Let<\/p>\n
At time \\(t\\), the present value of future benefits is
\n\\begin{eqnarray*}
\nPVFB_t &=& \\int_0^{\\infty} b_{t+s} \\frac{v(t+s)}{v(t)} ~_s p_{[x]+t+s} \\mu_{[x]+t+s} ds,
\n\\end{eqnarray*}
\nthe present value of future expenses is
\n\\begin{eqnarray*}
\nPVFE_t &=& \\int_0^{\\infty} E_{t+s} \\frac{v(t+s)}{v(t)} ~_s p_{[x]+t+s} \\mu_{[x]+t+s} ds\\\\
\n&+& \\int_0^{\\infty} e_{t+s} \\frac{v(t+s)}{v(t)} ~_s p_{[x]+t+s} ds,
\n\\end{eqnarray*}
\nand the present value of future premiums is
\n\\begin{eqnarray*}
\nPVFP_t &=& \\int_0^{\\infty} P_{t+s} \\frac{v(t+s)}{v(t)} ~_s p_{[x]+t+s} ds.
\n\\end{eqnarray*}
\nCombining these, we have the policy value at time \\(t\\) is
\n\\begin{eqnarray*}
\n~_t V &=& PVFB_t + PVFE_t – PVFP_t \\\\
\n&=& \\int_0^{\\infty} (b_{t+s}+E_{t+s}) \\frac{v(t+s)}{v(t)} ~_s p_{[x]+t+s} \\mu_{[x]+t+s} ds \\\\
\n&-& \\int_0^{\\infty} (P_{t+s}+e_{t+s}) \\frac{v(t+s)}{v(t)} ~_s p_{[x]+t+s} ds
\n\\end{eqnarray*}<\/p>\n