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Title: Stochastic claims reserving in insurance with regression models
Author: Portugal da Costa Lobo Rodrigues dos Santos, L.
ISNI:       0000 0004 7656 5198
Awarding Body: University of Liverpool
Current Institution: University of Liverpool
Date of Award: 2018
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The insurance business and the claims process features, as two sources of uncertainty and risk, are presented and the importance of a correct reserving to overcome this is highlighted. The claims reserving framework with data in triangle format is summarized and the main method used for reserving, the Chain-Ladder (CL), is explained. The assumptions of the latter are also presented and criticized. An attempt is made to explain why actuaries use the CL. The current methods for claims reserving are also summarized, both the deterministic and the stochastic. The relation of some of these methods with regression analysis is highlighted and an historical summary of the use of regression models in claims reserving is presented. The definition of the prediction error (as the square root of the mean square error of prediction) and the formulas for the confidence intervals are shown. Some conclusions about reserving models and the use of regression techniques are summarized. A first method, as an alternative to the traditional stochastic CL, Mack (1993a, 1993b, 1994), is presented. The new method, the stochastic Vector Projection (VP), is based on regression techniques with heteroscedastic errors and is shown, on the survey conducted, to produce lower prediction errors. A numerical analysis with regular and irregular data is performed and a method selection is done with errors inspection and back-testing calculations. The conclusions from these two tools for method selection are compared with the obtained prediction errors. A second method is presented, the stochastic generalized link ratios (GLR). The latter can replicate the VP, the CL, and the Simple Average (SA), as cases with a specific parameter. It also shows that other methods may be obtained through this specific parameter. The parameter is defined so that we get the method with the lowest prediction error. The method is also able to show an alternative to the prediction error estimation from the stochastic CL, from Mack (1993a). This GLR method presents the prediction errors with an analytical formula, not recursive, as was traditional with some similar approaches, such as the one from Murphy (1994). The GLR method highlights the importance of the heteroscedasticity assumption (nonconstant variance of the errors) in some claims reserving methods. A homoscedastic (constant variance of the errors) GLR is also developed, the homoscedastic VP. Using this GLR method a third method is presented with stochastic multivariate regressions inside the claims triangle, the multivariate generalized link ratios (MGLR). This method considers the contemporaneous correlations between all the regressions inside the triangle and brings light to other issues known in practice, such as the speed of payments that affects reserve estimation. This approach contrasts with the methods on multivariate claims reserving that estimate several triangles at the same time with the traditional CL, see for example Prohl and Schmidt (2005), Wüthrich and Merz (2007b), and Zhang (2010). With MGLR, we just have one triangle and the multivariate approach comes from the contemporaneous correlations considered inside that triangle. Using a specific parameter (as in the GLR), the MGLR will also present, in particular cases, the multivariate versions from VP, CL and SA. Other multivariate methods may be obtained for other values from this parameter. Numerical results are presented for irregular and regular datasets and a survey of 114 triangles is summarized. Heteroscedasticity tests are conducted as well as tests on the correlations between triangle equations. Serial correlation inside each equation is also analysed. In a fourth and fifth method, GLR and MGLR are extended, and we will consider the estimation of several triangles at the same time. The new methods, the portfolio generalized link ratios (PGLR) and the portfolio multivariate generalized link ratios (PMGLR) consider the estimation of several triangles at the same time. The PMGLR allow the consideration of contemporaneous correlations between those triangles and between equations inside each triangle. The PGLR and the MPGLR will also present, in particular cases, the portfolio versions (univariate and multivariate) for VP, CL, and SA. As with GLR and MGLR, a specific parameter is used to identify these methods. Other portfolio methods may be obtained for other values from this parameter, following the same procedures used with GLR and MGLR. Numerical results are presented using the three triangles considered in this thesis, either as portfolio data (the three triangles estimated at the same time with their correlations) or as aggregated data (the three triangles sum in just one triangle). A test for the possibility of having pooled data is also conducted. Finally, several general conclusions are presented about the thesis. Most of them respect the CL, the five alternative methods presented, and the decrease of the prediction errors when the latter is considered. The absence of heteroscedasticity in most insurers' triangles is emphasized. The existence of heteroscedasticity in irregular data triangles is not excluded. The relation between prediction errors and two other method selection techniques, errors analysis and back-testing, is emphasized and the importance of some regression tests to help for method selection is also highlighted. The need to consider multivariate regressions in claims reserving, with correlations between the equations, is explained, as well as the advantage of working with portfolio data and triangle's correlations.
Supervisor: Pantelous, Athanasios ; Assa, Hirbod Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral