curtate vs complete future lifetime

How long someone will still live can be measured two ways, and the choice changes the arithmetic. The complete future lifetime is the exact remaining time, decimals and all — a person aged 65 who dies at age 84 years and 7 months had a complete future lifetime of about 19.58 years. This is the continuous random variable T(x). The curtate future lifetime simply throws away the fractional part and counts whole years completed: that same person completed 19 full years, so the curtate future lifetime K(x) = 19.

Why bother with the cruder, rounded-down version? Because insurance is mostly paid on a yearly rhythm. Premiums arrive once a year, annuities pay once a year, and a death benefit may be deemed paid at the end of the year of death. For those calculations you only need to know how many whole years a life survives, which is exactly K(x) = the integer part of T(x). The continuous T(x) matters for benefits paid at the precise moment of death and for theoretical work. K(x) is a discrete (whole-number) random variable; T(x) is continuous.

The two are closely linked: K(x) = floor of T(x), and on average T(x) is roughly K(x) plus about a half, because a death is, loosely, equally likely anywhere within its final year. Keeping the distinction straight prevents a classic error — mixing a curtate annuity formula with a complete-lifetime expectation, which double-counts or omits that final half-year of life.