Spot-rate actuarial present value functions — chapter15_spot_interest

Chapter 15 functions for actuarial present values when discounting uses spot rates by maturity. If $z_t$ denotes the annual effective spot rate for maturity $t$, then the discount factor is $(1+z_t)^{-t}$.

Computes $$ {}_nE_x = (1+z_n)^{-n}\cdot {}_np_x. $$

Computes $$ A_{x:\overline{n}|}^1 = \sum_{t=1}^{n}(1+z_t)^{-t}\cdot {}_{t-1}p_x \cdot q_{x+t-1}. $$

Computes $$ A_{x:\overline{n}|} = A_{x:\overline{n}|}^1 + {}_nE_x $$ using spot-rate discount factors.

For an immediate annuity, $$ a_{x:\overline{n}|} = \sum_{t=1}^{n}(1+z_t)^{-t}\cdot {}_tp_x. $$

Usage

nEx_spot(qx, z, benefit = 1)

Axn1_spot(qx, z, benefit = 1)

Axn_spot(qx, z, benefit = 1)

axn_spot(qx, z, type = c("immediate", "due"), benefit = 1)

Arguments

qx: Numeric vector of one-year mortality rates.
z: Numeric vector of annual effective spot rates for maturities $1,\dots,n$.
benefit: Amount of each annuity payment.
type: Either "immediate" or "due".

Value

A numeric scalar.

Details

For an annuity-due, $$ \ddot{a}_{x:\overline{n}|} = \sum_{t=0}^{n-1}(1+z_t)^{-t}\cdot {}_tp_x, $$ where the time-0 discount factor is 1.

Examples

qx <- c(.02, .03, .04, .05, .06)
z  <- c(.03, .04, .05, .06, .07)
nEx_spot(qx, z, benefit = 1000000)
#> [1] 581034.1

qx <- c(.02, .03, .04, .05, .06)
z  <- c(.03, .04, .05, .06, .07)
Axn1_spot(qx, z)
#> [1] 0.1526756

qx <- c(.02, .03, .04, .05, .06)
z  <- c(.03, .04, .05, .06, .07)
Axn_spot(qx, z)
#> [1] 0.7337096

qx <- c(.02, .03, .04, .05, .06)
z  <- c(.03, .04, .05, .06, .07)
axn_spot(qx, z, type = "due")
#> [1] 4.30536