multidecrins: Multiple decrement insurances and annuities

In lifecontingencies: Financial and Actuarial Mathematics for Life Contingencies

Multiple decrement insurances and annuities

Description

Axn.mdt gives the actuarial present value (APV) of a term insurance on a multiple decrement table, paying at the end of the year of decrement a benefit that may depend on the cause of decrement. axn.mdt gives the APV of an annuity payable while the insured is still in the active (no decrement yet) state.

Usage

Axn.mdt(object, x, n, i, decrement, benefits = 1, m = 0)

axn.mdt(object, x, n, i, m = 0, k = 1, payment = "advance")

Arguments

object	an mdt object
x	policyholder's age (an integer age tabulated in object)
n	contract duration in years. If missing, the cover runs to the end of the table, i.e. n = \omega + 1 - x - m.
i	interest rate
decrement	decrement(s) covered: one or more names (or column indices) among getDecrements(object). If missing, every decrement is covered (insurance on the total decrement (\tau)).
benefits	benefit amounts, one per element of decrement (recycled). Default 1, i.e. a unit benefit for every covered cause.
m	deferment period in years (default 0).
k	number of annuity payments per year (default 1). Survival probabilities at fractional durations are interpolated linearly in l^{(\tau)}_x.
payment	"advance" (or "due", default) or "arrears" (or "immediate").

Details

With v = (1+i)^{-1} the insurance APV is

\sum_{j \in J} b_j \sum_{h=0}^{n-1} v^{m+h+1}\,{}_{m+h}p^{(\tau)}_x\, q^{(j)}_{x+m+h},

which for a single cause reduces to the historical Axn.mdt. The annuity APV is \frac1k \sum_{h} v^{t_h}\, {}_{t_h}p^{(\tau)}_x, with payment times t_h = m, m+1/k, \ldots, m+n-1/k (in advance) or m+1/k, \ldots, m+n (in arrears). Benefit premiums and reserves follow from the equivalence principle as ratios/differences of the two.

Value

A numeric vector of APVs (one per element of x).

References

Finan, M. B. (2014). A Reading of the Theory of Life Contingency Models: A Preparation for Exam MLC/3L, Sections 68-69.

Examples

# Finan (2014), Example 69.1: 3-year term on (16), i = 10%
myTable <- data.frame(x = 16:18, lx = c(20000, 17600, 14520),
                      da = c(1300, 1870, 2380), doc = c(1100, 1210, 1331))
myMdt <- new("mdt", table = myTable, name = "Finan 69.1")
A <- Axn.mdt(myMdt, x = 16, n = 3, i = 0.10, decrement = "doc")
a <- axn.mdt(myMdt, x = 16, n = 3, i = 0.10)
20000 * A / a   # level annual premium: 1250

# Finan (2014), Example 68.1: benefit 1 for cause 1, 2 for cause 2
t681 <- data.frame(x = 50:51, lx = c(1200, 800),
                   d1 = c(100, 200), d2 = c(300, 300))
m681 <- new("mdt", table = t681)
Axn.mdt(m681, x = 50, n = 2, i = 0.5, decrement = c("d1", "d2"),
        benefits = c(1, 2))  # 0.6852

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