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EPV of term life insurance payable at end of 1/m year in which death occurs

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  • EPV of term life insurance payable at end of 1/m year in which death occurs

    Why do text books always leave the EPV of an n year life insurance benefit of one unit of money payable at the end of the 1/m year (m =, for example, 1,2,4 or 12) in which death occurs as the sum, from k=0 to m*n minus one inclusive, of (1-d(m)/m)^(k+1)*(l_x+k/m minus l_x+k/m+1/m)/l_x and not show that this becomes one minus (1-d(m)/m)^(m*n)*l_x+n/l_x minus the sum, from k=0 to m*n minus one inclusive, of d(m)*(1-d(m)/m)^k*l_x+k/m/(m*l_x), which can, of course, also be expressed as one minus (1+i(m)/m)^(-m*n)*l_x+n/l_x minus the sum, from k=0 to m*n minus one inclusive, of i(m)*(1+i(m)/m)^(-k-1)*l_x+k/m/(m*l_x).
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