延期之終身壽險 — 保單契約有效期間內身故時將退還已繳保費

Antonietta Brandeis (1848-1926) was a Czech-born Italian painter.

本頁有以下小節

• 年繳純保費
• 真實保險費
• 年賦保險費
• 比例分攤保險費

延期 \(m\)年之終身壽險係指保單契約約定被保險人若於 \(m\)年末仍生存時，則當被保險人在 \(m\)年之後身故時，保險公司需給付 \(\Lambda\)元保險金，給付方式可分為身故當年末給付或即刻給付兩種。若被保險人於 \(m\)年內身故時，保險公司將以不計息方式退還已繳之保險費。

\begin{equation}{\label{a}}\tag{A}\mbox{}\end{equation}

年繳純保費.

可分為年末給付與即刻給付保險金方式。另外，又可分為延期 \(z\)年繳費且繳費至終身與延期 \(z\)年繳費且繳費期間為 \(q\)年，其中 \(z\leq m\)。將依身故當年末退還已繳之保險費與身故即刻退還已繳之保險費分別討論。

年末給付.

採用身故當年末給付保險金方式，

延期繳費且繳費至終身.

考慮身故當年末退還已繳之保險費，現年 \(x\)歲的被保險人，假設延期 \(z\)年繳費且繳費至終身，其年繳純保費以符號 \({}_{m,z|}P_{x}^{\Lambda}\)表示之，若 \(z=0\)，則簡單表為 \({}_{m|}P_{x}^{\Lambda}\)。保險公司的支出狀況如下所述:

• 被保險人於第 \(1\)年內身故，則保險公司須於當年末退還 \({}_{m,z|}P_{x}^{\Lambda}\)
• 若被保險人於第 \(2\)年內身故，則保險公司須於當年末退還 \(2\cdot{}_{m,z|}P_{x}^{\Lambda}\)
• 依此類推，若被保險人於第 \(m\)年內身故，則保險公司須於當年末退還 \(n\cdot{}_{m,z|}P_{x}^{\Lambda}\)
• 若$m$年後仍然生存時，則當被保險人在 \(m\)年之後身故時，保險公司須於身故當年末給付 \(\Lambda\)元保險金。

若被保險人於 \(m\)年內身故時，其年末退還保險費方式類似年末給付之遞增型 \(m\)年定期壽險，也就是說，被保險人若於第 \(1\)年內身故時，則保險公司需於當年末給付 \({}_{m,z|}P_{x}^{\Lambda}\)元，若於第 \(2\)年內身故時，則保險公司需於當年末給付 \(2\cdot{}_{m,z|}P_{x}^{\Lambda}\)元，依此類推，每年增加 \({}_{m,z|}P_{x}^{\Lambda}\)元至第 \(m\)年為止，以後不再給付。因此保險公司總支出現值為
\[{}_{m,z|}P_{x}^{\Lambda}\cdot (IA)_{x:m\!\rceil}^{1}+\Lambda\cdot {}_{m|}A_{x}\mbox{。}\]
因保費收入現值為
\[{}_{m,z|}P_{x}^{\Lambda}\cdot {}_{z}E_{x}+{}_{m,z|}P_{x}^{\Lambda}\cdot {}_{z+1}E_{x}
+{}_{m,z|}P_{x}^{\Lambda}\cdot {}_{z+2}E_{x}+\cdots +{}_{m,z|}P_{x}^{\Lambda}\cdot
{}_{\omega -x}E_{x}={}_{m,z|}P_{x}^{\Lambda}\cdot {}_{z|}\ddot{a}_{x}\mbox{。}\]
依據收支平衡原則，可得
\[{}_{m,z|}P_{x}^{\Lambda}\cdot {}_{z|}\ddot{a}_{x}={}_{m,z|}P_{x}^{\Lambda}\cdot (IA)_{x:m\!\rceil}^{1}
+\Lambda\cdot {}_{m|}A_{x}\mbox{。}\]
因此可解得
\[{}_{m,z|}P_{x}^{\Lambda}=\frac{\Lambda\cdot {}_{m|}A_{x}}{{}_{z|}\ddot{a}_{x}-(IA)_{x:m\!\rceil}^{1}}
=\frac{\Lambda\cdot\left ({}_{m|}E_{x}-d\cdot {}_{m|}\ddot{a}_{x}\right )}
{{}_{z|}\ddot{a}_{x}-(IA)_{x:m\!\rceil}^{1}}=\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}
{{}_{z|}\ddot{a}_{x}-(IA)_{x:m\!\rceil}^{1}}\]
及
\[{}_{m|}P_{x}^{\Lambda}=\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}
{\ddot{a}_{x}-(IA)_{x:m\!\rceil}^{1}}\mbox{。}\]

考慮身故即刻退還已繳之保險費，其年繳純保費以符號 \({}_{m,z}{}_{|}^{*}P_{x}^{\Lambda}\)表示之，若 \(z=0\)，則簡單表為 \({}_{m}{}_{|}^{*}P_{x}^{\Lambda}\)。若被保險人於 \(m\)年內身故時，其即刻退還保險費方式類似即刻給付之遞增型 \(m\)年定期壽險。因此可解得
\[{}_{m,z}{}_{|}^{*}P_{x}^{\Lambda}=\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}
{{}_{z|}\ddot{a}_{x}-(I\bar{A})_{x:m\!\rceil}^{1}}\mbox{ 及 }{}_{m}{}_{|}^{*}P_{x}^{\Lambda}
=\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}{\ddot{a}_{x}-(I\bar{A})_{x:m\!\rceil}^{1}}\mbox{。}\]

延期繳費且繳費期間為q年.

考慮身故當年末退還已繳之保險費，現年 \(x\)歲的被保險人，假設延期 \(z\)年繳費且繳費至終身，其年繳純保費以符號 \({}_{m,z|}P_{x,q}^{\Lambda}\)表示之，若 \(z=0\)，則簡單表為 \({}_{m|}P_{x,q}^{\Lambda}\)。依據之前類似論點可知保險公司總支出現值為
\[{}_{m,z|}P_{x,q}^{\Lambda}\cdot (IA)_{x:m\!\rceil}^{1}+\Lambda\cdot {}_{m|}A_{x}\mbox{。}\]
而保費收入現值為
\[{}_{m,z|}P_{x,q}\cdot {}_{z}E_{x}+{}_{m,z|}P_{x,q}\cdot {}_{z+1}E_{x}
+{}_{m,z|}P_{x,q}\cdot {}_{z+2}E_{x}+\cdots +{}_{m,z|}P_{x,q}\cdot {}_{z+q-1}E_{x}
={}_{m,z|}P_{x,q}\cdot {}_{z|}\ddot{a}_{x:q\!\rceil}\mbox{。}\]
依據收支平衡原則，可得
\[{}_{m,z|}P_{x,q}^{\Lambda}\cdot {}_{z|}\ddot{a}_{x:q\!\rceil}
={}_{m,z|}P_{x,q}^{\Lambda}\cdot (IA)_{x:m\!\rceil}^{1}+\Lambda\cdot {}_{m|}A_{x}\mbox{。}\]
因此可解得
\[{}_{m,z|}P_{x,q}^{\Lambda}=\frac{\Lambda\cdot {}_{m|}A_{x}}{{}_{z|}\ddot{a}_{x:q\!\rceil}-(IA)_{x:m\!\rceil}^{1}}
=\frac{\Lambda\cdot\left ({}_{m|}E_{x}-d\cdot {}_{m|}\ddot{a}_{x}\right )}{{}_{z|}\ddot{a}_{x:q\!\rceil}-(IA)_{x:m\!\rceil}^{1}}
=\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}{{}_{z|}\ddot{a}_{x:q\!\rceil}-(IA)_{x:m\!\rceil}^{1}}\]
及
\[{}_{m|}P_{x,q}^{\Lambda}=\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}
{{}_{z|}\ddot{a}_{x:q\!\rceil}-(IA)_{x:m\!\rceil}^{1}}\mbox{。}\]

考慮身故即刻退還已繳之保險費，其年繳純保費以符號 \({}_{m,z}{}_{|}^{*}P_{x,q}^{\Lambda}\)表示之，若 \(z=0\)，則簡單表為 \({}_{m}{}_{|}^{*}P_{x,q}^{\Lambda}\)。若被保險人於 \(m\)年內身故時，其即刻退還保險費方式類似即刻給付之遞增型 \(m\)年定期壽險。因此可解得
\[{}_{m,z}{}_{|}^{*}P_{x,q}^{\Lambda}=\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}
{{}_{z|}\ddot{a}_{x:q\!\rceil}-(I\bar{A})_{x:m\!\rceil}^{1}}\mbox{ 及 }{}_{m}{}_{|}^{*}P_{x,q}^{\Lambda}
=\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}
{{}_{z|}\ddot{a}_{x:q\!\rceil}-(I\bar{A})_{x:m\!\rceil}^{1}}\mbox{。}\]

即刻給付.

採用身故即刻給付保險金方式。

延期繳費且繳費至終身.

考慮身故當年末退還已繳之保險費，現年 \(x\)歲的被保險人，假設延期 \(z\)年繳費且繳費至終身，其年繳純保費以符號 \({}_{z|}P^{\Lambda}({}_{m|}\bar{A}_{x})\)表示之，若 \(z=0\)，則簡單表為 \(P^{\Lambda}({}_{m|}\bar{A}_{x})\)。依據收支平衡原則，可得
\[{}_{z|}P^{\Lambda}({}_{m|}\bar{A}_{x})\cdot {}_{z|}\ddot{a}_{x}
={}_{z|}P^{\Lambda}({}_{m|}\bar{A}_{x})\cdot (IA)_{x:m\!\rceil}^{1}+\Lambda\cdot {}_{m|}\bar{A}_{x}\mbox{。}\]
因此可解得
\begin{align*}
{}_{z|}P^{\Lambda}({}_{m|}\bar{A}_{x}) & =\frac{\Lambda\cdot {}_{m|}\bar{A}_{x}}{{}_{z|}\ddot{a}_{x}-(IA)_{x:m\!\rceil}^{1}}
=\frac{\Lambda\cdot\left [\left (1+\frac{\delta}{2}\right )\cdot {}_{m}E_{x}-\delta\cdot {}_{m|}\ddot{a}_{x}\right ]}
{{}_{z|}\ddot{a}_{x}-(IA)_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda}{{}_{z|}\ddot{a}_{x}-(IA)_{x:m\!\rceil}^{1}}\cdot\left [\left (1-\frac{\delta}{2}\right )\cdot {}_{m|}\ddot{a}_{x}
-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]
\end{align*}
及
\[P^{\Lambda}({}_{m|}\bar{A}_{x})=\frac{\Lambda}{\ddot{a}_{x}-(IA)_{x:m\!\rceil}^{1}}\cdot
\left [\left (1-\frac{\delta}{2}\right )\cdot {}_{m|}\ddot{a}_{x}-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]\mbox{。}\]

考慮身故即刻退還已繳之保險費，其年繳純保費以符號 \({}_{z}{}_{|}^{*}P^{\Lambda}({}_{m|}\bar{A}_{x})\)表示之，若 \(z=0\)，則簡單表為 \({}^{*}P^{\Lambda}({}_{m|}\bar{A}_{x})\)。因此可解得
\[{}_{z}{}_{|}^{*}P^{\Lambda}({}_{m|}\bar{A}_{x})=\frac{\Lambda}{{}_{z|}\ddot{a}_{x}-(I\bar{A})_{x:m\!\rceil}^{1}}\cdot
\left [\left (1-\frac{\delta}{2}\right )\cdot {}_{m|}\ddot{a}_{x}-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]\]
及
\[{}^{*}P^{\Lambda}({}_{m|}\bar{A}_{x})=\frac{\Lambda}{\ddot{a}_{x}-(I\bar{A})_{x:m\!\rceil}^{1}}\cdot
\left [\left (1-\frac{\delta}{2}\right )\cdot {}_{m|}\ddot{a}_{x}-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]\mbox{。}\]

延期繳費且繳費期間為q年.

