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
begin{eqnarray*}
_k V &=& bar{A}_{x+k:overline{n-k|}} –
P(bar{A}_{x:overline{n|}}) ddot{a}_{x+k:overline{n-k|}} .
end{eqnarray*}
To evaluate this, under UDD we have
begin{eqnarray*}
bar{A}_{x+k:overline{n-k|}} &=&
bar{A}_{x+k:overline{n-k|}}^{~~1}
+A_{x+k:overline{n-k|}}^{~~~~~~1} \
&=& frac{i}{delta} A_{x+k:overline{n-k|}}^{~~1}
+A_{x+k:overline{n-k|}}^{~~~~~~1}
end{eqnarray*}
and
begin{eqnarray*}
P(bar{A}_{x:overline{n|}})
=frac{bar{A}_{x:overline{n|}}}{ddot{a}_{x:overline{n|}}}
&=&
frac{frac{i}{delta} A_{x:overline{n|}}^{1}
+A_{x:overline{n|}}^{~~1} }{ddot{a}_{x:overline{n|}}}\
&=& frac{i}{delta}P_{x:overline{n|}}^{1}
+P_{x:overline{n|}}^{~~1} .
end{eqnarray*}
This yields
begin{eqnarray*}
_k V &=& frac{i}{delta} A_{x+k:overline{n-k|}}^{~~1}
+A_{x+k:overline{n-k|}}^{~~~~~~1} – left(
frac{i}{delta}P_{x:overline{n|}}^{1} +P_{x:overline{n|}}^{~~1}
right) ddot{a}_{x+k:overline{n-k|}} \
&=& frac{i}{delta} ~_k V_{x:overline{n|}}^{1} + ~_k
V_{x:overline{n|}}^{~~1},
end{eqnarray*}
where (~_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.