Relations between Life Annuities and Life Insurances (One Life)

F. Etienne De Vylder

Chapter Chapter 7 in Life Insurance Theory, 1997, pp 51-56 from Springer

Abstract: Abstract By an integration by parts, $$\begin{array}{*{20}{c}} {_{{s|t}}\overline {{{A}_{x}}} = \int {_{{\left( {s,s + t} \right)\,{{v}^{\tau }}{{d}_{\tau }}{{q}_{x}} = - \int {_{{\left( {s,s + t} \right)\,{{V}^{\tau }}d{{\,}_{\tau }}{{p}_{x}}}}} }}} \,} \\ {\, = - {{{\left[ {{{v}^{\tau }}_{\tau }{{p}_{x}}} \right]}}_{s}}^{{s + t}} + \int {_{{\left( {s,s + t} \right)\,\tau {{p}_{x}}d{{v}^{\tau }} = {{v}^{s}}_{s}{{p}_{x}} - {{v}^{{s + t}}}_{{s + t}}{{p}_{x}} - \delta \int {_{{\left( {s,s + t} \right)\,\tau {{p}_{x}}{{v}^{\tau }}d\tau .}}} }}} } \\ \end{array}$$ Hence, 1 $$_{s|t}\overline {{A_x}} { = _s}{E_x}{ - _{s + t}}{E_x} - {\delta _{s|t}}\overline {{a_x}} .$$

Date: 1997
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DOI: 10.1007/978-1-4757-2616-9_7

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