考慮身故當年末退還已繳之保險費，現年 \(x\)歲的被保險人，假設延期 \(z\)年繳費且繳費至終身，其年繳純保費以符號 \({}_{z|}P_{q}^{\Lambda}({}_{m|}\bar{A}_{x})\)表示之，若 \(z=0\)，則簡單表為 \(P_{q}^{\Lambda}({}_{m|}\bar{A}_{x})\)。依據收支平衡原則，可得
\[{}_{z|}P_{q}^{\Lambda}({}_{m|}\bar{A}_{x})\cdot {}_{z|}\ddot{a}_{x:q\!\rceil}
={}_{z|}P_{q}^{\Lambda}({}_{m|}\bar{A}_{x})\cdot (IA)_{x:m\!\rceil}^{1}+\Lambda\cdot {}_{m|}\bar{A}_{x}\mbox{。}\]
因此可解得
\begin{align*}
{}_{z|}P_{q}^{\Lambda}({}_{m|}\bar{A}_{x}) & =\frac{\Lambda\cdot {}_{m|}\bar{A}_{x}}{{}_{z|}\ddot{a}_{x:q\!\rceil}-(IA)_{x:m\!\rceil}^{1}}=\frac{\Lambda\cdot\left [\left (1+\frac{\delta}{2}\right )\cdot
{}_{m}E_{x}-\delta\cdot {}_{m|}\ddot{a}_{x}\right ]}{{}_{z|}\ddot{a}_{x:q\!\rceil}-(IA)_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda}{{}_{z|}\ddot{a}_{x:q\!\rceil}-(IA)_{x:m\!\rceil}^{1}}\cdot
\left [\left (1-\frac{\delta}{2}\right )\cdot {}_{m|}\ddot{a}_{x}-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]
\end{align*}
及
\[P_{q}^{\Lambda}({}_{m|}\bar{A}_{x})=\frac{\Lambda}{\ddot{a}_{x:q\!\rceil}-(IA)_{x:m\!\rceil}^{1}}\cdot
\left [\left (1-\frac{\delta}{2}\right )\cdot {}_{m|}\ddot{a}_{x}
-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]\mbox{。}\]

考慮身故即刻退還已繳之保險費，其年繳純保費以符號 \({}_{z}{}_{|}^{*}P_{q}^{\Lambda}({}_{m|}\bar{A}_{x})\)表示之，若 \(z=0\)，則簡單表為 \({}^{*}P_{x,q}^{\Lambda}\)。因此可解得
\[{}_{z}{}_{|}^{*}P_{q}^{\Lambda}({}_{m|}\bar{A}_{x})
=\frac{\Lambda}{{}_{z|}\ddot{a}_{x:q\!\rceil}-(I\bar{A})_{x:m\!\rceil}^{1}}\cdot
\left [\left (1-\frac{\delta}{2}\right )\cdot {}_{m|}\ddot{a}_{x}-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]\]
及
\[{}^{*}P_{q}^{\Lambda}({}_{m|}\bar{A}_{x})
=\frac{\Lambda}{\ddot{a}_{x:q\!\rceil}-(I\bar{A})_{x:m\!\rceil}^{1}}\cdot
\left [\left (1-\frac{\delta}{2}\right )\cdot {}_{m|}\ddot{a}_{x}
-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]\mbox{。}\]

\begin{equation}{\label{b}}\tag{B}\mbox{}\end{equation}

真實保險費.

假設將每年分成 \(h\)期，可分為年末給付與即刻給付保險金方式。又可分為延期 \(z\)年繳費且繳費至終身與延期 \(z\)年繳費且繳費期間為 \(q\)年，其中 \(z\leq m\)。將依身故當年末退還已繳之保險費、身故當年期末退還已繳之保險費與身故即刻退還已繳之保險費分別討論。

年末給付.

採用身故當年末給付保險金方式。

延期繳費且繳費至終身.

考慮身故當年末退還已繳之保險費，其真實保險費以符號 \({}_{m,z|}P_{x}^{\Lambda (h)}\)表示，若 \(z=0\)，則簡單表為 \({}_{m|}P_{x}^{\Lambda (h)}\)。因此每期應繳之保險費為 \({}_{m,z|}P_{x}^{\Lambda (h)}/h\)。依據收支平衡原則，可得
\begin{equation}{\label{eq481}}\tag{1}
{}_{m,z|}P_{x}^{\Lambda (h)}\cdot{}_{z|}\ddot{a}_{x}^{(h)}
={}_{m,z|}P_{x}^{\Lambda (h)}\cdot (IA)_{x:m\!\rceil}^{1}+\Lambda\cdot {}_{m|}A_{x}\mbox{。}
\end{equation}
因此可解得
\begin{align*}
{}_{m,z|}P_{x}^{\Lambda (h)} & =\frac{\Lambda\cdot {}_{m|}A_{x}}{{}_{z|}\ddot{a}_{x}^{(h)}-(IA)_{x:m\!\rceil}^{1}}
=\frac{\Lambda\cdot {}_{m|}A_{x}}{{}_{z|}\ddot{a}_{x}-\frac{h-1}{2h}\cdot {}_{z}E_{x}-(IA)_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}
{\frac{h+1}{2h}\cdot {}_{z|}\ddot{a}_{x}+\frac{h-1}{2h}\cdot {}_{z|}a_{x}-(IA)_{x:m\!\rceil}^{1}}
\end{align*}
及
\[{}_{m|}P_{x}^{\Lambda (h)}=\frac{\Lambda\cdot {}_{m|}A_{x}}{\ddot{a}_{x}^{(h)}-(IA)_{x:m\!\rceil}^{1}}
=\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}{\ddot{a}_{x}-\frac{h-1}{2h}-(IA)_{x:m\!\rceil}^{1}}\mbox{。}\]

考慮身故即刻退還已繳之保險費，其真實保險費以符號 \({}_{m,z}{}_{|}^{*}P_{x}^{\Lambda (h)}\)表示，若 \(z=0\)，則簡單表為 \({}_{m}{}_{|}^{*}P_{x}^{\Lambda (h)}\)。因此每期應繳之保險費為 \({}_{m,z}{}_{|}^{*}P_{x}^{\Lambda (h)}/h\)。依據收支平衡原則，可解得
\[{}_{m,z}{}_{|}^{*}P_{x}^{\Lambda (h)}=\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}
{\frac{h+1}{2h}\cdot {}_{z|}\ddot{a}_{x}+\frac{h-1}{2h}\cdot {}_{z|}a_{x}-(I\bar{A})_{x:m\!\rceil}^{1}}\]
及
\[{}_{m}{}_{|}^{*}P_{x}^{\Lambda (h)}=\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}
{\ddot{a}_{x}-\frac{h-1}{2h}-(I\bar{A})_{x:m\!\rceil}^{1}}\mbox{。}\]

考慮身故當期末退還已繳之保險費，其真實保險費以符號 \({}_{m,z|}P_{x}^{\Lambda (hh)}\)表示，若 \(z=0\)，則簡單表為 \({}_{m|}P_{x}^{\Lambda (hh)}\)。因此每期應繳之保險費為 \({}_{m,z|}P_{x}^{\Lambda (hh)}/h\)。保險公司的支出狀況如下所述:

• 若被保險人於第 \(1/h\)年內身故，則保險公司須於第 \(1/h\)年末退還 \({}_{m,z|}P_{x}^{\Lambda (hh)}/h\)
• 若被保險人於第 \(1/h\)至 \(2/h\)年內身故，則保險公司須於第 \(2/h\)年末退還 \(2\cdot{}_{m,z|}P_{x}^{\Lambda (hh)}/h\)
• 依此類推，若被保險人於第 \(t+\frac{s}{h}\)至 \(t+\frac{s+1}{h}\)年內身故，則保險公司須於 \(t+\frac{s+1}{h}\)年末退還 \(\left (t+\frac{s+1}{h}\right )\cdot{}_{m,z|}P_{x}^{\Lambda (hh)}\)，其中 \(t=0,1,\cdots ,n-1\)及 \(s=0,1,\cdots ,h-1\)
• 若 \(m\)年後仍然生存時，可得到保險公司給付 \(\Lambda\)元保險金。

若被保險人於 \(m\)年內身故時，其保險費退還方式類似每年遞增 \(h\)次之 \(m\)年定期壽險。因此保險公司總支出現值為
\[{}_{m,z|}P_{x}^{\Lambda (hh)}\cdot (I^{(h)}A)_{x:m\!\rceil}^{1}+\Lambda\cdot {}_{m|}A_{x}\mbox{。}\]
依據收支平衡原則，可得
\[{}_{m,z|}P_{x}^{\Lambda (hh)}\cdot{}_{z|}\ddot{a}_{x}^{(h)}
={}_{m,z|}P_{x}^{\Lambda (hh)}\cdot (I^{(h)}A)_{x:m\!\rceil}^{1}+\Lambda\cdot {}_{m|}A_{x}\mbox{。}\]
由每年遞增數次之人壽保險金頁之(31)式，亦可得
\[{}_{m,z|}P_{x}^{\Lambda (hh)}\cdot{}_{z|}\ddot{a}_{x}^{(h)}={}_{m,z|}P_{x}^{\Lambda (hh)}\cdot
\left ((IA)_{x:m\!\rceil}^{1}-\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}\right )+\Lambda\cdot {}_{m|}A_{x}\mbox{。}\]
因此可解得
\begin{align*}
{}_{m,z|}P_{x}^{\Lambda (hh)} & =\frac{\Lambda\cdot {}_{m|}A_{x}}
{{}_{z|}\ddot{a}_{x}^{(h)}-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}
{\frac{h+1}{2h}\cdot {}_{z|}\ddot{a}_{x}+\frac{h-1}{2h}\cdot {}_{z|}a_{x}-(IA)_{x:m\!\rceil}^{1}
+\frac{h-1}{2h}\cdot\left (v\cdot\ddot{a}_{x:m\!\rceil}-a_{x:m\!\rceil}\right )}
\end{align*}
及
\begin{align*}
{}_{m|}P_{x}^{\Lambda (hh)} & =\frac{\Lambda\cdot {}_{m|}A_{x}}
{\ddot{a}_{x}^{(h)}-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}{\ddot{a}_{x}-\frac{h-1}{2h}-(IA)_{x:m\!\rceil}^{1}
+\frac{h-1}{2h}\cdot\left (v\cdot\ddot{a}_{x:m\!\rceil}-a_{x:m\!\rceil}\right )}\mbox{。}
\end{align*}

延期繳費且繳費期間為q年.

考慮身故當年末退還已繳之保險費，其真實保險費以符號 \({}_{m,z|}P_{x,q}^{\Lambda (h)}\)表示，若 \(z=0\)，則簡單表為 \({}_{m|}P_{x,q}^{\Lambda (h)}\)。因此每期應繳之保險費為 \({}_{m,z|}P_{x,q}^{\Lambda (h)}/h\)。依據收支平衡原則，可得

\[{}_{m,z|}P_{x,q}^{\Lambda (h)}\cdot {}_{z|}\ddot{a}_{x:q\!\rceil}^{(h)}
={}_{m,z|}P_{x,q}^{\Lambda (h)}\cdot (IA)_{x:m\!\rceil}^{1}+\Lambda\cdot {}_{m|}A_{x}\mbox{。}\]
因此可解得
\begin{align*}
{}_{m,z|}P_{x,q}^{\Lambda (h)} & =\frac{\Lambda\cdot {}_{m|}A_{x}}{{}_{z|}\ddot{a}_{x:q\!\rceil}^{(h)}-(IA)_{x:m\!\rceil}^{1}}
=\frac{\Lambda\cdot {}_{m|}A_{x}}{{}_{z|}\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}
\cdot\left ({}_{z}E_{x}-{}_{z+q}E_{x}\right )-(IA)_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}{{}_{z|}\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}
\cdot\left ({}_{z}E_{x}-{}_{z+q}E_{x}\right )-(IA)_{x:m\!\rceil}^{1}}
\end{align*}
及
\[{}_{m|}P_{x,q}^{\Lambda (h)}
=\frac{\Lambda\cdot {}_{m|}A_{x}}{\ddot{a}_{x:q\!\rceil}^{(h)}-(IA)_{x:m\!\rceil}^{1}}
=\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}{\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}
\cdot\left (1-{}_{q}E_{x}\right )-(IA)_{x:m\!\rceil}^{1}}\mbox{。}\]

考慮身故即刻退還已繳之保險費，其真實保險費以符號 \({}_{m,z}{}_{|}^{*}P_{x,q}^{\Lambda (h)}\)表示，若 \(z=0\)，則簡單表為 \({}_{m}{}_{|}^{*}P_{x,q}^{\Lambda (h)}\)。因此每期應繳之保險費為 \({}_{m,z}{}_{|}^{*}P_{x,q}^{\Lambda (h)}/h\)。依據收支平衡原則，可解得
\[{}_{m,z}{}_{|}^{*}P_{x,q}^{\Lambda (h)}=\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}
{{}_{z|}\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}\cdot\left ({}_{z}E_{x}-{}_{z+q}E_{x}\right )-(I\bar{A})_{x:m\!\rceil}^{1}}\]
及
\[{}_{m}{}_{|}^{*}P_{x,q}^{\Lambda (h)}=\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}
{\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}\cdot\left (1-{}_{q}E_{x}\right )-(I\bar{A})_{x:m\!\rceil}^{1}}\mbox{。}\]

考慮身故當期末退還已繳之保險費，其真實保險費以符號 \({}_{m,z|}P_{x,q}^{\Lambda (hh)}\)表示，若 \(z=0\)，則簡單表為 \({}_{m|}P_{x,q}^{\Lambda (hh)}\)。因此每期應繳之保險費為 \({}_{m,z|}P_{x,q}^{\Lambda (hh)}/h\)。若被保險人於 \(m\)年內身故時，其保險費退還方式類似每年遞增 \(h\)次之 \(m\)年定期壽險。依據收支平衡原則，可得
\[{}_{m,z|}P_{x,q}^{\Lambda (hh)}\cdot{}_{z|}\ddot{a}_{x:q\!\rceil}^{(h)}
={}_{m,z|}P_{x,q}^{\Lambda (hh)}\cdot (I^{(h)}A)_{x:m\!\rceil}^{1}+\Lambda\cdot {}_{m|}A_{x}\mbox{。}\]
由每年遞增數次之人壽保險金頁之(31)式，亦可得
\[{}_{m,z|}P_{x,q}^{\Lambda (hh)}\cdot{}_{z|}\ddot{a}_{x:q\!\rceil}^{(h)}={}_{m,z|}P_{x,q}^{\Lambda (hh)}\cdot
\left ((IA)_{x:m\!\rceil}^{1}-\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}\right )+\Lambda\cdot {}_{m|}A_{x}\mbox{。}\]
因此可解得
\begin{align*}
{}_{m,z|}P_{x,q}^{\Lambda (hh)} & =\frac{\Lambda\cdot {}_{m|}A_{x}}
{{}_{z|}\ddot{a}_{x:q\!\rceil}^{(h)}-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}{{}_{z|}\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}
\cdot\left ({}_{z}E_{x}-{}_{z+q}E_{x}\right )-(IA)_{x:m\!\rceil}^{1}
+\frac{h-1}{2h}\cdot\left (v\cdot\ddot{a}_{x:m\!\rceil}-a_{x:m\!\rceil}\right )}
\end{align*}
及
\begin{align*}
{}_{m|}P_{x,q}^{\Lambda (hh)} & =\frac{\Lambda\cdot {}_{m|}A_{x}}{\ddot{a}_{x:q\!\rceil}^{(h)}-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}
{\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}\cdot\left (1-{}_{q}E_{x}\right )-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot
\left (v\cdot\ddot{a}_{x:m\!\rceil}-a_{x:m\!\rceil}\right )}\mbox{。}
\end{align*}

即刻給付.

