Curtate Future Lifetime¶

Definition¶

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

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.

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.

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$$\sum _{k=1}^{\infty }{}_k{p}_x$.
• $\displaystyle \mathbb{E}[K_x^2]=\sum_{k=1}^{\infty}(2k-1){}_kp_x$$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¶

• Future Lifetime
• Actuarial Survival and Death Probabilities

Related Concepts¶

• Complete and Curtate Expectation of Life
• Whole Life Insurance
• Floor and Ceiling Functions

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