European University at St. Petersburg THE METHOD OF QUANTILE REGRESSION, A NEW APPROACH TO ACTUARIAL MATHEMATICS Authors: Ruslan Abduramanov Andrey Kudryavtsev Saint Petersburg State University

INTRODUCTION How correctly estimate a net premium? Classical approaches: An expected net premium. Risk loadings. Premium principles. Disadvantages of classical approaches: The problem of outliers. Fitting the theoretical distribution function. Independence of losses.

Advantages of the method of quantile regression: In a series of observations, there is a small portion of “outliers”. The distribution of the sample can be accurately fitted. Observations from the sample are not independent. General model of quantile regression: QUANTILE REGRESSION

Quantile regression, as a linear programming problem: Several important consequences: It is guaranteed that the computing procedure will be finished with a finite number of iterations. Without changing estimates of s, can be increased/decreased to, robust. QUANTILE REGRESSION

-level of significance, then using well known quantile approach to rating: where In this case So problem is to estimate a conditional distribution function A NEW APPROACH TO ESTIMATE NET PREMIUM RATES

The general model is as follow: so, we actually build inverse conditional distribution function: where А is all possible As it is more convenient to take logarithmic function Then receive: A NEW APPROACH TO ESTIMATE NET PREMIUM RATES

The idea of new approach is to estimate quantile regression of type with parameter - Quantile is seen as a good tool to separate good and bad risks, thus, the method proposed gives adequate unbiased estimates. - The avoidance of dividing the problem of the estimation of net premium rates into two subtasks – the assessment of expected net premium and the evaluation of risk loadings A NEW APPROACH TO ESTIMATE NET PREMIUM RATES

Conditional premium rate calculated by the formula: as a function, it is sufficient to take the expectation: Then premium rate for the whole portfolio will be calculated with the following formula: A NEW APPROACH TO ESTIMATE NET PREMIUM RATES

Analyzed data provided by one of the insurance companies operating on the motor insurance market in St. Petersburg (theft of vehicles): all contracts = 11790, where claims = Let so Characteristics of the object insured: the type of car, the colour, the region of theft and the date of theft. -it is important to describe dependence of loss (claim size) from the characteristics of the object insured. EXAMPLE

Histogram of losses.

Assumptions about theoretical distribution function of losses: 1.Losses are LogNormally distributed. 2.Losses are Gamma-distributed. -for both, the net premium estimates were achieved as a third quartile for appropriate constructed theoretical distribution. By the method introduced in the paper, the model should be evaluated by quantile regression with parameter -in the case of the method of quantile regression, it is enough to take expectation of the theoretical distribution. EXAMPLE

The results of estimating by all methods, are shown in the table :

The new method is free from some important assumptions which are critical for classical methods. The method proposed is based on quantiles which are popular for risk assessment in finance. The method proposed gives greater confidence to the actuary in the results received. The working capability and power of the method is shown with practical example on the real data. CONCLUSION