採用身故即刻給付保險金方式。

延期繳費且繳費至終身.

考慮身故當年末退還已繳之保險費，現年 \(x\)歲的被保險人，其真實保險費以符號 \({}_{z|}P^{\Lambda (h)}({}_{m|}\bar{A}_{x})\)表示之，若 \(z=0\)，則簡單表為 \(P^{\Lambda (h)}({}_{m|}\bar{A}_{x})\)。因此每期應繳納之保費為 \({}_{z|}P^{\Lambda (h)}({}_{m|}\bar{A}_{x})/h\)。依據收支平衡原則，可得
\[{}_{z|}P^{\Lambda (h)}({}_{m|}\bar{A}_{x})\cdot {}_{z|}\ddot{a}_{x}^{(h)}
={}_{z|}P^{\Lambda (h)}({}_{m|}\bar{A}_{x})\cdot (IA)_{x:m\!\rceil}^{1}+\Lambda\cdot {}_{m|}\bar{A}_{x}\mbox{。}\]
因此可解得
\begin{align*}
{}_{z|}P^{\Lambda (h)}({}_{m|}\bar{A}_{x}) & =\frac{\Lambda\cdot {}_{m|}\bar{A}_{x}}
{{}_{z|}\ddot{a}_{x}^{(h)}-(IA)_{x:m\!\rceil}^{1}}=\frac{\Lambda\cdot\left [\left (1+\frac{\delta}{2}\right )\cdot
{}_{m}E_{x}-\delta\cdot {}_{m|}\ddot{a}_{x}\right ]}{{}_{z|}\ddot{a}_{x}-\frac{h-1}{2h}\cdot {}_{z}E_{x}-(IA)_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda}{\frac{h+1}{2h}\cdot {}_{z|}\ddot{a}_{x}+\frac{h-1}{2h}\cdot {}_{z|}a_{x}-(IA)_{x:m\!\rceil}^{1}}\cdot
\left [\left (1-\frac{\delta}{2}\right )\cdot {}_{m|}\ddot{a}_{x}-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]
\end{align*}
及
\[P^{\Lambda (h)}({}_{m|}\bar{A}_{x})=\frac{\Lambda\cdot {}_{m|}\bar{A}_{x}}{\ddot{a}_{x}^{(h)}-(IA)_{x:m\!\rceil}^{1}}
=\frac{\Lambda}{\ddot{a}_{x}-\frac{h-1}{2h}-(IA)_{x:m\!\rceil}^{1}}\cdot\left [\left (1-\frac{\delta}{2}\right )\cdot {}_{m|}\ddot{a}_{x}-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]\mbox{。}\]

考慮身故即刻退還已繳之保險費，其真實保險費以符號 \({}_{z}{}_{|}^{*}P^{\Lambda (h)}({}_{m|}\bar{A}_{x})\)表示之，若 \(z=0\)，則簡單表為 \({}^{*}P^{\Lambda (h)}({}_{m|}\bar{A}_{x})\)。因此每期應繳納之保費為 \({}_{z}{}_{|}^{*}P^{\Lambda (h)}({}_{m|}\bar{A}_{x})/h\)。依據收支平衡原則，可解得
\[{}_{z}{}_{|}^{*}P^{\Lambda (h)}({}_{m|}\bar{A}_{x})=\frac{\Lambda}{\frac{h+1}{2h}\cdot {}_{z|}\ddot{a}_{x}
+\frac{h-1}{2h}\cdot {}_{z|}a_{x}-(I\bar{A})_{x:m\!\rceil}^{1}}\cdot\left [\left (1-\frac{\delta}{2}\right )\cdot {}_{m|}\ddot{a}_{x}
-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]\]
及
\[{}^{*}P^{\Lambda (h)}({}_{m|}\bar{A}_{x})=\frac{\Lambda}{\ddot{a}_{x}-\frac{h-1}{2h}-(I\bar{A})_{x:m\!\rceil}^{1}}\cdot
\left [\left (1-\frac{\delta}{2}\right )\cdot {}_{m|}\ddot{a}_{x}-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]\mbox{。}\]

考慮身故當期末退還已繳之保險費，其真實保險費以符號 \({}_{z}{}_{|}^{*}P^{\Lambda (hh)}({}_{m|}\bar{A}_{x})\)表示，若 \(z=0\)，則簡單表為 \({}^{*}P^{\Lambda (hh)}({}_{m|}\bar{A}_{x})\)。因此每期應繳之保險費為 \({}_{z}{}_{|}^{*}P^{\Lambda (hh)}({}_{m|}\bar{A}_{x})/h\)。若被保險人於 \(m\)年內身故時，其保險費退還方式類似每年遞增 \(h\)次之 \(m\)年定期壽險。依據收支平衡原則，可得
\[{}_{z}{}_{|}^{*}P^{\Lambda (hh)}({}_{m|}\bar{A}_{x})\cdot{}_{z|}\ddot{a}_{x}^{(h)}
={}_{z}{}_{|}^{*}P^{\Lambda (hh)}({}_{m|}\bar{A}_{x})\cdot
(I^{(h)}A)_{x:m\!\rceil}^{1}+\Lambda\cdot {}_{m|}\bar{A}_{x}\mbox{。}\]
由每年遞增數次之人壽保險金頁之(31)式，亦可得
\[{}_{z}{}_{|}^{*}P^{\Lambda (hh)}({}_{m|}\bar{A}_{x})\cdot
{}_{z|}\ddot{a}_{x}^{(h)}
={}_{z}{}_{|}^{*}P^{\Lambda (hh)}({}_{m|}\bar{A}_{x})\cdot
\left ((IA)_{x:m\!\rceil}^{1}-\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}\right )+\Lambda\cdot {}_{m|}\bar{A}_{x}\mbox{。}\]
因此可解得
\begin{align*}
{}_{z}{}_{|}^{*}P^{\Lambda (hh)}({}_{m|}\bar{A}_{x}) & =\frac{\Lambda\cdot {}_{m|}\bar{A}_{x}}
{{}_{z|}\ddot{a}_{x}^{(h)}-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left [\left (1-\frac{\delta}{2}\right )\cdot
{}_{m|}\ddot{a}_{x}-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]}
{\frac{h+1}{2h}\cdot {}_{z|}\ddot{a}_{x}+\frac{h-1}{2h}\cdot {}_{z|}a_{x}-(IA)_{x:m\!\rceil}^{1}
+\frac{h-1}{2h}\cdot\left (v\cdot\ddot{a}_{x:m\!\rceil}-a_{x:m\!\rceil}\right )}
\end{align*}
及
\begin{align*}
{}^{*}P^{\Lambda (hh)}({}_{m|}\bar{A}_{x}) & =\frac{\Lambda\cdot {}_{m|}\bar{A}_{x}}
{\ddot{a}_{x}^{(h)}-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left [\left (1-\frac{\delta}{2}\right )\cdot
{}_{m|}\ddot{a}_{x}-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]}
{\ddot{a}_{x}-\frac{h-1}{2h}-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot
\left (v\cdot\ddot{a}_{x:m\!\rceil}-a_{x:m\!\rceil}\right )}\mbox{。}
\end{align*}

延期繳費且繳費期間為q年

考慮身故當年末退還已繳之保險費，現年 \(x\)歲的被保險人，其真實保險費以符號 \({}_{z|}P_{q}^{\Lambda (h)}({}_{m|}\bar{A}_{x})\)表示之，若 \(z=0\)，則簡單表為 \(P_{q}^{\Lambda (h)}({}_{m|}\bar{A}_{x})\)。因此每期應繳納之保費為 \({}_{z|}P_{q}^{\Lambda (h)}({}_{m|}\bar{A}_{x})/h\)。依據收支平衡原則，可得
\[{}_{z|}P_{q}^{\Lambda (h)}({}_{m|}\bar{A}_{x})\cdot{}_{z|}\ddot{a}_{x:q\!\rceil}^{(h)}
={}_{z|}P_{q}^{\Lambda (h)}({}_{m|}\bar{A}_{x})\cdot (IA)_{x:m\!\rceil}^{1}+\Lambda\cdot {}_{m|}\bar{A}_{x}\mbox{。}\]
因此可解得
\[{}_{z|}P_{q}^{\Lambda (h)}({}_{m|}\bar{A}_{x})=\frac{\Lambda\cdot {}_{m|}\bar{A}_{x}}
{{}_{z|}\ddot{a}_{x:q\!\rceil}^{(h)}-(IA)_{x:m\!\rceil}^{1}}=\frac{\Lambda\cdot\left [\left (1+\frac{\delta}{2}\right )\cdot
{}_{m}E_{x}-\delta\cdot {}_{m|}\ddot{a}_{x}\right ]}{{}_{z|}\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}
\cdot\left ({}_{z}E_{x}-{}_{z+q}E_{x}\right )-(IA)_{x:m\!\rceil}^{1}}\]
及
\[P_{q}^{\Lambda (h)}({}_{m|}\bar{A}_{x})=\frac{\Lambda\cdot {}_{m|}\bar{A}_{x}}
{\ddot{a}_{x:q\!\rceil}^{(h)}-(IA)_{x:m\!\rceil}^{1}}=\frac{\Lambda\cdot\left [\left (1+\frac{\delta}{2}\right )\cdot
{}_{m}E_{x}-\delta\cdot {}_{m|}\ddot{a}_{x}\right ]}{\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}\cdot\left (1-{}_{q}E_{x}\right )
-(IA)_{x:m\!\rceil}^{1}}\mbox{。}\]

考慮身故即刻退還已繳之保險費，其真實保險費以符號 \({}_{z}{}_{|}^{*}P_{q}^{\Lambda (h)}({}_{m|}\bar{A}_{x})\)表示之，
若 \(z=0\)，則簡單表為 \({}^{*}P_{q}^{\Lambda (h)}({}_{m|}\bar{A}_{x})\)。因此每期應繳納之保費為 \({}_{z}{}_{|}^{*}P_{q}^{\Lambda (h)}({}_{m|}\bar{A}_{x})/h\)。依據收支平衡原則，可解得
\[{}_{z}{}_{|}^{*}P_{q}^{\Lambda (h)}({}_{m|}\bar{A}_{x})=\frac{\Lambda\cdot\left [\left (1+\frac{\delta}{2}\right )\cdot
{}_{m}E_{x}-\delta\cdot {}_{m|}\ddot{a}_{x}\right ]}{{}_{z|}\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}
\cdot\left ({}_{z}E_{x}-{}_{z+q}E_{x}\right )-(I\bar{A})_{x:m\!\rceil}^{1}}\]
及
\[{}^{*}P_{q}^{\Lambda (h)}({}_{m|}\bar{A}_{x})=\frac{\Lambda\cdot\left [\left (1+\frac{\delta}{2}\right )\cdot
{}_{m}E_{x}-\delta\cdot {}_{m|}\ddot{a}_{x}\right ]}
{\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}\cdot\left (1-{}_{q}E_{x}\right )-(I\bar{A})_{x:m\!\rceil}^{1}}\mbox{。}\]

