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
\n\\begin{eqnarray*}
\n~_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 \\\\
\n&=& b (1+r_b)^t \\int_0^{\\infty} (1+r_b)^s v^s ~_s p_{[x]+t+s} \\mu_{[x]+t+s} ds \\\\
\n&-& P (1+r_P)^t \\int_0^{\\infty} (1+r_P)^s v^s ~_s p_{[x]+t+s} ds \\\\
\n&=& b (1+r_b)^t \\bar{A}_{[x]+t}^{b} – P (1+r_P)^t\\bar{a}_{[x]+t}^{P} ,
\n\\end{eqnarray*}
\nwhere \\(\\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)\\).<\/p>\n
To illustrate, suppose that \\(i=6\\%\\), \\(r_b=3\\%\\), and \\(r_P= 1\\%\\). Then, we would calculate policy values as
\n\\begin{eqnarray*}
\n~_t V = b (1.03)^t \\bar{A}_{[x]+t}^{@i=2.92\\%} – P (1.01)^t \\bar{a}_{[x]+t}^{@i=4.95\\%} .
\n\\end{eqnarray*}<\/p>\n