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RESEARCH PRODUCT

Using Wavelet Techniques to Approximate the Subjacent Risk of Death

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In Actuarial science, graduation techniques have been used extensively: the large number of scientific papers and technical documentation published evidences this fact (see Ayuso M et al. (Estadistica Actuarial Vida. UBe, Barcelona (2007)), Baeza Sampere and Morillas Jurado (Rev. Anales del Instituto de Actuarios Espanoles 135–164 (2011)), London (Graduation: The Revision of Estimates. ACTEX Publications, Connecticut (1985)), Cairns, et al. (Scand Actuar J 2(3):79–113 (2008)) and the references therein). Graduation techniques are defined by Haberman and Renshaw J Inst Actuar 110:135–156 (1983) as a set of principles and techniques for use that are used on raw data so that a more appropriate basis is obtained to make inferences and calculations of premiums, reserves and other variables of interest in the financial and insurance sector. Solvency II (Directive 2009/138/EC) is the regulatory framework used for risk management and for the supervision of insurance companies. This normative is effective from 1/2016 and it establishes the technical exigencies to be applied on some procedures such as mathematical provisioning or pricing. Related to these procedures, this normative introduces the concepts of best-estimate and margin-risk (see also CEIOPS: QIS5 Technical specifications. Technical Report. European Commissions-Internal market and services DG (2010)). The purpose of the former is to approximate the expected loss, and the latter to control the deviation from the best-estimate. In life actuarial methodologies, the probabilities of death are used explicitly and so the estimation of qx has a great impact on best-estimate. A widely recognized technique is that of wavelet techniques which have been applied in several fields such as engineering, digital processing of images, medicine, economy, finances, etc. (see Mallat (A wavelet tour of signal processing. Elsevier, Oxford (2009)). In this paper, we use wavelet techniques to achieve a good approximation of qx with the aim of obtaining a good estimation of the best-estimate. Therefore, as we can deduce from Standards IFRS 17 (view [16]), the knowledge of future scenarios is an important key point. In this research, the best-estimate is obtained using a wavelet-based graduation technique similar to Meneu et al. El Factor de Sostenibilidad: Disenos alternativos y valoracion financiero-actuarial de sus efectos sobre los parametros del sistema. Economia Espanola y Proteccion Social, V, 63–96 (2013). Then, using resampling techniques over the deviations between the graduated values and the observed values, we can obtain an approximation to the variability of the estimation and go on to construct several series of alternative scenarios of death. Complementarily, we note that the wavelet techniques present some problems when the amount of data is small; for this reason, we countercheck the cases where we have low-frequency series (as is usually the case in the context of Life Insurers).