考慮身故當期末退還已繳之保險費，其真實保險費以符號 \({}_{z}{}_{|}^{*}P_{q}^{\Lambda (hh)}({}_{m|}\bar{A}_{x})\)表示，
若 \(z=0\)，則簡單表為 \({}^{*}P_{q}^{\Lambda (hh)}({}_{m|}\bar{A}_{x})\)。因此每期應繳之保險費為 \({}_{z}{}_{|}^{*}P_{q}^{\Lambda (hh)}({}_{m|}\bar{A}_{x})/h\)。若被保險人於 \(m\)年內身故時，其保險費退還方式類似每年遞增 \(h\)次之 \(m\)年定期壽險。依據收支平衡原則，可得
\[{}_{z}{}_{|}^{*}P_{q}^{\Lambda (hh)}({}_{m|}\bar{A}_{x})\cdot{}_{z|}\ddot{a}_{x:q\!\rceil}^{(h)}
={}_{z}{}_{|}^{*}P_{q}^{\Lambda (hh)}({}_{m|}\bar{A}_{x})\cdot
(I^{(h)}A)_{x:m\!\rceil}^{1}+\Lambda\cdot {}_{m|}\bar{A}_{x}\mbox{。}\]
由每年遞增數次之人壽保險金頁之(31)式，亦可得
\[{}_{z}{}_{|}^{*}P_{q}^{\Lambda (hh)}({}_{m|}\bar{A}_{x})\cdot{}_{z|}\ddot{a}_{x:q\!\rceil}^{(h)}
={}_{z}{}_{|}^{*}P_{q}^{\Lambda (hh)}({}_{m|}\bar{A}_{x})\cdot
\left ((IA)_{x:m\!\rceil}^{1}-\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}\right )+\Lambda\cdot {}_{m|}\bar{A}_{x}\mbox{。}\]
因此可解得
\begin{align*}
{}_{z}{}_{|}^{*}P_{q}^{\Lambda (hh)}({}_{m|}\bar{A}_{x}) & =\frac{\Lambda\cdot {}_{m|}\bar{A}_{x}}
{{}_{z|}\ddot{a}_{x:q\!\rceil}^{(h)}-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left [\left (1+\frac{\delta}{2}\right )\cdot{}_{m}E_{x}-\delta\cdot {}_{m|}\ddot{a}_{x}\right ]}
{{}_{z|}\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}\cdot\left ({}_{z}E_{x}-{}_{z+q}E_{x}\right )-(IA)_{x:m\!\rceil}^{1}
+\frac{h-1}{2h}\cdot\left (v\cdot\ddot{a}_{x:m\!\rceil}-a_{x:m\!\rceil}\right )}
\end{align*}
及
\begin{align*}
{}^{*}P_{q}^{\Lambda (hh)}({}_{m|}\bar{A}_{x}) & =\frac{\Lambda\cdot {}_{m|}\bar{A}_{x}}
{\ddot{a}_{x}^{(h)}-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left [\left (1+\frac{\delta}{2}\right )\cdot {}_{m}E_{x}-\delta\cdot {}_{m|}\ddot{a}_{x}\right ]}
{\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}\cdot\left (1-{}_{q}E_{x}\right )-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot
\left (v\cdot\ddot{a}_{x:m\!\rceil}-a_{x:m\!\rceil}\right )}\mbox{。}
\end{align*}

\begin{equation}{\label{c}}\tag{C}\mbox{}\end{equation}

年賦保險費.

假設將每年分成 \(h\)期，可分為年末給付與即刻給付保險金方式。又可分為延期 \(z\)年繳費且繳費至終身與延期$z$年繳費且繳費期間為 \(q\)年，其中 \(z\leq m\)。將依身故當年末退還已繳之保險費、身故當年期末退還已繳之保險費與身故即刻退還已繳之保險費分別討論。

年末給付.

採用身故當年末給付保險金方式。

延期繳費且繳費至終身.

考慮身故當年末退還已繳之保險費，其年賦保險費以符號 \({}_{m,z|}P_{x}^{\Lambda [h]}\)表示，若 \(z=0\)，則簡單表為 \({}_{m|}P_{x}^{\Lambda [h]}\)。因此每期應繳之保險費為 \({}_{m,z|}P_{x}^{\Lambda [h]}/h\)。依據收支平衡原則，可得
\[{}_{m,z|}P_{x}^{\Lambda [h]}\cdot{}_{z|}\ddot{a}_{x}^{(h)}={}_{m,z|}P_{x}^{\Lambda [h]}\cdot (IA)_{x:m\!\rceil}^{1}
+\Lambda\cdot {}_{m|}A_{x}\cdot\left (1-\frac{h-1}{2h}\cdot {}_{m,z|}P_{x}^{\Lambda [h]}\right )\mbox{。}\]
因此可解得
\begin{align*}
{}_{m,z|}P_{x}^{\Lambda [h]} & =\frac{\Lambda\cdot {}_{m|}A_{x}}{{}_{z|}\ddot{a}_{x}^{(h)}
+\frac{h-1}{2h}\cdot \Lambda\cdot {}_{m|}A_{x}-(IA)_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot {}_{m|}A_{x}}{{}_{z|}\ddot{a}_{x}-\frac{h-1}{2h}
\cdot {}_{z}E_{x}+\frac{h-1}{2h}\cdot \Lambda\cdot {}_{m|}A_{x}-(IA)_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}
{\frac{h+1}{2h}\cdot {}_{z|}\ddot{a}_{x}+\frac{h-1}{2h}\cdot {}_{z|}a_{x}
+\frac{h-1}{2h}\cdot\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )-(IA)_{x:m\!\rceil}^{1}}
\end{align*}
及
\begin{align*}
{}_{m|}P_{x}^{\Lambda [h]} & =\frac{\Lambda\cdot {}_{m|}A_{x}}{\ddot{a}_{x}^{(h)}
+\frac{h-1}{2h}\cdot \Lambda\cdot {}_{m|}A_{x}-(IA)_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}{\ddot{a}_{x}-\frac{h-1}{2h}
+\frac{h-1}{2h}\cdot\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )-(IA)_{x:m\!\rceil}^{1}}\mbox{。}
\end{align*}

考慮身故即刻退還已繳之保險費，其年賦保險費以符號 \({}_{m,z}{}_{|}^{*}P_{x}^{\Lambda [h]}\)表示，若 \(z=0\)，則簡單表為 \({}_{m}{}_{|}^{*}P_{x}^{\Lambda [h]}\)。因此每期應繳之保險費為 \({}_{m,z}{}_{|}^{*}P_{x}^{\Lambda [h]}/h\)。依據收支平衡原則，可解得
\[{}_{m,z}{}_{|}^{*}P_{x}^{\Lambda [h]}=\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}
{\frac{h+1}{2h}\cdot {}_{z|}\ddot{a}_{x}+\frac{h-1}{2h}\cdot {}_{z|}a_{x}
+\frac{h-1}{2h}\cdot\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )-(I\bar{A})_{x:m\!\rceil}^{1}}\]
及
\[{}_{m}{}_{|}^{*}P_{x}^{\Lambda [h]}=\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}
{\ddot{a}_{x}-\frac{h-1}{2h}+\frac{h-1}{2h}\cdot\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )
-(I\bar{A})_{x:m\!\rceil}^{1}}\mbox{。}\]

考慮身故當期末退還已繳之保險費，其年賦保險費以符號 \({}_{m,z|}P_{x}^{\Lambda [hh]}\)表示，若 \(z=0\)，則簡單表為 \({}_{m|}P_{x}^{\Lambda [hh]}\)。因此每期應繳之保險費為 \({}_{m,z|}P_{x}^{\Lambda [hh]}/h\)。若被保險人於 \(m\)年內身故時，其保險費退還方式類似每年遞增 \(h\)次之 \(m\)年定期壽險。依據收支平衡原則，可得
\[{}_{m,z|}P_{x}^{\Lambda [hh]}\cdot{}_{z|}\ddot{a}_{x}^{(h)}
={}_{m,z|}P_{x}^{\Lambda [hh]}\cdot (I^{(h)}A)_{x:m\!\rceil}^{1}
+\Lambda\cdot {}_{m|}A_{x}\cdot\left (1-\frac{h-1}{2h}\cdot{}_{m,z|}P_{x}^{\Lambda [hh]}\right )\mbox{。}\]
由每年遞增數次之人壽保險金頁之(31)式，亦可得
\[{}_{m,z|}P_{x}^{\Lambda [hh]}\cdot{}_{z|}\ddot{a}_{x}^{(h)}={}_{m,z|}P_{x}^{\Lambda [hh]}\cdot
\left ((IA)_{x:m\!\rceil}^{1}-\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}\right )
+\Lambda\cdot {}_{m|}A_{x}\cdot\left (1-\frac{h-1}{2h}\cdot{}_{m,z|}P_{x}^{\Lambda [hh]}\right )\mbox{。}\]
因此可解得
\begin{align*}
{}_{m,z|}P_{x}^{\Lambda [hh]} & =\frac{\Lambda\cdot {}_{m|}A_{x}}
{{}_{z|}\ddot{a}_{x}^{(h)}+\frac{h-1}{2h}\cdot\Lambda\cdot {}_{m|}A_{x}
-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}
{\frac{h+1}{2h}\cdot {}_{z|}\ddot{a}_{x}+\frac{h-1}{2h}\cdot {}_{z|}a_{x}
+\frac{h-1}{2h}\cdot\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )-(IA)_{x:m\!\rceil}^{1}
+\frac{h-1}{2h}\cdot\left (v\cdot\ddot{a}_{x:m\!\rceil}-a_{x:m\!\rceil}\right )}
\end{align*}
及
\begin{align*}
{}_{m|}P_{x}^{\Lambda [hh]} & =\frac{\Lambda\cdot {}_{m|}A_{x}}
{\ddot{a}_{x}^{(h)}+\frac{h-1}{2h}\cdot\Lambda\cdot {}_{m|}A_{x}
-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}{\ddot{a}_{x}-\frac{h-1}{2h}
+\frac{h-1}{2h}\cdot\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )
-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot\left (v\cdot\ddot{a}_{x:m\!\rceil}-a_{x:m\!\rceil}\right )}\mbox{。}
\end{align*}

延期繳費且繳費期間為q年.

考慮身故當年末退還已繳之保險費，其年賦保險費以符號 \({}_{m,z|}P_{x,q}^{\Lambda [h]}\)表示，若 \(z=0\)，則簡單表為 \({}_{m|}P_{x,q}^{\Lambda [h]}\)。因此每期應繳之保險費為 \({}_{m,z|}P_{x,q}^{\Lambda [h]}/h\)。依據收支平衡原則，可得
\[{}_{m,z|}P_{x,q}^{\Lambda [h]}\cdot {}_{z|}\ddot{a}_{x:q\!\rceil}^{(h)}
={}_{m,z|}P_{x,q}^{\Lambda [h]}\cdot (IA)_{x:m\!\rceil}^{1}
+\Lambda\cdot {}_{m|}A_{x}\cdot\left (1-\frac{h-1}{2h}\cdot{}_{m,z|}P_{x,q}^{\Lambda [h]}\right )\mbox{。}\]
因此可解得
\begin{align*}
{}_{m,z|}P_{x,q}^{\Lambda [h]} & =\frac{\Lambda\cdot {}_{m|}A_{x}}{{}_{z|}\ddot{a}_{x:q\!\rceil}^{(h)}
+\frac{h-1}{2h}\cdot\Lambda\cdot {}_{m|}A_{x}-(IA)_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot {}_{m|}A_{x}}{{}_{z|}\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}\cdot\left ({}_{z}E_{x}-{}_{z+q}E_{x}\right )
+\frac{h-1}{2h}\cdot\Lambda\cdot {}_{m|}A_{x}-(IA)_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}{{}_{z|}\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}
\cdot\left ({}_{z}E_{x}-{}_{z+q}E_{x}\right )
+\frac{h-1}{2h}\cdot\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )-(IA)_{x:m\!\rceil}^{1}}
\end{align*}
及
\begin{align*}
{}_{m|}P_{x,q}^{\Lambda [h]} & =\frac{\Lambda\cdot {}_{m|}A_{x}}{\ddot{a}_{x:q\!\rceil}^{(h)}
+\frac{h-1}{2h}\cdot\Lambda\cdot {}_{m|}A_{x}-(IA)_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}{\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}
\cdot\left (1-{}_{q}E_{x}\right )
+\frac{h-1}{2h}\cdot\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )-(IA)_{x:m\!\rceil}^{1}}\mbox{。}
\end{align*}

考慮身故即刻退還已繳之保險費，其年賦保險費以符號 \({}_{m,z}{}_{|}^{*}P_{x,q}^{\Lambda [h]}\)表示，若 \(z=0\)，則簡單表為 \({}_{m}{}_{|}^{*}P_{x,q}^{\Lambda [h]}\)。因此每期應繳之保險費為 \({}_{m,z}{}_{|}^{*}P_{x,q}^{\Lambda [h]}/h\)。依據收支平衡原則，可解得
\[{}_{m,z}{}_{|}^{*}P_{x,q}^{\Lambda [h]}=\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}
{{}_{z|}\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}\cdot\left ({}_{z}E_{x}-{}_{z+q}E_{x}\right )
+\frac{h-1}{2h}\cdot\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )-(I\bar{A})_{x:m\!\rceil}^{1}}\]
及
\[{}_{m}{}_{|}^{*}P_{x,q}^{\Lambda [h]}=\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}
{\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}\cdot\left (1-{}_{q}E_{x}\right )
+\frac{h-1}{2h}\cdot\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )-(I\bar{A})_{x:m\!\rceil}^{1}}\mbox{。}\]

