For continuous varying increasing whole life, the present value is given by the integral ∫_0^∞ t v^t tpx μ_{x+t} dt. For a term policy with finite term n, what is the corresponding integral?

• A. ∫_0^n t v^t tpx μ_{x+t} dt
• B. ∫_0^n v^t μ_{x+t} dt
• C. ∫_0^n t v^t μ_{x} dt
• D. ∫_0^n t v^t tpx dt

Answer: (A)

The key idea is that the present value of a payment that occurs at the moment of death depends on the time of death through the discount factor, the probability of surviving to that time, and the instantaneous mortality rate at that age. For this continuous model, the density of death at time t is μ_{x+t} times tpx, and discounting is v^t. Since the benefit at death is equal to the time t itself (a linearly increasing payment), the payment amount contributed at time t is t. Put together, the present value is ∫ t v^t tpx μ_{x+t} dt over the relevant time horizon.

For a term policy that pays only if death occurs within the finite term n, the horizon is restricted to 0 to n. Hence the corresponding integral is ∫0^n t v^t tpx μ{x+t} dt.