The continuous Ax integral value is commonly expressed by which integral?

• A. ∫_0^t v^s · n_p_x ds
• B. ∫_0^t v^s · p_x ds
• C. ∫_0^t e^{−δ s} ds
• D. ∫_0^t q_x ds

Answer: (A)

The main idea is that the present value of a continuous life annuity depends on both time valuation and mortality, so you integrate over time the discounted payment that is only received if the life is still alive at that moment. If payments occur at a steady rate of 1 per year, a tiny payment ds at time s is worth v^s ds at time 0, and it should be weighted by the probability that the life aged x is alive at time s, which is _s p_x. Integrating from 0 to t gives the standard expression for the continuous annuity value: ∫_0^t v^s · _s p_x ds. This captures how time value of money and survival probabilities taper the expected present value as time progresses. The other forms either omit the time-dependent survival factor or replace it with a quantity that doesn’t reflect how survival changes with s, so they don’t represent the continuous, life-contingent value as accurately.