Insurance benefits payable at the moment of failure and premiums payable at the beginning of the year. In this case, the policy value at duration \\(k\\) is
\n\\begin{eqnarray*}
\n_k V &=& \\bar{A}_{x+k:\\overline{n-k|}} –
\nP(\\bar{A}_{x:\\overline{n|}}) \\ddot{a}_{x+k:\\overline{n-k|}} .
\n\\end{eqnarray*}
\nTo evaluate this, under UDD we have
\n\\begin{eqnarray*}
\n\\bar{A}_{x+k:\\overline{n-k|}} &=&
\n\\bar{A}_{x+k:\\overline{n-k|}}^{~~1}
\n+A_{x+k:\\overline{n-k|}}^{~~~~~~1} \\\\
\n&=& \\frac{i}{\\delta} A_{x+k:\\overline{n-k|}}^{~~1}
\n+A_{x+k:\\overline{n-k|}}^{~~~~~~1}
\n\\end{eqnarray*}
\nand
\n\\begin{eqnarray*}
\nP(\\bar{A}_{x:\\overline{n|}})
\n=\\frac{\\bar{A}_{x:\\overline{n|}}}{\\ddot{a}_{x:\\overline{n|}}}
\n&=&
\n\\frac{\\frac{i}{\\delta} A_{x:\\overline{n|}}^{1}
\n+A_{x:\\overline{n|}}^{~~1} }{\\ddot{a}_{x:\\overline{n|}}}\\\\
\n&=& \\frac{i}{\\delta}P_{x:\\overline{n|}}^{1}
\n+P_{x:\\overline{n|}}^{~~1} .
\n\\end{eqnarray*}
\nThis yields
\n\\begin{eqnarray*}
\n_k V &=& \\frac{i}{\\delta} A_{x+k:\\overline{n-k|}}^{~~1}
\n+A_{x+k:\\overline{n-k|}}^{~~~~~~1} – \\left(
\n\\frac{i}{\\delta}P_{x:\\overline{n|}}^{1} +P_{x:\\overline{n|}}^{~~1}
\n\\right) \\ddot{a}_{x+k:\\overline{n-k|}} \\\\
\n&=& \\frac{i}{\\delta} ~_k V_{x:\\overline{n|}}^{1} + ~_k
\nV_{x:\\overline{n|}}^{~~1},
\n\\end{eqnarray*}
\nwhere \\(~_k V_{x:\\overline{n|}}^{1}\\) is the policy value at duration \\(k\\) of an \\(n\\)-year term policy and \\(~_k V_{x:\\overline{n|}}^{~~1}\\) is the policy value at duration \\(k\\) of an \\(n\\)-year pure endowment policy.<\/p>\n