A theoretical summation-integral scheme involving commutation function in life insurance business

<p>The aim of this paper is to analytically extend Euler’s summation-integral quadrature to core actuarial functions based on sound judgement of numerical analytics. Specifically, the objectives are to theoretically (i) Obtain the value of a continuous commutation function relating the value o...

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Main Author: M. G. Ogungbenle
Format: Article
Language:English
Published: Faculty of Science, University of Peradeniya, Sri Lanka 2022-06-01
Series:Ceylon Journal of Science
Subjects:
Online Access:https://cjs.sljol.info/articles/8009
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author M. G. Ogungbenle
author_facet M. G. Ogungbenle
author_sort M. G. Ogungbenle
collection DOAJ
description <p>The aim of this paper is to analytically extend Euler’s summation-integral quadrature to core actuarial functions based on sound judgement of numerical analytics. Specifically, the objectives are to theoretically (i) Obtain the value of a continuous commutation function relating the value of a discrete sum to its integral (ii) Estimate the n-term life annuity due based on the estimated force of mortality and force of interest using alternative mathematical technique (iii) Estimate the temporary life expectancy based on the estimated force of mortality and force of interest. One of the most relevant applications of this paper is to provide a sound estimate of commutation functional values used in life and pension funds valuation. We used the elementary summation-integral formula of Euler-Maclaurin to relate the value of a discrete sum  <em><sub>t=n</sub></em><em><sup>k</sup></em><em></em><em>Ʃ</em><em>c</em>+<em>x</em> to its integral <sup>ω−<em>x</em></sup><em><sub>t=n</sub></em>∫C<em>x</em>+<em>t dt</em> in terms of the derivatives of a continuous commutation function (<em>C</em>=<em>x</em>,<em>t</em>) at two distinct points a and b specified in the integral and a remainder term (<em>R</em>=<em>x</em>,<em>t</em>). We assume the remainder term (<em>R</em>=<em>x</em>,<em>t</em>)→ <em>o</em>(1) where <em><sub>0</sub></em>(1)is small tending to zero as b grows very large and this can be used to compute asymptotic expansions for sums.</p>
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spelling doaj.art-be684b575fd444eca62945d658d8d3302022-12-22T02:43:58ZengFaculty of Science, University of Peradeniya, Sri LankaCeylon Journal of Science2513-28142513-230X2022-06-0151214715410.4038/cjs.v51i2.80095929A theoretical summation-integral scheme involving commutation function in life insurance businessM. G. Ogungbenle0University of Jos, Plateau State<p>The aim of this paper is to analytically extend Euler’s summation-integral quadrature to core actuarial functions based on sound judgement of numerical analytics. Specifically, the objectives are to theoretically (i) Obtain the value of a continuous commutation function relating the value of a discrete sum to its integral (ii) Estimate the n-term life annuity due based on the estimated force of mortality and force of interest using alternative mathematical technique (iii) Estimate the temporary life expectancy based on the estimated force of mortality and force of interest. One of the most relevant applications of this paper is to provide a sound estimate of commutation functional values used in life and pension funds valuation. We used the elementary summation-integral formula of Euler-Maclaurin to relate the value of a discrete sum  <em><sub>t=n</sub></em><em><sup>k</sup></em><em></em><em>Ʃ</em><em>c</em>+<em>x</em> to its integral <sup>ω−<em>x</em></sup><em><sub>t=n</sub></em>∫C<em>x</em>+<em>t dt</em> in terms of the derivatives of a continuous commutation function (<em>C</em>=<em>x</em>,<em>t</em>) at two distinct points a and b specified in the integral and a remainder term (<em>R</em>=<em>x</em>,<em>t</em>). We assume the remainder term (<em>R</em>=<em>x</em>,<em>t</em>)→ <em>o</em>(1) where <em><sub>0</sub></em>(1)is small tending to zero as b grows very large and this can be used to compute asymptotic expansions for sums.</p>https://cjs.sljol.info/articles/8009asymptotic, annuity, continuous, indicator, euler-maclaurin
spellingShingle M. G. Ogungbenle
A theoretical summation-integral scheme involving commutation function in life insurance business
Ceylon Journal of Science
asymptotic, annuity, continuous, indicator, euler-maclaurin
title A theoretical summation-integral scheme involving commutation function in life insurance business
title_full A theoretical summation-integral scheme involving commutation function in life insurance business
title_fullStr A theoretical summation-integral scheme involving commutation function in life insurance business
title_full_unstemmed A theoretical summation-integral scheme involving commutation function in life insurance business
title_short A theoretical summation-integral scheme involving commutation function in life insurance business
title_sort theoretical summation integral scheme involving commutation function in life insurance business
topic asymptotic, annuity, continuous, indicator, euler-maclaurin
url https://cjs.sljol.info/articles/8009
work_keys_str_mv AT mgogungbenle atheoreticalsummationintegralschemeinvolvingcommutationfunctioninlifeinsurancebusiness
AT mgogungbenle theoreticalsummationintegralschemeinvolvingcommutationfunctioninlifeinsurancebusiness