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# Moment problem and its applications to risk assessment

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<title>Moment problem and its applications to risk assessment</title>
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<name type="personal" usage="primary">
<namePart>Tian, Ruilin</namePart>
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<name type="personal">
<namePart>Cox, Samuel H.</namePart>
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<name type="personal">
<namePart>Zuluaga, Luis F.</namePart>
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<abstract>This article discusses how to assess risk by computing the best upper and lower bounds on the expected value E[f(X)], subject to the constraints E[Xi] = µi for i = 0, 1, 2, , n. f(x) can take the form of the indicator function  in which the bounds on  are calculated and the form f(x) = (?(x) - K)+ in which the bounds on financial payments are found. We solve the moment bounds on  through three methods: the semidefinite programming method, the moment-matching method, and the linear approximation method. We show that for practical purposes, these methods provide numerically equivalent results. We explore the accuracy of bounds in terms of the number of moments considered. We investigate the usefulness of the moment method by comparing the moment bounds with the point estimate provided by the Johnson system of distributions. In addition, we propose a simpler formulation for the unimodal bounds on  compared to the existing formulations in the literature. For those problems that could be solved both analytically and numerically given the first few moments, our comparisons between the numerical and analytical results call attention to the potential differences between these two methodologies. Our analysis indicates the numerical bounds could deviate from their corresponding analytical counterparts. The accuracy of numerical bounds is sensitive to the volatility of X. The more volatile the random variable X is, the looser the numerical bounds are, compared to their closed-form solutions.</abstract>
<note type="statement of responsibility">Ruilin Tian, Samuel H. Cox, Luis F. Zuluaga</note>
<subject>
<topic>Cálculo actuarial</topic>
</subject>
<subject>
<topic>Matemática del seguro</topic>
</subject>
<subject>
<topic>Análisis actuarial</topic>
</subject>
<classification authority="">6</classification>
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<titleInfo>
<title>North American actuarial journal</title>
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<originInfo>
<publisher>Schaumburg : Society of Actuaries, 1997-</publisher>
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<identifier type="issn">1092-0277</identifier>
<identifier type="local">MAP20077000239</identifier>
<part>
<text>05/06/2017 Tomo 21 Número 2 - 2017 , p. 242-266</text>
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<recordIdentifier source="MAP">MAP20170033233</recordIdentifier>