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Chapter 9 premium, loss, and expense functions

premium_ch9.Rd

Functions in this file implement Chapter 9 funding-plan formulas, including:

  • net annual premiums under the equivalence principle,

  • limited-payment premiums,

  • continuous-payment premium rates,

  • fully continuous premium rates,

  • true fractional premiums,

  • present-value-of-loss means and variances,

  • a basic gross premium formula for whole life insurance.

Usage

Px(x, i, tbl = NULL, model = NULL, ...)

Pxn1(x, n, i, tbl = NULL, model = NULL, ...)

PnEx(x, n, i, tbl = NULL, model = NULL, ...)

Pxn(x, n, i, tbl = NULL, model = NULL, ...)

tPx(x, t, i, tbl = NULL, model = NULL, ...)

tPxn1(x, n, t, i, tbl = NULL, model = NULL, ...)

tPnEx(x, n, t, i, tbl = NULL, model = NULL, ...)

tPxn(x, n, t, i, tbl = NULL, model = NULL, ...)

PnAx(x, n, i, tbl = NULL, model = NULL, ...)

tPnAx(x, n, t, i, tbl = NULL, model = NULL, ...)

Pbarx(x, i, model, ..., tol = 1e-10)

Pbarxn1(x, n, i, model, ...)

Pbarxn(x, n, i, model, ...)

PbarAbarx(x, i, model, ..., tol = 1e-10)

PbarAbarxn1(x, n, i, model, ...)

PbarAbarxn(x, n, i, model, ...)

Px_m(x, m, i, tbl = NULL, model = NULL, ...)

Pxn1_m(x, n, m, i, tbl = NULL, model = NULL, ...)

Pxn_m(x, n, m, i, tbl = NULL, model = NULL, ...)

PnAx_m(x, n, m, i, tbl = NULL, model = NULL, ...)

EL0x(x, P, i, tbl = NULL, model = NULL, ...)

varL0x(x, P, i, tbl = NULL, model = NULL, ..., tol = 1e-12, k_max = 5000)

EL0xn1(x, n, P, i, tbl = NULL, model = NULL, ...)

varL0xn1(x, n, P, i, tbl = NULL, model = NULL, ...)

EL0xn(x, n, P, i, tbl = NULL, model = NULL, ...)

varL0xn(x, n, P, i, tbl = NULL, model = NULL, ...)

EL0barAbarx(x, P, i, model, ..., tol = 1e-10)

varL0barAbarx(x, P, i, model, ...)

Gx(
  x,
  i,
  benefit = 1,
  first_premium_pct = 0,
  renewal_premium_pct = 0,
  first_policy_exp = 0,
  renewal_policy_exp = 0,
  settlement_exp = 0,
  tbl = NULL,
  model = NULL,
  ...
)

Arguments

x

Age.

i

Effective annual interest rate.

tbl

Optional life table object.

model

Optional parametric survival model name.

...

Additional arguments passed to survival-model functions.

n

Term.

t

Premium-paying period.

tol

Numerical tolerance for functions that truncate infinite sums.

m

Number of payments per year.

P

Premium amount or premium rate.

k_max

Maximum summation horizon for functions that truncate infinite sums.

benefit

Benefit amount.

first_premium_pct

First-year premium expense proportion.

renewal_premium_pct

Renewal premium expense proportion.

first_policy_exp

First-year fixed expense.

renewal_policy_exp

Renewal fixed expense each year after the first.

settlement_exp

Settlement expense incurred at benefit payment.

Value

Numeric vector.

Numeric vector.

Numeric vector.

Numeric vector.

Numeric vector.

Numeric vector.

Numeric vector.

Numeric vector.

Numeric vector.

Numeric vector.

Numeric vector.

Numeric vector.

Numeric vector.

Numeric vector.

Numeric vector.

Numeric vector.

Numeric vector.

Numeric vector.

Numeric vector.

Numeric vector.

Numeric vector.

Numeric vector.

Numeric vector.

Numeric vector.

Details

Naming follows Chapter 9 notation as closely as possible:

  • Px() = whole life annual premium

  • Pxn1() = term insurance annual premium

  • PnEx() = pure endowment annual premium

  • Pxn() = endowment insurance annual premium

  • tPx() = limited-payment whole life annual premium

  • tPxn1() = limited-payment term insurance annual premium

  • tPnEx() = limited-payment pure endowment annual premium

  • tPxn() = limited-payment endowment insurance annual premium

  • PnAx() = deferred insurance annual premium

  • tPnAx() = limited-payment deferred insurance annual premium

  • Pbarx() = continuous-payment premium for discrete whole life insurance

  • Pbarxn1() = continuous-payment premium for discrete term insurance

  • Pbarxn() = continuous-payment premium for discrete endowment insurance

  • PbarAbarx() = fully continuous premium for continuous whole life insurance

  • PbarAbarxn1() = fully continuous premium for continuous term insurance

  • PbarAbarxn() = fully continuous premium for continuous endowment insurance

  • Px_m() = true fractional whole life annual premium

  • Pxn1_m() = true fractional term insurance annual premium

  • Pxn_m() = true fractional endowment insurance annual premium

  • PnAx_m() = true fractional deferred insurance annual premium

The discrete premium functions can be evaluated from either a life table via tbl = ... or from a parametric model via model = ....

The continuous-premium-rate functions use the continuous annuity functions already in the package, so they are written for the parametric survival model framework.