考慮身故當期末退還已繳之保險費，其年賦保險費以符號 \({}_{m,z|}P_{x,q}^{\Lambda [hh]}\)表示，若 \(z=0\)，則簡單表為 \({}_{m|}P_{x,q}^{\Lambda [hh]}\)。因此每期應繳之保險費為 \({}_{m,z|}P_{x,q}^{\Lambda [hh]}/h\)。若被保險人於$m$年內身故時，其保險費退還方式類似每年遞增$h$次之$m$年定期壽險。依據收支平衡原則，可得
\[{}_{m,z|}P_{x,q}^{\Lambda [hh]}\cdot{}_{z|}\ddot{a}_{x:q\!\rceil}^{(h)}
={}_{m,z|}P_{x,q}^{\Lambda [hh]}\cdot (I^{(h)}A)_{x:m\!\rceil}^{1}
+\Lambda\cdot {}_{m|}A_{x}\cdot\left (1-\frac{h-1}{2h}\cdot{}_{m,z|}P_{x,q}^{\Lambda [hh]}\right )\mbox{。}\]
由每年遞增數次之人壽保險金頁之(31)式，亦可得
\[{}_{m,z|}P_{x,q}^{\Lambda [hh]}\cdot{}_{z|}\ddot{a}_{x:q\!\rceil}^{(h)}={}_{m,z|}P_{x,q}^{\Lambda [hh]}\cdot
\left ((IA)_{x:m\!\rceil}^{1}-\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}\right )+\Lambda\cdot {}_{m|}A_{x}\cdot\left (1-\frac{h-1}{2h}\cdot{}_{m,z|}P_{x,q}^{\Lambda [hh]}\right )\mbox{。}\]
因此可解得
\begin{align*}
{}_{m,z|}P_{x,q}^{\Lambda [hh]} & =\frac{\Lambda\cdot {}_{m|}A_{x}}{{}_{z|}\ddot{a}_{x:q\!\rceil}^{(h)}
+\frac{h-1}{2h}\cdot\Lambda\cdot {}_{m|}A_{x}-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}{{}_{z|}\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}
\cdot\left ({}_{z}E_{x}-{}_{z+q}E_{x}\right )+\frac{h-1}{2h}\cdot\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )
-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot\left (v\cdot\ddot{a}_{x:m\!\rceil}-a_{x:m\!\rceil}\right )}
\end{align*}
及
\begin{align*}
{}_{m|}P_{x,q}^{\Lambda [hh]} & =\frac{\Lambda\cdot {}_{m|}A_{x}}
{\ddot{a}_{x:q\!\rceil}^{(h)}+\frac{h-1}{2h}\cdot\Lambda\cdot {}_{m|}A_{x}
-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}
{\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}\cdot\left (1-{}_{q}E_{x}\right )
+\frac{h-1}{2h}\cdot\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )
-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot\left (v\cdot\ddot{a}_{x:m\!\rceil}-a_{x:m\!\rceil}\right )}\mbox{。}
\end{align*}

即刻給付.

採用身故即刻給付保險金方式。

延期繳費且繳費至終身.

考慮身故當年末退還已繳之保險費，現年 \(x\)歲的被保險人，其年賦保險費以符號 \({}_{z|}P^{\Lambda [h]}({}_{m|}\bar{A}_{x})\)表示之，若 \(z=0\)，則簡單表為 \(P^{\Lambda [h]}({}_{m|}\bar{A}_{x})\)。因此每期應繳納之保費為 \({}_{z|}P^{\Lambda [h]}({}_{m|}\bar{A}_{x})/h\)。依據收支平衡原則，可得
\[{}_{z|}P^{\Lambda [h]}({}_{m|}\bar{A}_{x})\cdot {}_{z|}\ddot{a}_{x}^{(h)}
={}_{z|}P^{\Lambda [h]}({}_{m|}\bar{A}_{x})\cdot (IA)_{x:m\!\rceil}^{1}
+\Lambda\cdot {}_{m|}\bar{A}_{x}\cdot\left (1-\frac{h-1}{2h}\cdot
{}_{z|}P^{\Lambda [h]}({}_{m|}\bar{A}_{x})\right )\mbox{。}\]
因此可解得
\begin{align*}
{}_{z|}P^{\Lambda [h]}({}_{m|}\bar{A}_{x}) & =\frac{\Lambda\cdot {}_{m|}\bar{A}_{x}}
{{}_{z|}\ddot{a}_{x}^{(h)}+\frac{h-1}{2h}\cdot\Lambda\cdot {}_{m|}\bar{A}_{x}-(IA)_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left [\left (1-\frac{\delta}{2}\right )\cdot {}_{m|}\ddot{a}_{x}
-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]}
{\frac{h+1}{2h}\cdot {}_{z|}\ddot{a}_{x}+\frac{h-1}{2h}\cdot {}_{z|}a_{x}+\frac{h-1}{2h}\cdot\Lambda\cdot
\left [\left (1-\frac{\delta}{2}\right )\cdot {}_{m|}\ddot{a}_{x}
-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]-(IA)_{x:m\!\rceil}^{1}}
\end{align*}
及
\begin{align*}
P^{\Lambda [h]}({}_{m|}\bar{A}_{x}) & =\frac{\Lambda\cdot {}_{m|}\bar{A}_{x}}{\ddot{a}_{x}^{(h)}
+\frac{h-1}{2h}\cdot\Lambda\cdot {}_{m|}\bar{A}_{x}-(IA)_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left [\left (1-\frac{\delta}{2}\right )\cdot {}_{m|}\ddot{a}_{x}
-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]}
{\ddot{a}_{x}-\frac{h-1}{2h}+\frac{h-1}{2h}\cdot\Lambda\cdot
\left [\left (1-\frac{\delta}{2}\right )\cdot {}_{m|}\ddot{a}_{x}
-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]-(IA)_{x:m\!\rceil}^{1}}\mbox{。}
\end{align*}

考慮身故即刻退還已繳之保險費，其年賦保險費以符號 \({}_{z}{}_{|}^{*}P^{\Lambda [h]}({}_{m|}\bar{A}_{x})\)表示之，若 \(z=0\)，則簡單表為 \({}^{*}P^{\Lambda [h]}({}_{m|}\bar{A}_{x})\)。因此每期應繳納之保費為 \({}_{z}{}_{|}^{*}P^{\Lambda [h]}({}_{m|}\bar{A}_{x})/h\)。依據收支平衡原則，可解得
\[{}_{z}{}_{|}^{*}P^{\Lambda [h]}({}_{m|}\bar{A}_{x})=\frac{\Lambda\cdot
\left [\left (1-\frac{\delta}{2}\right )\cdot {}_{m|}\ddot{a}_{x}-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]}
{\frac{h+1}{2h}\cdot {}_{z|}\ddot{a}_{x}+\frac{h-1}{2h}\cdot {}_{z|}a_{x}+\frac{h-1}{2h}\cdot\Lambda\cdot
\left [\left (1-\frac{\delta}{2}\right )\cdot {}_{m|}\ddot{a}_{x}
-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]-(I\bar{A})_{x:m\!\rceil}^{1}}\]
及
\[{}^{*}P^{\Lambda [h]}({}_{m|}\bar{A}_{x})=\frac{\Lambda\cdot\left [\left (1-\frac{\delta}{2}\right )\cdot {}_{m|}\ddot{a}_{x}
-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]}{\ddot{a}_{x}-\frac{h-1}{2h}+\frac{h-1}{2h}\cdot\Lambda\cdot
\left [\left (1-\frac{\delta}{2}\right )\cdot {}_{m|}\ddot{a}_{x}
-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]-(I\bar{A})_{x:m\!\rceil}^{1}}\mbox{。}\]

考慮身故當期末退還已繳之保險費，其年賦保險費以符號 \({}_{z}{}_{|}^{*}P^{\Lambda [hh]}({}_{m|}\bar{A}_{x})\)表示，若 \(z=0\)，則簡單表為 \({}^{*}P^{\Lambda [hh]}({}_{m|}\bar{A}_{x})\)。因此每期應繳之保險費為 \({}_{z}{}_{|}^{*}P^{\Lambda [hh]}({}_{m|}\bar{A}_{x})/h\)。若被保險人於 \(m\)年內身故時，其保險費退還方式類似每年遞增 \(h\)次之 \(m\)年定期壽險。依據收支平衡原則，可得
\[{}_{z}{}_{|}^{*}P^{\Lambda [hh]}({}_{m|}\bar{A}_{x})\cdot{}_{z|}\ddot{a}_{x}^{(h)}
={}_{z}{}_{|}^{*}P^{\Lambda [hh]}({}_{m|}\bar{A}_{x})\cdot
(I^{(h)}A)_{x:m\!\rceil}^{1}+\Lambda\cdot {}_{m|}\bar{A}_{x}\cdot\left (1-\frac{h-1}{2h}\cdot
{}_{z|}P^{\Lambda [hh]}({}_{m|}\bar{A}_{x})\right )\mbox{。}\]
由每年遞增數次之人壽保險金頁之(31)式，亦可得
\[{}_{z}{}_{|}^{*}P^{\Lambda [hh]}({}_{m|}\bar{A}_{x})\cdot {}_{z|}\ddot{a}_{x}^{(h)}
={}_{z}{}_{|}^{*}P^{\Lambda [hh]}({}_{m|}\bar{A}_{x})\cdot
\left ((IA)_{x:m\!\rceil}^{1}-\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}\right )+\Lambda\cdot {}_{m|}\bar{A}_{x}\cdot\left (1-\frac{h-1}{2h}\cdot{}_{z|}P^{\Lambda [hh]}({}_{m|}\bar{A}_{x})\right )\mbox{。}\]
因此可解得
\begin{align*}
{}_{z}{}_{|}^{*}P^{\Lambda [hh]}({}_{m|}\bar{A}_{x}) & =\frac{\Lambda\cdot {}_{m|}\bar{A}_{x}}
{{}_{z|}\ddot{a}_{x}^{(h)}+\frac{h-1}{2h}\cdot\Lambda\cdot {}_{m|}\bar{A}_{x}
-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left [\left (1-\frac{\delta}{2}\right )\cdot
{}_{m|}\ddot{a}_{x}-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]}{\frac{h+1}{2h}\cdot {}_{z|}\ddot{a}_{x}
+\frac{h-1}{2h}\cdot {}_{z|}a_{x}+\frac{h-1}{2h}\cdot\Lambda\cdot\left [\left (1-\frac{\delta}{2}\right )\cdot
{}_{m|}\ddot{a}_{x}-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]-(IA)_{x:m\!\rceil}^{1}
+\frac{h-1}{2h}\cdot\left (v\cdot\ddot{a}_{x:m\!\rceil}-a_{x:m\!\rceil}\right )}
\end{align*}
及
\begin{align*}
{}^{*}P^{\Lambda [hh]}({}_{m|}\bar{A}_{x}) & =\frac{\Lambda\cdot {}_{m|}\bar{A}_{x}}
{\ddot{a}_{x}^{(h)}+\frac{h-1}{2h}\cdot\Lambda\cdot {}_{m|}\bar{A}_{x}
-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left [\left (1-\frac{\delta}{2}\right )\cdot
{}_{m|}\ddot{a}_{x}-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]}
{\ddot{a}_{x}-\frac{h-1}{2h}+\frac{h-1}{2h}\cdot\Lambda\cdot\left [\left (1-\frac{\delta}{2}\right )\cdot
{}_{m|}\ddot{a}_{x}-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]
-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot\left (v\cdot\ddot{a}_{x:m\!\rceil}-a_{x:m\!\rceil}\right )}\mbox{。}
\end{align*}

延期繳費且繳費期間為q年.

