The policy value, also known as the reserve, is the expected future loss variable, typically denoted as \(V = \mathrm{E~} L\). Sometimes we write \(_t V = \mathrm{E~} (L_t | T>t)\) to emphasize the conditional expectation, that is, we are only computing policy values for individuals who have survived \(t\) future years. In this basic case, the constant premium rate \(P\) is determined at contract initiation, e.g., at \(t=0\).
Policy Value. Specifically, at time \(t\), the EPV of future benefits is \(\mathrm{E~} (v^{T-t}|T>t )= \bar{A}_{x+t}\). The EPV of future premiums is \(\mathrm{E~} (P \bar{a}_{\overline{T-t}|}|T>t )= P \bar{a}_{x+t}\). With these, the net premium policy value is
\begin{eqnarray*}
_t V^n = \bar{A}_{x+t} – P^n \bar{a}_{x+t}.
\end{eqnarray*}
In the premium modules, we learned that we could derive net premiums through the equivalence principle, that is, find \(P^n\) such that \(\mathrm{E~} L_0^n = ~_0 V^n = 0\). Thus, we have \(P^n = \bar{A}_{x+t}/\bar{a}_{x+t}.\)
Variance. With the net premium and the relation \(\bar{a}_{\overline{T-t}|}=(1-v^{T-t})/\delta\), we write the net premium loss variable as
\begin{eqnarray*}
L_t^n =v^{T-t} – P^n \frac{1-v^{T-t}}{\delta}= \left(1+\frac{P^n}{\delta}\right) – \frac{P^n}{\delta}.
\end{eqnarray*}
Thus, we have
\begin{eqnarray*}
\textrm{Var}(L_t^n|T>t) &=& \left(1+\frac{P^n}{\delta}\right)^2 \textrm{Var}(v^{T-t}|T>t) \\
&=& \left(1+\frac{P^n}{\delta}\right)^2 \left( ~^2 \bar{A}_{x+t} – \bar{A}_{x+t}^2 \right) .
\end{eqnarray*}
Here, recall the notation \(~^2 \bar{A}\) means compute the EPV of a whole life insurance at force of interest \(2\delta\).