Which statement correctly contrasts discrete and continuous varying life insurance valuations?

• A. Discrete uses sums with year indices and qx; continuous uses integrals with μ and tpx
• B. Discrete uses integrals; continuous uses sums
• C. Both use integrals
• D. Both use sums

Answer: (A)

In discrete-time valuations, time is sliced into whole-year steps, so probabilities are taken from year-by-year life tables using qx, the probability of dying within a specific year. The value is then built up as a sum over future years, with each term discounted and weighted by the appropriate survival/death probabilities derived from qx (and related p_x values). This year-by-year summation is the natural way to model payments that occur in whole-year intervals.

In continuous-time valuations, time is treated as a continuum and the force of mortality μ(t) describes the instantaneous rate of dying at time t. The survival to time t is tpx = exp(-∫0^t μ(s) ds), and the instantaneous probability density of death at time t is μ(t) tpx. The present value is obtained by integrating over all possible death times, using the continuous-time density and the discount factor e^{-δ t} or v^t. This integral form reflects the continuous nature of time and hazard.

So the statement that discrete valuations use sums with year indices and qx, while continuous valuations use integrals with μ and tpx, captures the fundamental distinction between the two frameworks. The other options mix up the roles of sums and integrals and don’t reflect the standard discrete versus continuous modeling.