exn: Expected residual life.

In lifecontingencies: Financial and Actuarial Mathematics for Life Contingencies

Expected residual life.

Description

Expected future lifetime of a life aged x, either over the whole remaining lifespan or over a temporary period of n years.

Usage

exn(object, x, n, type = "curtate", fxt = 0.5, axOmega = 1 - fxt)

Arguments

object	A lifetable/actuarialtable object.
x	Attained age
n	Length (in years) of the period over which the expected lifetime is computed, i.e. a temporary expectation. Assumed omega - x + 1 (the whole remaining lifespan) whether missing.
type	Either "Tx", "complete" or "continuous" for the complete (continuous) future lifetime, "Kx" or "curtate" for the curtate future lifetime (can be abbreviated). Default is "curtate".
fxt	fraction of the year of age lived by those who die within the year, used by the "complete" branch (see Lxt and Tx). Defaults to 0.5 (uniform distribution of deaths); ignored by the "curtate" branch.
axOmega	mean number of years lived in the last, open-ended age interval, used by the "complete" branch when the period reaches the last tabulated age (see Tx). Defaults to 1 - fxt, which reproduces the historical closed-interval behaviour.

Details

For type = "curtate" the function returns the (temporary) curtate expectation of life

e_{x:\overline{n}|} = \sum_{k=1}^{n} {}_k p_x ,

that is the expected number of complete future years lived by (x) within the next n years. With n missing it is the curtate expectation of life e_x = E[K_x].

For type = "complete" the function returns the (temporary) complete expectation of life

\mathring{e}_{x:\overline{n}|} = \int_0^n {}_t p_x \, dt = \frac{{}_nL_x}{l_x} ,

evaluated under the uniform distribution of deaths (UDD) assumption within each year, i.e. L_x = l_x - 0.5 d_x (see Lxt). With n missing it is \mathring{e}_x = T_x / l_x = E[T_x].

Under UDD the two quantities are related by \mathring{e}_{x:\overline{n}|} = e_{x:\overline{n}|} + 0.5\,(1 - {}_n p_x), which reduces to \mathring{e}_x = e_x + 0.5 when n covers the whole remaining lifespan.

The last tabulated age \omega is treated as a closed interval: those alive at \omega are assumed to die on average half a year later. Published tables that close the table with an open interval (e.g. L_{\omega} = l_{\omega}/m_{\omega}, as in the NCHS tables) can therefore show a slightly larger complete life expectancy.

Value

A numeric value representing the expected life span.

Author(s)

Giorgio Alfredo Spedicato

References

Actuarial Mathematics (Second Edition), 1997, by Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J.

See Also

lifetable, Tx, Lxt

Examples

#loads and show
data(soa08Act)
#curtate expectation of life at birth
exn(object=soa08Act, x=0)
#complete expectation of life at birth (curtate + 0.5 under UDD)
exn(object=soa08Act, x=0,type="complete")
#temporary 20-year expectations at age 50
exn(object=soa08Act, x=50, n=20)
exn(object=soa08Act, x=50, n=20, type="complete")

lifecontingencies documentation built on Oct. 10, 2026, 9:06 a.m.