考慮身故當年末退還已繳之保險費，現年 \(x\)歲的被保險人，其年賦保險費以符號 \({}_{z|}P_{q}^{\Lambda [h]}({}_{m|}\bar{A}_{x})\)表示之，若 \(z=0\)，則簡單表為 \(P_{q}^{\Lambda [h]}({}_{m|}\bar{A}_{x})\)。因此每期應繳納之保費為 \({}_{z|}P_{q}^{\Lambda [h]}({}_{m|}\bar{A}_{x})/h\)。依據收支平衡原則，可得
\[{}_{z|}P_{q}^{\Lambda [h]}({}_{m|}\bar{A}_{x})\cdot{}_{z|}\ddot{a}_{x:q\!\rceil}^{(h)}
={}_{z|}P_{q}^{\Lambda [h]}({}_{m|}\bar{A}_{x})\cdot (IA)_{x:m\!\rceil}^{1}
+\Lambda\cdot {}_{m|}\bar{A}_{x}\cdot\left (1-\frac{h-1}{2h}\cdot
{}_{z|}P_{q}^{\Lambda [h]}({}_{m|}\bar{A}_{x})\right )\mbox{。}\]
因此可解得
\begin{align*}
{}_{z|}P_{q}^{\Lambda [h]}({}_{m|}\bar{A}_{x}) & =\frac{\Lambda\cdot {}_{m|}\bar{A}_{x}}{{}_{z|}\ddot{a}_{x:q\!\rceil}^{(h)}
+\frac{h-1}{2h}\cdot\Lambda\cdot {}_{m|}\bar{A}_{x}-(IA)_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left [\left (1+\frac{\delta}{2}\right )\cdot{}_{m}E_{x}-\delta\cdot {}_{m|}\ddot{a}_{x}\right ]}
{{}_{z|}\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}\cdot\left ({}_{z}E_{x}-{}_{z+q}E_{x}\right )
+\frac{h-1}{2h}\cdot\Lambda\cdot\left [\left (1+\frac{\delta}{2}\right )\cdot
{}_{m}E_{x}-\delta\cdot {}_{m|}\ddot{a}_{x}\right ]-(IA)_{x:m\!\rceil}^{1}}
\end{align*}
及
\begin{align*}
P_{q}^{\Lambda [h]}({}_{m|}\bar{A}_{x}) & =\frac{\Lambda\cdot {}_{m|}\bar{A}_{x}}
{\ddot{a}_{x:q\!\rceil}^{(h)}+\frac{h-1}{2h}\cdot\Lambda\cdot {}_{m|}\bar{A}_{x}-(IA)_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left [\left (1+\frac{\delta}{2}\right )\cdot{}_{m}E_{x}-\delta\cdot {}_{m|}\ddot{a}_{x}\right ]}
{\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}\cdot\left (1-{}_{q}E_{x}\right )
+\frac{h-1}{2h}\cdot\Lambda\cdot\left [\left (1+\frac{\delta}{2}\right )\cdot
{}_{m}E_{x}-\delta\cdot {}_{m|}\ddot{a}_{x}\right ]-(IA)_{x:m\!\rceil}^{1}}\mbox{。}
\end{align*}

考慮身故即刻退還已繳之保險費，其年賦保險費以符號 \({}_{z}{}_{|}^{*}P_{q}^{\Lambda [h]}({}_{m|}\bar{A}_{x})\)表示之，若 \(z=0\)，則簡單表為 \({}^{*}P_{q}^{\Lambda [h]}({}_{m|}\bar{A}_{x})\)。因此每期應繳納之保費為 \({}_{z}{}_{|}^{*}P_{q}^{\Lambda [h]}({}_{m|}\bar{A}_{x})/h\)。依據收支平衡原則，可解得
\[{}_{z}{}_{|}^{*}P_{q}^{\Lambda [h]}({}_{m|}\bar{A}_{x})=\frac{\Lambda\cdot\left [\left (1+\frac{\delta}{2}\right )\cdot
{}_{m}E_{x}-\delta\cdot {}_{m|}\ddot{a}_{x}\right ]}{{}_{z|}\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}
\cdot\left ({}_{z}E_{x}-{}_{z+q}E_{x}\right )+\frac{h-1}{2h}\cdot\Lambda\cdot\left [\left (1+\frac{\delta}{2}\right )\cdot
{}_{m}E_{x}-\delta\cdot {}_{m|}\ddot{a}_{x}\right ]-(I\bar{A})_{x:m\!\rceil}^{1}}\]
及
\[{}^{*}P_{q}^{\Lambda [h]}({}_{m|}\bar{A}_{x})=\frac{\Lambda\cdot\left [\left (1+\frac{\delta}{2}\right )\cdot
{}_{m}E_{x}-\delta\cdot {}_{m|}\ddot{a}_{x}\right ]}{\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}\cdot\left (1-{}_{q}E_{x}\right )
+\frac{h-1}{2h}\cdot\Lambda\cdot\left [\left (1+\frac{\delta}{2}\right )\cdot
{}_{m}E_{x}-\delta\cdot {}_{m|}\ddot{a}_{x}\right ]-(I\bar{A})_{x:m\!\rceil}^{1}}\mbox{。}\]

考慮身故當期末退還已繳之保險費，其年賦保險費以符號 \({}_{z}{}_{|}^{*}P_{q}^{\Lambda [hh]}({}_{m|}\bar{A}_{x})\)表示，若 \(z=0\)，則簡單表為 \({}^{*}P_{q}^{\Lambda [hh]}({}_{m|}\bar{A}_{x})\)。因此每期應繳之保險費為 \({}_{z}{}_{|}^{*}P_{q}^{\Lambda [hh]}({}_{m|}\bar{A}_{x})/h\)。若被保險人於 \(m\)年內身故時，其保險費退還方式類似每年遞增 \(h\)次之 \(m\)年定期壽險。依據收支平衡原則，可得
\[{}_{z}{}_{|}^{*}P_{q}^{\Lambda [hh]}({}_{m|}\bar{A}_{x})\cdot{}_{z|}\ddot{a}_{x:q\!\rceil}^{(h)}
={}_{z}{}_{|}^{*}P_{q}^{\Lambda [hh]}({}_{m|}\bar{A}_{x})\cdot
(I^{(h)}A)_{x:m\!\rceil}^{1}+\Lambda\cdot {}_{m|}\bar{A}_{x}\cdot\left (1-\frac{h-1}{2h}\cdot
{}_{z|}P_{q}^{\Lambda [hh]}({}_{m|}\bar{A}_{x})\right )\mbox{。}\]
由每年遞增數次之人壽保險金頁之(31)式，亦可得
\begin{align*}
{}_{z}{}_{|}^{*}P_{q}^{\Lambda [hh]}({}_{m|}\bar{A}_{x})\cdot {}_{z|}\ddot{a}_{x:q\!\rceil}^{(h)}
& ={}_{z}{}_{|}^{*}P_{q}^{\Lambda [hh]}({}_{m|}\bar{A}_{x})\cdot
\left ((IA)_{x:m\!\rceil}^{1}-\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}\right )\\
& \quad +\Lambda\cdot {}_{m|}\bar{A}_{x}\cdot\left (1-\frac{h-1}{2h}\cdot
{}_{z|}P_{q}^{\Lambda [hh]}({}_{m|}\bar{A}_{x})\right )\mbox{。}
\end{align*}
因此可解得
\begin{align*}
{}_{z}{}_{|}^{*}P_{q}^{\Lambda [hh]}({}_{m|}\bar{A}_{x}) & =\frac{\Lambda\cdot {}_{m|}\bar{A}_{x}}
{{}_{z|}\ddot{a}_{x:q\!\rceil}^{(h)}+\frac{h-1}{2h}\cdot\Lambda\cdot {}_{m|}\bar{A}_{x}
-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left [\left (1+\frac{\delta}{2}\right )\cdot{}_{m}E_{x}-\delta\cdot {}_{m|}\ddot{a}_{x}\right ]}
{{}_{z|}\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}\cdot\left ({}_{z}E_{x}-{}_{z+q}E_{x}\right )
+\frac{h-1}{2h}\cdot\Lambda\cdot\left [\left (1+\frac{\delta}{2}\right )\cdot
{}_{m}E_{x}-\delta\cdot {}_{m|}\ddot{a}_{x}\right ]-(IA)_{x:m\!\rceil}^{1}
+\frac{h-1}{2h}\cdot\left (v\cdot\ddot{a}_{x:m\!\rceil}-a_{x:m\!\rceil}\right )}
\end{align*}
及
\begin{align*}
{}^{*}P_{q}^{\Lambda [hh]}({}_{m|}\bar{A}_{x}) & =\frac{\Lambda\cdot {}_{m|}\bar{A}_{x}}
{\ddot{a}_{x}^{(h)}+\frac{h-1}{2h}\cdot\Lambda\cdot {}_{m|}\bar{A}_{x}
-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left [\left (1+\frac{\delta}{2}\right )\cdot
{}_{m}E_{x}-\delta\cdot {}_{m|}\ddot{a}_{x}\right ]}
{\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}\cdot\left (1-{}_{q}E_{x}\right )
+\frac{h-1}{2h}\cdot\Lambda\cdot\left [\left (1+\frac{\delta}{2}\right )\cdot
{}_{m}E_{x}-\delta\cdot {}_{m|}\ddot{a}_{x}\right ]-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot
\left (v\cdot\ddot{a}_{x:m\!\rceil}-a_{x:m\!\rceil}\right )}\mbox{。}
\end{align*}

\begin{equation}{\label{d}}\tag{D}\mbox{}\end{equation}

比例分攤保險費.

假設將每年分成 \(h\)期，可分為年末給付與即刻給付保險金方式。又可分為延期 \(z\)年繳費且繳費至終身與延期 \(z\)年繳費且繳費期間為 \(q\)年，其中 \(z\leq m\)。將依身故當年末退還已繳之保險費、身故當年期末退還已繳之保險費與身故即刻退還已繳之保險費分別討論。

年末給付.

採用身故當年末給付保險金方式。

延期繳費且繳費至終身.

考慮身故當年末退還已繳之保險費，其比例分攤保險費以符號 \({}_{m,z|}P_{x}^{\Lambda\{h\}}\)表示，若 \(z=0\)，則簡單表為 \({}_{m|}P_{x}^{\Lambda\{h\}}\)。因此每期應繳之保險費為 \({}_{m,z|}P_{x}^{\Lambda\{h\}}/h\)。依據收支平衡原則，可得
\[{}_{m,z|}P_{x}^{\Lambda\{h\}}\cdot{}_{z|}\ddot{a}_{x}^{(h)}={}_{m,z|}P_{x}^{\Lambda\{h\}}\cdot (IA)_{x:m\!\rceil}^{1}
+\Lambda\cdot {}_{m|}A_{x}\cdot\left (1+\frac{{}_{m,z|}P_{x}^{\Lambda\{h\}}}{2h}\cdot\right )\mbox{。}\]
因此可解得
\begin{align*}
{}_{m,z|}P_{x}^{\Lambda\{h\}} & =\frac{\Lambda\cdot {}_{m|}A_{x}}{{}_{z|}\ddot{a}_{x}^{(h)}
-\frac{1}{2h}\cdot\Lambda\cdot {}_{m|}A_{x}-(IA)_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot {}_{m|}A_{x}}{{}_{z|}\ddot{a}_{x}-\frac{h-1}{2h}
\cdot {}_{z}E_{x}-\frac{1}{2h}\cdot\Lambda\cdot {}_{m|}A_{x}-(IA)_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}
{\frac{h+1}{2h}\cdot {}_{z|}\ddot{a}_{x}+\frac{h-1}{2h}\cdot {}_{z|}a_{x}
-\frac{1}{2h}\cdot\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )-(IA)_{x:m\!\rceil}^{1}}
\end{align*}
及
\begin{align*}
{}_{m|}P_{x}^{\Lambda\{h\}} & =\frac{\Lambda\cdot {}_{m|}A_{x}}{\ddot{a}_{x}^{(h)}
-\frac{1}{2h}\cdot\Lambda\cdot {}_{m|}A_{x}-(IA)_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}{\ddot{a}_{x}-\frac{h-1}{2h}
-\frac{1}{2h}\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )-(IA)_{x:m\!\rceil}^{1}}\mbox{。}
\end{align*}

考慮身故即刻退還已繳之保險費，其比例分攤保險費以符號 \({}_{m,z}{}_{|}^{*}P_{x}^{\Lambda\{h\}}\)表示，若 \(z=0\)，則簡單表為 \({}_{m}{}_{|}^{*}P_{x}^{\Lambda\{h\}}\)。因此每期應繳之保險費為 \({}_{m,z}{}_{|}^{*}P_{x}^{\Lambda\{h\}}/h\)。依據收支平衡原則，可解得
\[{}_{m,z}{}_{|}^{*}P_{x}^{\Lambda\{h\}}=\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}
{\frac{h+1}{2h}\cdot {}_{z|}\ddot{a}_{x}+\frac{h-1}{2h}\cdot {}_{z|}a_{x}
-\frac{1}{2h}\cdot\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )-(I\bar{A})_{x:m\!\rceil}^{1}}\]
及
\[{}_{m}{}_{|}^{*}P_{x}^{\Lambda\{h\}}=\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}
{\ddot{a}_{x}-\frac{h-1}{2h}-\frac{1}{2h}\cdot\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )
-(I\bar{A})_{x:m\!\rceil}^{1}}\mbox{。}\]

考慮身故當期末退還已繳之保險費，其比例分攤保險費以符號 \({}_{m,z|}P_{x}^{\Lambda [hh]}\)表示，若 \(z=0\)，則簡單表為 \({}_{m|}P_{x}^{\Lambda [hh]}\)。因此每期應繳之保險費為 \({}_{m,z|}P_{x}^{\Lambda [hh]}/h\)。若被保險人於 \(m\)年內身故時，其保險費退還方式類似每年遞增 \(h\)次之 \(m\)年定期壽險。依據收支平衡原則，可得
\[{}_{m,z|}P_{x}^{\Lambda [hh]}\cdot{}_{z|}\ddot{a}_{x}^{(h)}
={}_{m,z|}P_{x}^{\Lambda [hh]}\cdot (I^{(h)}A)_{x:m\!\rceil}^{1}+\Lambda\cdot {}_{m|}A_{x}\cdot
\left (1+\frac{{}_{m,z|}P_{x}^{\Lambda [hh]}}{2h}\right )\mbox{。}\]
由每年遞增數次之人壽保險金頁之(31)式，亦可得
\[{}_{m,z|}P_{x}^{\Lambda [hh]}\cdot{}_{z|}\ddot{a}_{x}^{(h)}={}_{m,z|}P_{x}^{\Lambda [hh]}\cdot
\left ((IA)_{x:m\!\rceil}^{1}-\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}\right )
+\Lambda\cdot {}_{m|}A_{x}\cdot\left (1+\frac{{}_{m,z|}P_{x}^{\Lambda [hh]}}{2h}\right )\mbox{。}\]
因此可解得
\begin{align*}
{}_{m,z|}P_{x}^{\Lambda [hh]} & =\frac{\Lambda\cdot {}_{m|}A_{x}}
{{}_{z|}\ddot{a}_{x}^{(h)}-\frac{1}{2h}\cdot\Lambda\cdot {}_{m|}A_{x}
-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}
{\frac{h+1}{2h}\cdot {}_{z|}\ddot{a}_{x}+\frac{h-1}{2h}\cdot {}_{z|}a_{x}
-\frac{1}{2h}\cdot\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )-(IA)_{x:m\!\rceil}^{1}
+\frac{h-1}{2h}\cdot\left (v\cdot\ddot{a}_{x:m\!\rceil}-a_{x:m\!\rceil}\right )}
\end{align*}
及
\begin{align*}
{}_{m|}P_{x}^{\Lambda [hh]} & =\frac{\Lambda\cdot {}_{m|}A_{x}}
{\ddot{a}_{x}^{(h)}-\frac{1}{2h}\cdot\Lambda\cdot {}_{m|}A_{x}
-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}{\ddot{a}_{x}-\frac{h-1}{2h}
-\frac{1}{2h}\cdot\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )
-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot\left (v\cdot\ddot{a}_{x:m\!\rceil}-a_{x:m\!\rceil}\right )}\mbox{。}
\end{align*}

延期繳費且繳費期間為q年.

