Actuarial present values under variable annual interest rates

Computes life-contingent actuarial present values using a specified sequence of annual effective interest rates.

Usage

nEx_var(qx, i, benefit = 1)

Axn1_var(qx, i, benefit = 1)

Axn_var(qx, i, benefit = 1)

axn_var(qx, i, type = c("immediate", "due"), benefit = 1)

Arguments

qx

Numeric vector of one-year mortality probabilities.

i

Numeric vector of annual effective interest rates. Each value must be greater than -1.

benefit

Nonnegative scalar benefit or annuity payment amount.

type

Character string equal to "immediate" or "due".

Value

A numeric scalar.

Details

The vectors qx and i represent one valuation scenario and must have the same positive length.

nEx_var() computes a pure endowment.

Axn1_var() computes term insurance payable at the end of the year of death.

Axn_var() computes endowment insurance.

axn_var() computes a temporary annuity-immediate or annuity-due.

Each year's payment is discounted using the cumulative product of the annual effective interest rates supplied in i.

The pure endowment is $$ {}_nE = {}_np_x\,v_n, $$ where \(v_n\) is the cumulative discount factor implied by the sequence of annual effective interest rates.

Term insurance is obtained by discounting each possible death benefit using the cumulative discount factor applicable to its payment year.

Endowment insurance equals the sum of the corresponding term insurance and pure endowment.

Temporary annuities discount each payment using the cumulative discount factors derived from the interest-rate sequence.

Examples

qx <- c(0.03, 0.04, 0.05, 0.06, 0.07)
rates <- c(0.06, 0.07, 0.08, 0.09, 0.10)

nEx_var(qx, rates, benefit = 1000)
#> [1] 526.5563
Axn1_var(qx, rates)
#> [1] 0.1799082
Axn_var(qx, rates)
#> [1] 0.7064644
axn_var(qx, rates, type = "due")
#> [1] 4.081115