Continuous multiple-decrement insurance APV \(\overline{A}_{x}^{(j)}\)

Computes the actuarial present value of a benefit payable at the moment of decrement by Cause \(j\), matching Equation (14.4) in Chapter 14.

Usage

Abarxj_md(t, ptau, muj, delta, benefit = 1)

Arguments

t

Numeric vector of time points.

ptau

Numeric vector of values \({}_{t}p_{x}^{(\tau)}\).

muj

Numeric vector of values \(\mu_{x+t}^{(j)}\).

delta

Force of interest.

benefit

Benefit amount payable on decrement by Cause \(j\).

Value

A numeric scalar.

Details

The integral is evaluated numerically by the trapezoidal rule: $$ \overline{A}_{x}^{(j)} = \int_0^T v^t {}_{t}p_{x}^{(\tau)} \mu_{x+t}^{(j)} dt $$

Examples

t <- seq(0, 20, by = 0.01)
ptau <- exp(-0.012 * t)
mu_ac <- rep(0.002, length(t))
Abarxj_md(t, ptau, mu_ac, delta = 0.05, benefit = 2000)
#> [1] 45.84618