varxn: Variance and standard deviation of the future lifetime

In lifecontingencies: Financial and Actuarial Mathematics for Life Contingencies

Variance and standard deviation of the future lifetime

Description

varxn and sdxn return the variance and the standard deviation of the future lifetime of a life aged x, as the natural second-moment companions of exn (which returns the mean).

Usage

varxn(object, x, n, type = "Kx")

sdxn(object, x, n, type = "Kx")

Arguments

object	A lifetable or actuarialtable object.
x	Attained age. Defaults to 0.
n	Length (in years) of a temporary period. If missing, the whole remaining lifespan is used.
type	Either "Kx"/"curtate" for the curtate future lifetime K_x or "Tx"/"complete"/"continuous" for the complete future lifetime T_x (can be abbreviated). Default is "Kx".

Details

For the curtate future lifetime the (temporary) variance is computed exactly from the life table as

\mathrm{Var}(\min(K_x, n)) = \sum_{k=1}^{n}(2k-1)\,{}_k p_x - \left(\sum_{k=1}^{n}{}_k p_x\right)^2 .

For the complete future lifetime the variance is returned only for the whole remaining lifespan (n missing), using the uniform-distribution-of-deaths relation \mathrm{Var}(T_x) = \mathrm{Var}(K_x) + 1/12 (because, under UDD, T_x = K_x + U with U\sim\mathrm{Unif}(0,1) independent of K_x). A temporary complete variance is not provided; use type = "Kx" for a temporary period.

Value

A numeric value: the variance (varxn) or standard deviation (sdxn) of the (temporary) future lifetime.

Author(s)

Giorgio Alfredo Spedicato

References

Dickson, D.C.M., Hardy, M.R., Waters, H.R. (2013), Actuarial Mathematics for Life Contingent Risks, 2nd ed., Cambridge University Press.

See Also

exn

Examples

data(soa08Act)
# variance and sd of the curtate future lifetime at birth
varxn(soa08Act, x = 0)
sdxn(soa08Act, x = 0)
# complete future lifetime: Var(T) = Var(K) + 1/12
varxn(soa08Act, x = 65, type = "complete")
# temporary (20-year) curtate future lifetime at age 40
varxn(soa08Act, x = 40, n = 20)

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