Curtate Future Lifetime¶

Definition¶

The curtate future lifetime of a life aged $x$ is $K_x=\lfloor T_x\rfloor$ $⌊ T_{x} ⌋$ , the number of complete future years lived. It takes values $0,1,2,\ldots$ . Under the continuous lifetime assumption,

\mathbb{P}(K_x=k)={}_kp_x-{}_{k+1}p_x={}_kp_xq_{x+k}.

Interpretation¶

Annual insurance payments often depend on the year of death rather than its exact time. A death after 3.7 years has $K_x=3$ and an end-of-year death payment at time 4.

Time axis divided into years labelled with curtate lifetimes 0 to 4; a life ending at time 3.7 falls in the shaded year from 3 to 4, and an end-of-year payment is made at time 4.

Each year $[k,k+1)$ of the time axis is labelled with the value of $K_x$ for a death inside it. The life in the figure ends at $T_x=3.7$ $3.7$ , inside the shaded year $[3,4)$ , so $K_x=3$ and an end-of-year death payment is made at $K_x+1=4$ $4$ . The position of the death within the year is discarded: $\mathbb{P}(K_x=3)$ is the probability that the death falls anywhere in the shaded year.

Move $T_x$ : every death in the year $[k,k+1)$ gives $K_x=k$ $k$ , and the bar over that year has area $\mathbb P(K_x=k)={}_kp_x-{}_{k+1}p_x$ $_{k + 1} p_{x}$ , the area under the density over the same year.

Taking the floor of future lifetime $T_x$ records only completed years. The actuarial survival and death probabilities at successive integer durations give the probability of death during year $k$ : survive $k$ years, then die before the next anniversary.

Key Results¶

$\{K_x\ge k\}=\{T_x\ge k\}$ for integer $k$ ; strict versus weak inequalities have the same probability under continuous lifetimes.
$\displaystyle \mathbb{E}[K_x]=\sum_{k=1}^{\infty}{}_kp_x$ $k = 1 \sum \infty_{k} p_{x}$ .
$\displaystyle \mathbb{E}[K_x^2]=\sum_{k=1}^{\infty}(2k-1){}_kp_x$ $1)_{k} p_{x}$ , when finite.
A mass function for $K_x$ gives probabilities; a density for $T_x$ does not.

Examples¶

At constant force of mortality $\lambda$ , $\mathbb{P}(K_x=k)=\mathrm{e}^{-\lambda k}(1-\mathrm{e}^{-\lambda})$ and $\mathbb{E}[K_x]=1/(\mathrm{e}^\lambda-1)$ $1)$ .

Prerequisites¶

Sources¶

Life Insurance · Lecture 1

Aliases: completed future years, curtate lifetime, integer part of future lifetime Keywords: floor of T x, complete years lived before death, P(K = k) = k p x q x+k, year of death, end-of-year death payment at K + 1 Symbols: infinity, lambda, sum

Last updated: 14 October 2026 08:21