Which statement best distinguishes discrete from continuous varying life insurance valuation?

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Multiple Choice

Which statement best distinguishes discrete from continuous varying life insurance valuation?

The essential distinction is how time and mortality are modeled: discrete valuations accumulate value by summing annual contributions, while continuous valuations accumulate by integrating over time. In the discrete case, payments and mortality are tied to whole-year intervals. The probability that death occurs during a given year is derived from the life-table quantities qx (the probability of dying within year x) and tpx (the probability of surviving to the end of a year). You take the payment for that year and multiply it by the probability of death in that year, then discount and sum across years. This is why sums over years with qx appear. In the continuous case, time is treated as a continuum, and mortality is described by the force of mortality μ(t) (often written as μ_{x+t} for someone currently aged x). The probability density of death at exact time t is μ_{x+t} times the probability of surviving to time t (tpx). The present value then becomes an integral over time of the discount factor times this instantaneous death density, which is why integrals over time with μ and tpx appear. So the statement captures the fundamental modeling difference: discrete uses sums over years with qx; continuous uses integrals over time with μ and tpx.

The essential distinction is how time and mortality are modeled: discrete valuations accumulate value by summing annual contributions, while continuous valuations accumulate by integrating over time.

In the discrete case, payments and mortality are tied to whole-year intervals. The probability that death occurs during a given year is derived from the life-table quantities qx (the probability of dying within year x) and tpx (the probability of surviving to the end of a year). You take the payment for that year and multiply it by the probability of death in that year, then discount and sum across years. This is why sums over years with qx appear.

In the continuous case, time is treated as a continuum, and mortality is described by the force of mortality μ(t) (often written as μ_{x+t} for someone currently aged x). The probability density of death at exact time t is μ_{x+t} times the probability of surviving to time t (tpx). The present value then becomes an integral over time of the discount factor times this instantaneous death density, which is why integrals over time with μ and tpx appear.

So the statement captures the fundamental modeling difference: discrete uses sums over years with qx; continuous uses integrals over time with μ and tpx.

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