We can illustrate many important concepts in terms of this basic traditional life insurance policy. For concreteness, we assume the fully continuous case, that pays 1 immediately upon the failure of the policyholder and where level premiums are payable continuously throughout the year until the moment of failure.
As before, we assume that contract initiation age is (x). We now wish to establish the value of the policy (t) years later when the policyholder is age (x+t) (if alive).
Figure 1. Timeline for a Whole Life Insurance Policy
At (policy) time (t), we define the future (net) loss variable (to the insurer) to be the excess of the present value of future insurance benefits (PVFB) over the present value of future net premiums (PVFNP)
begin{eqnarray*}
L_t^n = PVFB – PVFNP = v^{T-t} – P^n bar{a}_{overline{T-t}|}.
end{eqnarray*}
Because (T) is random, (L_t^n) is a random variable. Although we focus on its mean, the variance and distribution function are also important for some applications.
To get a step closer to reality, we can easily add expenses into the mix. As before, we think of a “gross premium,” (P^G) as an income that accounts for benefits and expenses. With this, define the future gross loss variable to be
begin{eqnarray*}
L_t^g = PVFB + PVFE – PVFGP,
end{eqnarray*}
where (PFVE) represents the present value of future insurance benefits and (PVFGP) represents the present value of future gross premiums.