Let
- (P_t) be the annual rate of premium payable at time (t)
- (e_t) be the annual rate of premium-related expense payable at time (t)
- (b_t) be the benefit, or sum insured, payable at time (t) for failure at time (t)
- (E_t) be the expense of paying the benefit at time (t)
- (mu_{[x]+t}) be the force of mortality at age ([x]+t)
- (delta_t) be the force of interest at time (t)
- (v(t) = expleft(-int_0^t delta_s ds right)) is the present value (at time 0) of 1 at time (t)
- (~_t V) be the policy value at time (t)
At time (t), the present value of future benefits is
begin{eqnarray*}
PVFB_t &=& int_0^{infty} b_{t+s} frac{v(t+s)}{v(t)} ~_s p_{[x]+t+s} mu_{[x]+t+s} ds,
end{eqnarray*}
the present value of future expenses is
begin{eqnarray*}
PVFE_t &=& int_0^{infty} E_{t+s} frac{v(t+s)}{v(t)} ~_s p_{[x]+t+s} mu_{[x]+t+s} ds\
&+& int_0^{infty} e_{t+s} frac{v(t+s)}{v(t)} ~_s p_{[x]+t+s} ds,
end{eqnarray*}
and the present value of future premiums is
begin{eqnarray*}
PVFP_t &=& int_0^{infty} P_{t+s} frac{v(t+s)}{v(t)} ~_s p_{[x]+t+s} ds.
end{eqnarray*}
Combining these, we have the policy value at time (t) is
begin{eqnarray*}
~_t V &=& PVFB_t + PVFE_t – PVFP_t \
&=& 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 \
&-& int_0^{infty} (P_{t+s}+e_{t+s}) frac{v(t+s)}{v(t)} ~_s p_{[x]+t+s} ds
end{eqnarray*}