Consider a fully continuous whole life insurance with both benefits and premiums indexed to inflation rates. Specifically, for failure at policy time (t), the benefit payment is (b_t = b (1+r_b)^t). Premiums also increased continuously, with premium payment rate at time (t) being (P_t = P (1+r_P)^t). Thus, we allow the premium rate increase ((r_P)) to differ from the benefit rate increase ((r_b)). Assuming a constant force of interest and no expenses, the policy value at time (t) may be expressed as
begin{eqnarray*}
~_t V &=& int_0^{infty} b_{t+s} v^s ~_s p_{[x]+t+s} mu_{[x]+t+s} ds – int_0^{infty} P_{t+s} v^s ~_s p_{[x]+t+s} ds \
&=& b (1+r_b)^t int_0^{infty} (1+r_b)^s v^s ~_s p_{[x]+t+s} mu_{[x]+t+s} ds \
&-& P (1+r_P)^t int_0^{infty} (1+r_P)^s v^s ~_s p_{[x]+t+s} ds \
&=& b (1+r_b)^t bar{A}_{[x]+t}^{b} – P (1+r_P)^tbar{a}_{[x]+t}^{P} ,
end{eqnarray*}
where (bar{A}_{[x]+t}^{b}) is evaluated at discount rate (v^b = (1+r_b)v). This corresponds to a new force of interest (delta^b = – ln v^b = – ln (v(1+r_b)) = delta – ln(1+r_b)). Similarly, for (bar{a}_{[x]+t}^{P}), the new force of interest is (delta^P = delta – ln(1+r_P)).
To illustrate, suppose that (i=6%), (r_b=3%), and (r_P= 1%). Then, we would calculate policy values as
begin{eqnarray*}
~_t V = b (1.03)^t bar{A}_{[x]+t}^{@i=2.92%} – P (1.01)^t bar{a}_{[x]+t}^{@i=4.95%} .
end{eqnarray*}