Variable-interest actuarial present value functions — chapter15_variable_interest

Chapter 15 functions for actuarial present values under variable annual effective interest rates interpreted as a yearly scenario $i_1, i_2, \dots, i_n$.

Computes the APV of an $n$-year pure endowment under a variable annual interest scenario: $$ {}_nE_x = v_n \cdot {}_np_x. $$

Computes the APV of an $n$-year term insurance with benefit paid at the end of the year of death under a variable annual interest scenario: $$ A_{x:\overline{n}|}^1 = \sum_{t=1}^{n} v_t \cdot {}_{t-1}p_x \cdot q_{x+t-1}. $$

Computes the APV of an $n$-year endowment insurance under a variable annual interest scenario: $$ A_{x:\overline{n}|} = A_{x:\overline{n}|}^1 + {}_nE_x. $$

Computes the APV of an $n$-year temporary life annuity under a variable annual interest scenario.

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 rates $q_x, q_{x+1}, \dots, q_{x+n-1}$.
i: Numeric vector of annual effective interest rates $i_1, i_2, \dots, i_n$.
benefit: Amount of each annuity payment.
type: Either "immediate" or "due".

Value

A numeric scalar.

Details

If a benefit amount is supplied, the function returns that benefit times the APV factor.

For an immediate annuity, $$ a_{x:\overline{n}|} = \sum_{t=1}^{n} v_t \cdot {}_tp_x. $$

For an annuity-due, $$ \ddot{a}_{x:\overline{n}|} = \sum_{t=0}^{n-1} v_t \cdot {}_tp_x, $$ with $v_0 = 1$.

Examples

qx <- c(.03, .04, .05, .06, .07)
nEx_var(qx = qx, i = c(.06, .07, .08, .09, .10), benefit = 1000)
#> [1] 526.5563

qx <- c(.03, .04, .05, .06, .07)
Axn1_var(qx = qx, i = c(.06, .07, .08, .09, .10))
#> [1] 0.1799082

qx <- c(.03, .04, .05, .06, .07)
Axn_var(qx = qx, i = c(.06, .07, .08, .09, .10))
#> [1] 0.7064644

qx <- rep(.02, 5)
axn_var(qx = qx, i = c(.06, .05, .04, .03, .03), type = "immediate")
#> [1] 4.110256
axn_var(qx = qx, i = c(.03, .04, .05, .06, .07), type = "due")
#> [1] 4.458454