An Estimation of a Hybrid Log-Poisson Regression Using a Quadratic Optimization Program for Optimal Loss Reserving in Insurance

Abstract

In this article, we are interested in developing an alternative estimation method of the parameters of the hybrid log-Poisson regression model. In our previous paper, we have proposed a hybrid log-Poisson regression model where we have derived the analytical expression of the fuzzy parameters. We found that the hybrid model provide better results than the classical log-Poisson regression model according to the mean square error prediction and the goodness of fit index. However, nowhere we have taken into account the optimal value of h(α-cut) which is of greatest importance in fuzzy regressions literature. In this paper, we provide an alternative estimation method of our hybrid model using a quadratic optimization program and the optimized h-value (α-cut). The expected value of fuzzy number is used as a defuzzification procedure to move from fuzzy values to crisp values. We perform the hybrid model with the alternative estimation we are suggesting on two different numerical data to predict incremental payments in loss reserving. From the mean square error prediction, we prove that the alternative estimation of the new hybrid model with an optimized h-value predicts incremental payments better than the classical log-Poisson regression model as well as the same hybrid model with analytical estimation of parameters. Hence we have optimized the outstanding loss reserves.

Calculating Insurance Claim Reserves with an Intuitionistic Fuzzy Chain-Ladder Method

“…Heuristically [4,14,15,18] ------FR-MFP [3,13] [ 23,24] [ 19,20,25] FR-FLS [26] [21] [22] Note: FR-MFP stands for fuzzy regression with the minimum fuzziness principle, and FR-FLS stands for fuzzy least squares.…”

“…whose exact values are given in (20), can also be determined via Taylor expansion to the first grade from β = 0 to β ∈ (0, 1]. Therefore, for B * β , we state…”

“…As shown in Table 1, among the most commonly used schemes for quantifying loss reserves, in addition to the CL method, we can highlight the geometric separation method [16] and methods that model incremental claims in a two-way manner, such as in [17]. The methodology for adjusting the parameters governing the evolution of claims over time can be performed heuristically [4,14,15,18] or with fuzzy regression methods that apply both the principle of minimum fuzziness [19,20] and the fuzzy least-squares approach [21,22]. Table 1.…”

Fuzzy Markovian Bonus-Malus Systems in Non-Life Insurance

“…Option 3 Fuzzy Regression Methods (FRMs) have been applied in several actuarial issues to fit relevant variables [52] for a comprehensive description of application areas). In this way, [53] fits the term structure of interest rates, [54,55] predicts claim provisions, and [25,28] adjusts the Lee-Carter mortality law.…”

“…To fit , the fuzzy extension of the log-Poisson regression by [55] may be used. It combines the conventional Poisson GLM and the minimum fuzziness principle by [56].…”