考慮身故當年末退還已繳之保險費，其比例分攤保險費以符號 \({}_{m,z|}P_{x,q}^{\Lambda\{h\}}\)表示，若 \(z=0\)，則簡單表為 \({}_{m|}P_{x,q}^{\Lambda\{h\}}\)。因此每期應繳之保險費為 \({}_{m,z|}P_{x,q}^{\Lambda\{h\}}/h\)。依據收支平衡原則，可得
\[{}_{m,z|}P_{x,q}^{\Lambda\{h\}}\cdot {}_{z|}\ddot{a}_{x:q\!\rceil}^{(h)}
={}_{m,z|}P_{x,q}^{\Lambda\{h\}}\cdot (IA)_{x:m\!\rceil}^{1}+\Lambda\cdot {}_{m|}A_{x}\cdot
\left (1+\frac{{}_{m,z|}P_{x,q}^{\Lambda\{h\}}}{2h}\right )\mbox{。}\]
因此可解得
\begin{align*}
{}_{m,z|}P_{x,q}^{\Lambda\{h\}} & =\frac{\Lambda\cdot {}_{m|}A_{x}}{{}_{z|}\ddot{a}_{x:q\!\rceil}^{(h)}
-\frac{1}{2h}\cdot\Lambda\cdot {}_{m|}A_{x}-(IA)_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot {}_{m|}A_{x}}{{}_{z|}\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}
\cdot\left ({}_{z}E_{x}-{}_{z+q}E_{x}\right )-\frac{1}{2h}\cdot\Lambda\cdot {}_{m|}A_{x}-(IA)_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}{{}_{z|}\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}
\cdot\left ({}_{z}E_{x}-{}_{z+q}E_{x}\right )
-\frac{1}{2h}\cdot\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )-(IA)_{x:m\!\rceil}^{1}}
\end{align*}
及
\begin{align*}
{}_{m|}P_{x,q}^{\Lambda\{h\}} & =\frac{\Lambda\cdot {}_{m|}A_{x}}{\ddot{a}_{x:q\!\rceil}^{(h)}
-\frac{1}{2h}\cdot\Lambda\cdot {}_{m|}A_{x}-(IA)_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}{\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}
\cdot\left (1-{}_{q}E_{x}\right )-\frac{1}{2h}\cdot\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )
-(IA)_{x:m\!\rceil}^{1}}\mbox{。}
\end{align*}

考慮身故即刻退還已繳之保險費，其比例分攤保險費以符號 \({}_{m,z}{}_{|}^{*}P_{x,q}^{\Lambda\{h\}}\)表示，若 \(z=0\)，則簡單表為 \({}_{m}{}_{|}^{*}P_{x,q}^{\Lambda\{h\}}\)。因此每期應繳之保險費為 \({}_{m,z}{}_{|}^{*}P_{x,q}^{\Lambda\{h\}}/h\)。依據收支平衡原則，可解得
\[{}_{m,z}{}_{|}^{*}P_{x,q}^{\Lambda\{h\}}=\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}
{{}_{z|}\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}\cdot\left ({}_{z}E_{x}-{}_{z+q}E_{x}\right )
-\frac{1}{2h}\cdot\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )-(I\bar{A})_{x:m\!\rceil}^{1}}\]
及
\[{}_{m}{}_{|}^{*}P_{x,q}^{\Lambda\{h\}}=\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}
{\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}\cdot\left (1-{}_{q}E_{x}\right )
-\frac{1}{2h}\cdot\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )-(I\bar{A})_{x:m\!\rceil}^{1}}\mbox{。}\]

考慮身故當期末退還已繳之保險費，其比例分攤保險費以符號 \({}_{m,z|}P_{x,q}^{\Lambda [hh]}\)表示，若 \(z=0\)，則簡單表為 \({}_{m|}P_{x,q}^{\Lambda [hh]}\)。因此每期應繳之保險費為 \({}_{m,z|}P_{x,q}^{\Lambda [hh]}/h\)。若被保險人於 \(m\)年內身故時，其保險費退還方式類似每年遞增 \(h\)次之 \(m\)年定期壽險。依據收支平衡原則，可得
\[{}_{m,z|}P_{x,q}^{\Lambda [hh]}\cdot{}_{z|}\ddot{a}_{x:q\!\rceil}^{(h)}
={}_{m,z|}P_{x,q}^{\Lambda [hh]}\cdot (I^{(h)}A)_{x:m\!\rceil}^{1}+\Lambda\cdot {}_{m|}A_{x}\cdot
\left (1+\frac{{}_{m,z|}P_{x,q}^{\Lambda [hh]}}{2h}\right )\mbox{。}\]
由每年遞增數次之人壽保險金頁之(31)式，亦可得
\[{}_{m,z|}P_{x,q}^{\Lambda [hh]}\cdot{}_{z|}\ddot{a}_{x:q\!\rceil}^{(h)}={}_{m,z|}P_{x,q}^{\Lambda [hh]}\cdot
\left ((IA)_{x:m\!\rceil}^{1}-\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}\right )+\Lambda\cdot {}_{m|}A_{x}\cdot
\left (1+\frac{{}_{m,z|}P_{x,q}^{\Lambda [hh]}}{2h}\right )\mbox{。}\]
因此可解得
\begin{align*}
{}_{m,z|}P_{x,q}^{\Lambda [hh]} & =\frac{\Lambda\cdot {}_{m|}A_{x}}{{}_{z|}\ddot{a}_{x:q\!\rceil}^{(h)}
-\frac{1}{2h}\cdot\Lambda\cdot {}_{m|}A_{x}-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}{{}_{z|}\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}
\cdot\left ({}_{z}E_{x}-{}_{z+q}E_{x}\right )-\frac{1}{2h}\cdot\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )
-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot\left (v\cdot\ddot{a}_{x:m\!\rceil}-a_{x:m\!\rceil}\right )}
\end{align*}
及
\begin{align*}
{}_{m|}P_{x,q}^{\Lambda [hh]} & =\frac{\Lambda\cdot {}_{m|}A_{x}}
{\ddot{a}_{x:q\!\rceil}^{(h)}-\frac{1}{2h}\cdot\Lambda\cdot {}_{m|}A_{x}
-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )}
{\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}\cdot\left (1-{}_{q}E_{x}\right )
-\frac{1}{2h}\cdot\Lambda\cdot\left (v\cdot {}_{m|}\ddot{a}_{x}-{}_{m|}a_{x}\right )
-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot\left (v\cdot\ddot{a}_{x:m\!\rceil}-a_{x:m\!\rceil}\right )}\mbox{。}
\end{align*}

即刻給付.

採用身故即刻給付保險金方式。

延期繳費且繳費至終身.

考慮身故當年末退還已繳之保險費，現年 \(x\)歲的被保險人，其比例分攤保險費以符號 \({}_{z|}P^{\Lambda\{h\}}({}_{m|}\bar{A}_{x})\)表示之，若 \(z=0\)，則簡單表為 \(P^{\Lambda\{h\}}({}_{m|}\bar{A}_{x})\)。因此每期應繳納之保費為 \({}_{z|}P^{\Lambda\{h\}}({}_{m|}\bar{A}_{x})/h\)。依據收支平衡原則，可得
\[{}_{z|}P^{\Lambda\{h\}}({}_{m|}\bar{A}_{x})\cdot {}_{z|}\ddot{a}_{x}^{(h)}
={}_{z|}P^{\Lambda\{h\}}({}_{m|}\bar{A}_{x})\cdot (IA)_{x:m\!\rceil}^{1}
+\Lambda\cdot {}_{m|}\bar{A}_{x}\cdot\left (1+\frac{{}_{z|}P^{\Lambda\{h\}}({}_{m|}\bar{A}_{x})}{2h}\cdot\right )\mbox{。}\]
因此可解得
\begin{align*}
{}_{z|}P^{\Lambda\{h\}}({}_{m|}\bar{A}_{x}) & =\frac{\Lambda\cdot {}_{m|}\bar{A}_{x}}
{{}_{z|}\ddot{a}_{x}^{(h)}-\frac{1}{2h}\cdot\Lambda\cdot {}_{m|}\bar{A}_{x}-(IA)_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left [\left (1-\frac{\delta}{2}\right )\cdot {}_{m|}\ddot{a}_{x}
-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]}{\frac{h+1}{2h}\cdot {}_{z|}\ddot{a}_{x}+\frac{h-1}{2h}\cdot {}_{z|}a_{x}
-\frac{1}{2h}\cdot\Lambda\cdot\left [\left (1-\frac{\delta}{2}\right )\cdot {}_{m|}\ddot{a}_{x}
-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]-(IA)_{x:m\!\rceil}^{1}}
\end{align*}
及
\begin{align*}
P^{\Lambda\{h\}}({}_{m|}\bar{A}_{x}) & =\frac{\Lambda\cdot {}_{m|}\bar{A}_{x}}{\ddot{a}_{x}^{(h)}
-\frac{1}{2h}\cdot\Lambda\cdot {}_{m|}\bar{A}_{x}-(IA)_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left [\left (1-\frac{\delta}{2}\right )\cdot {}_{m|}\ddot{a}_{x}
-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]}{\ddot{a}_{x}-\frac{h-1}{2h}-\frac{1}{2h}\cdot\Lambda\cdot
\left [\left (1-\frac{\delta}{2}\right )\cdot {}_{m|}\ddot{a}_{x}-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]
-(IA)_{x:m\!\rceil}^{1}}\mbox{。}
\end{align*}

考慮身故即刻退還已繳之保險費，其比例分攤保險費以符號 \({}_{z}{}_{|}^{*}P^{\Lambda\{h\}}({}_{m|}\bar{A}_{x})\)表示之，若 \(z=0\)，則簡單表為 \({}^{*}P^{\Lambda\{h\}}({}_{m|}\bar{A}_{x})\)。因此每期應繳納之保費為 \({}_{z}{}_{|}^{*}P^{\Lambda\{h\}}({}_{m|}\bar{A}_{x})/h\)。依據收支平衡原則，可解得
\[{}_{z}{}_{|}^{*}P^{\Lambda\{h\}}({}_{m|}\bar{A}_{x})=\frac{\Lambda\cdot
\left [\left (1-\frac{\delta}{2}\right )\cdot {}_{m|}\ddot{a}_{x}-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]}
{\frac{h+1}{2h}\cdot {}_{z|}\ddot{a}_{x}+\frac{h-1}{2h}\cdot {}_{z|}a_{x}-\frac{1}{2h}\cdot\Lambda\cdot
\left [\left (1-\frac{\delta}{2}\right )\cdot {}_{m|}\ddot{a}_{x}
-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]-(I\bar{A})_{x:m\!\rceil}^{1}}\]
及
\[{}^{*}P^{\Lambda\{h\}}({}_{m|}\bar{A}_{x})=\frac{\Lambda\cdot
\left [\left (1-\frac{\delta}{2}\right )\cdot {}_{m|}\ddot{a}_{x}-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]}
{\ddot{a}_{x}-\frac{h-1}{2h}-\frac{1}{2h}\cdot\Lambda\cdot\left [\left (1-\frac{\delta}{2}\right )\cdot {}_{m|}\ddot{a}_{x}
-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]-(I\bar{A})_{x:m\!\rceil}^{1}}\mbox{。}\]

