega }}{\Gamma (\omega )}{x}^{\omega -1}{e}^{-\lambda x},\hspace{1em}x>0,$

where ω and λ denote the shape parameter and the rate parameter, respectively.

7 The structural parameters in the classical credibility model and q-credibility model are accordingly substituted by their nonparametric counterparts developed in Section 3.1.

Additional information

Funding

Pingping Zeng acknowledges support from the National Natural Science Foundation of China (Grant Nos. 12171228 and 11701266). Yiying Zhang acknowledges the National Natural Science Foundation of China (No. 12101336, 72311530687), Guangdong Basic and Applied Basic Research Foundation (No. 2023A1515011806), and Shenzhen Science and Technology Program (No. JCYJ20230807093312026).