Special Cases

Special Case 1. Ordinary Whole Life

Here, take (j = omega – x) and (_1 P_{[x]}^n = v _1 q_{[x]} = A_{[x]:overline{1|}}^1). Thus,
( A_{[x]:overline{1|}}^1+ P^{FPT} a_x =P_x ddot{a}_x ) and
begin{eqnarray*}
P^{FPT} &=& frac{P_x ddot{a}_x – A_{[x]:overline{1|}}^1}{a_x} = frac{A_x – A_{[x]:overline{1|}}^1}{a_x}
&=& frac{_1 E_x A_{x+1}}{ _1 E_x ddot{a}_{x+1}} = frac{A_{x+1}}{ddot{a}_{x+1}} = P_{x+1} .
end{eqnarray*}

Special Case 2. (n)-pay, (m)-year Endowment

Here, take (j = n). Then, use the same logic in Special Case 1 to check that (P^{FPT} = _{n-1} P_{x+1:overline{m-1|}}.) Further,
begin{eqnarray*}
V_k^{FPT} &=& A_{x+k:overline{m-k|}} – _{n-1} P_{x+1:overline{m-1|}} ddot{a}_{x+k:overline{m-k|}}
= ~_{k-1}^{n-1} V_{x+1:overline{m-1|}},
end{eqnarray*}
that is, a reserve for a life age ((x+1)) at duration (k-1) for an (n-1)-pay, (m-1)-year endowment policy.

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