考慮身故當期末退還已繳之保險費，其比例分攤保險費以符號 \({}_{z}{}_{|}^{*}P^{\Lambda [hh]}({}_{m|}\bar{A}_{x})\)表示，若 \(z=0\)，則簡單表為 \({}^{*}P^{\Lambda [hh]}({}_{m|}\bar{A}_{x})\)。因此每期應繳之保險費為 \({}_{z}{}_{|}^{*}P^{\Lambda [hh]}({}_{m|}\bar{A}_{x})/h\)。若被保險人於 \(m\)年內身故時，其保險費退還方式類似每年遞增 \(h\)次之 \(m\)年定期壽險。依據收支平衡原則，可得
\[{}_{z}{}_{|}^{*}P^{\Lambda [hh]}({}_{m|}\bar{A}_{x})\cdot{}_{z|}\ddot{a}_{x}^{(h)}
={}_{z}{}_{|}^{*}P^{\Lambda [hh]}({}_{m|}\bar{A}_{x})\cdot (I^{(h)}A)_{x:m\!\rceil}^{1}+\Lambda\cdot {}_{m|}\bar{A}_{x}
\cdot\left (1+\frac{{}_{z|}P^{\Lambda [hh]}({}_{m|}\bar{A}_{x})}{2h}\cdot\right )\mbox{。}\]
由每年遞增數次之人壽保險金頁之(31)式，亦可得
\begin{align*}
{}_{z}{}_{|}^{*}P^{\Lambda [hh]}({}_{m|}\bar{A}_{x})\cdot {}_{z|}\ddot{a}_{x}^{(h)}
& ={}_{z}{}_{|}^{*}P^{\Lambda [hh]}({}_{m|}\bar{A}_{x})\cdot
\left ((IA)_{x:m\!\rceil}^{1}-\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}\right )\\
& \quad +\Lambda\cdot {}_{m|}\bar{A}_{x}\cdot\left (1+\frac{{}_{z|}P^{\Lambda [hh]}({}_{m|}\bar{A}_{x})}{2h}\cdot\right )\mbox{。}
\end{align*}
因此可解得
\begin{align*}
{}_{z}{}_{|}^{*}P^{\Lambda [hh]}({}_{m|}\bar{A}_{x}) & =\frac{\Lambda\cdot {}_{m|}\bar{A}_{x}}
{{}_{z|}\ddot{a}_{x}^{(h)}-\frac{1}{2h}\cdot\Lambda\cdot {}_{m|}\bar{A}_{x}
-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left [\left (1-\frac{\delta}{2}\right )\cdot
{}_{m|}\ddot{a}_{x}-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]}{\frac{h+1}{2h}\cdot {}_{z|}\ddot{a}_{x}
+\frac{h-1}{2h}\cdot {}_{z|}a_{x}-\frac{1}{2h}\cdot\Lambda\cdot\left [\left (1-\frac{\delta}{2}\right )\cdot
{}_{m|}\ddot{a}_{x}-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]-(IA)_{x:m\!\rceil}^{1}
+\frac{h-1}{2h}\cdot\left (v\cdot\ddot{a}_{x:m\!\rceil}-a_{x:m\!\rceil}\right )}
\end{align*}
及
\begin{align*}
{}^{*}P^{\Lambda [hh]}({}_{m|}\bar{A}_{x}) & =\frac{\Lambda\cdot {}_{m|}\bar{A}_{x}}
{\ddot{a}_{x}^{(h)}-\frac{1}{2h}\cdot\Lambda\cdot {}_{m|}\bar{A}_{x}
-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left [\left (1-\frac{\delta}{2}\right )\cdot
{}_{m|}\ddot{a}_{x}-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]}{\ddot{a}_{x}-\frac{h-1}{2h}-\frac{1}{2h}\cdot
\Lambda\cdot\left [\left (1-\frac{\delta}{2}\right )\cdot
{}_{m|}\ddot{a}_{x}-\left (1+\frac{\delta}{2}\right )\cdot {}_{m|}a_{x}\right ]-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot
\left (v\cdot\ddot{a}_{x:m\!\rceil}-a_{x:m\!\rceil}\right )}\mbox{。}
\end{align*}

延期繳費且繳費期間為q年.

考慮身故當年末退還已繳之保險費，現年$x$歲的被保險人，其年賦保險費以符號 \({}_{z|}P_{q}^{\Lambda\{h\}}({}_{m|}\bar{A}_{x})\)表示之，若 \(z=0\)，則簡單表為 \(P_{q}^{\Lambda\{h\}}({}_{m|}\bar{A}_{x})\)。因此每期應繳納之保費為 \({}_{z|}P_{q}^{\Lambda\{h\}}({}_{m|}\bar{A}_{x})/h\)。依據收支平衡原則，可得
\[{}_{z|}P_{q}^{\Lambda\{h\}}({}_{m|}\bar{A}_{x})\cdot{}_{z|}\ddot{a}_{x:q\!\rceil}^{(h)}
={}_{z|}P_{q}^{\Lambda\{h\}}({}_{m|}\bar{A}_{x})\cdot (IA)_{x:m\!\rceil}^{1}+\Lambda\cdot {}_{m|}\bar{A}_{x}\cdot
\left (1+\frac{{}_{z|}P_{q}^{\Lambda\{h\}}({}_{m|}\bar{A}_{x})}{2h}\cdot\right )\mbox{。}\]
因此可解得
\begin{align*}
{}_{z|}P_{q}^{\Lambda\{h\}}({}_{m|}\bar{A}_{x}) & =\frac{\Lambda\cdot {}_{m|}\bar{A}_{x}}
{{}_{z|}\ddot{a}_{x:q\!\rceil}^{(h)}-\frac{1}{2h}\cdot\Lambda\cdot {}_{m|}\bar{A}_{x}-(IA)_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left [\left (1+\frac{\delta}{2}\right )\cdot{}_{m}E_{x}-\delta\cdot {}_{m|}\ddot{a}_{x}\right ]}
{{}_{z|}\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}\cdot\left ({}_{z}E_{x}-{}_{z+q}E_{x}\right )
-\frac{1}{2h}\cdot\Lambda\cdot\left [\left (1+\frac{\delta}{2}\right )\cdot
{}_{m}E_{x}-\delta\cdot {}_{m|}\ddot{a}_{x}\right ]-(IA)_{x:m\!\rceil}^{1}}
\end{align*}
及
\begin{align*}
P_{q}^{\Lambda\{h\}}({}_{m|}\bar{A}_{x}) & =\frac{\Lambda\cdot {}_{m|}\bar{A}_{x}}
{\ddot{a}_{x:q\!\rceil}^{(h)}-\frac{1}{2h}\cdot\Lambda\cdot {}_{m|}\bar{A}_{x}-(IA)_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left [\left (1+\frac{\delta}{2}\right )\cdot {}_{m}E_{x}-\delta\cdot {}_{m|}\ddot{a}_{x}\right ]}
{\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}\cdot\left (1-{}_{q}E_{x}\right )
-\frac{1}{2h}\cdot\Lambda\cdot\left [\left (1+\frac{\delta}{2}\right )\cdot
{}_{m}E_{x}-\delta\cdot {}_{m|}\ddot{a}_{x}\right ]-(IA)_{x:m\!\rceil}^{1}}\mbox{。}
\end{align*}

考慮身故即刻退還已繳之保險費，其年賦保險費以符號 \({}_{z}{}_{|}^{*}P_{q}^{\Lambda\{h\}}({}_{m|}\bar{A}_{x})\)表示之，若 \(z=0\)，則簡單表為 \({}^{*}P_{q}^{\Lambda\{h\}}({}_{m|}\bar{A}_{x})\)。因此每期應繳納之保費為 \({}_{z}{}_{|}^{*}P_{q}^{\Lambda\{h\}}({}_{m|}\bar{A}_{x})/h\)。依據收支平衡原則，可解得
\[{}_{z}{}_{|}^{*}P_{q}^{\Lambda\{h\}}({}_{m|}\bar{A}_{x})=\frac{\Lambda\cdot\left [\left (1+\frac{\delta}{2}\right )\cdot
{}_{m}E_{x}-\delta\cdot {}_{m|}\ddot{a}_{x}\right ]}{{}_{z|}\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}
\cdot\left ({}_{z}E_{x}-{}_{z+q}E_{x}\right )-\frac{1}{2h}\cdot\Lambda\cdot\left [\left (1+\frac{\delta}{2}\right )\cdot
{}_{m}E_{x}-\delta\cdot {}_{m|}\ddot{a}_{x}\right ]-(I\bar{A})_{x:m\!\rceil}^{1}}\]
及
\[{}^{*}P_{q}^{\Lambda\{h\}}({}_{m|}\bar{A}_{x})=\frac{\Lambda\cdot\left [\left (1+\frac{\delta}{2}\right )\cdot
{}_{m}E_{x}-\delta\cdot {}_{m|}\ddot{a}_{x}\right ]}{\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}\cdot\left (1-{}_{q}E_{x}\right )
-\frac{1}{2h}\cdot\Lambda\cdot\left [\left (1+\frac{\delta}{2}\right )\cdot
{}_{m}E_{x}-\delta\cdot {}_{m|}\ddot{a}_{x}\right ]-(I\bar{A})_{x:m\!\rceil}^{1}}\mbox{。}\]

考慮身故當期末退還已繳之保險費，其年賦保險費以符號 \({}_{z}{}_{|}^{*}P_{q}^{\Lambda [hh]}({}_{m|}\bar{A}_{x})\)表示，若 \(z=0\)，則簡單表為 \({}^{*}P_{q}^{\Lambda [hh]}({}_{m|}\bar{A}_{x})\)。因此每期應繳之保險費為 \({}_{z}{}_{|}^{*}P_{q}^{\Lambda [hh]}({}_{m|}\bar{A}_{x})/h\)。若被保險人於 \(m\)年內身故時，其保險費退還方式類似每年遞增 \(h\)次之 \(m\)年定期壽險。依據收支平衡原則，可得
\[{}_{z}{}_{|}^{*}P_{q}^{\Lambda [hh]}({}_{m|}\bar{A}_{x})\cdot{}_{z|}\ddot{a}_{x:q\!\rceil}^{(h)}
={}_{z}{}_{|}^{*}P_{q}^{\Lambda [hh]}({}_{m|}\bar{A}_{x})\cdot
(I^{(h)}A)_{x:m\!\rceil}^{1}+\Lambda\cdot {}_{m|}\bar{A}_{x}
\cdot\left (1+\frac{{}_{z|}P_{q}^{\Lambda [hh]}({}_{m|}\bar{A}_{x})}{2h}\cdot\right )\mbox{。}\]
由每年遞增數次之人壽保險金頁之(31)式，亦可得
\begin{align*}
{}_{z}{}_{|}^{*}P_{q}^{\Lambda [hh]}({}_{m|}\bar{A}_{x})\cdot {}_{z|}\ddot{a}_{x:q\!\rceil}^{(h)}
& ={}_{z}{}_{|}^{*}P_{q}^{\Lambda [hh]}({}_{m|}\bar{A}_{x})\cdot
\left ((IA)_{x:m\!\rceil}^{1}-\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}\right )\\
& \quad +\Lambda\cdot {}_{m|}\bar{A}_{x}\cdot
\left (1+\frac{{}_{z|}P_{q}^{\Lambda [hh]}({}_{m|}\bar{A}_{x})}{2h}\cdot\right )\mbox{。}
\end{align*}
因此可解得
\begin{align*}
{}_{z}{}_{|}^{*}P_{q}^{\Lambda [hh]}({}_{m|}\bar{A}_{x}) & =\frac{\Lambda\cdot {}_{m|}\bar{A}_{x}}
{{}_{z|}\ddot{a}_{x:q\!\rceil}^{(h)}-\frac{1}{2h}\cdot\Lambda\cdot {}_{m|}\bar{A}_{x}
-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left [\left (1+\frac{\delta}{2}\right )\cdot {}_{m}E_{x}-\delta\cdot {}_{m|}\ddot{a}_{x}\right ]}
{{}_{z|}\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}\cdot\left ({}_{z}E_{x}-{}_{z+q}E_{x}\right )
-\frac{1}{2h}\cdot\Lambda\cdot\left [\left (1+\frac{\delta}{2}\right )\cdot
{}_{m}E_{x}-\delta\cdot {}_{m|}\ddot{a}_{x}\right ]-(IA)_{x:m\!\rceil}^{1}
+\frac{h-1}{2h}\cdot\left (v\cdot\ddot{a}_{x:m\!\rceil}-a_{x:m\!\rceil}\right )}
\end{align*}
及
\begin{align*}
{}^{*}P_{q}^{\Lambda [hh]}({}_{m|}\bar{A}_{x}) & =\frac{\Lambda\cdot {}_{m|}\bar{A}_{x}}
{\ddot{a}_{x}^{(h)}-\frac{1}{2h}\cdot\Lambda\cdot {}_{m|}\bar{A}_{x}
-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot A_{x:m\!\rceil}^{1}}\\
& =\frac{\Lambda\cdot\left [\left (1+\frac{\delta}{2}\right )\cdot {}_{m}E_{x}-\delta\cdot {}_{m|}\ddot{a}_{x}\right ]}
{\ddot{a}_{x:q\!\rceil}-\frac{h-1}{2h}\cdot\left (1-{}_{q}E_{x}\right )
-\frac{1}{2h}\cdot\Lambda\cdot\left [\left (1+\frac{\delta}{2}\right )\cdot
{}_{m}E_{x}-\delta\cdot {}_{m|}\ddot{a}_{x}\right ]-(IA)_{x:m\!\rceil}^{1}+\frac{h-1}{2h}\cdot
\left (v\cdot\ddot{a}_{x:m\!\rceil}-a_{x:m\!\rceil}\right )}\mbox{。}
\end{align*}