Package 'mqriskR'

Title: Actuarial Risk Modeling and Life Contingencies
Description: Provides functions for actuarial risk modeling, including survival models, life annuities, multiple-decrement models, and mortality improvement projections. The package is designed to align with standard actuarial notation and supports teaching, exam preparation, and reproducible actuarial analysis. The methods are based on standard actuarial references including Camilli, Duncan and London (2014, ISBN:9781625423474) "Models for Quantifying Risk" and Dickson, Hardy and Waters (2020, ISBN:9781108478083) "Actuarial Mathematics for Life Contingent Risks".
Authors: Nii Okine [aut, cre]
Maintainer: Nii Okine <[email protected]>
License: MIT + file LICENSE
Version: 0.1.0
Built: 2026-05-19 10:38:19 UTC
Source: https://github.com/aokine/mqriskr

Help Index

Second moment of continuous whole life insurance PV

Description

Computes 2Aˉx{}^{2}\bar{A}_x by evaluating Aˉx\bar{A}_x at doubled force.

Usage

A2barx(x, i, model, ...)

Arguments

x

Age.
i

Effective annual interest rate.
model

Parametric survival model name.
...

Additional model parameters passed to survival-model functions.

Value

Numeric vector of second moments.

Second moment of continuous endowment insurance PV

Description

Computes 2Aˉx:n‾∣{}^{2}\bar{A}_{x:\overline{n}|}.

Usage

A2barxn(x, n, i, model, ...)

Arguments

x

Age.
n

Term.
i

Effective annual interest rate.
model

Parametric survival model name.
...

Additional model parameters passed to survival-model functions.

Value

Numeric vector of second moments.

Second moment of continuous term insurance PV

Description

Computes 2Aˉx:n‾∣1{}^{2}\bar{A}_{x:\overline{n}|}^{1} by evaluating Aˉx:n‾∣1\bar{A}_{x:\overline{n}|}^{1} at doubled force.

Usage

A2barxn1(x, n, i, model, ...)

Arguments

x

Age.
n

Term.
i

Effective annual interest rate.
model

Parametric survival model name.
...

Additional model parameters passed to survival-model functions.

Value

Numeric vector of second moments.

Second moment of continuous deferred insurance PV

Description

Computes 2n∣Aˉx{}^{2}{}_{n\mid}\bar{A}_x by evaluating n∣Aˉx{}_{n\mid}\bar{A}_x at doubled force.

Usage

A2nAbarx(x, n, i, model, ...)

Arguments

x

Age.
n

Deferral period.
i

Effective annual interest rate.
model

Parametric survival model name.
...

Additional model parameters passed to survival-model functions.

Value

Numeric vector of second moments.

Second moment of deferred insurance PV

Description

Computes 2n∣Ax{}^{2}{}_{n\mid}A_x by evaluating n∣Ax{}_{n\mid}A_x at doubled force.

Usage

A2nAx(x, n, i, tbl = NULL, model = NULL, ..., tol = 1e-12, k_max = 5000)

Arguments

x

Age.
n

Deferral period.
i

Effective annual interest rate.
tbl

Optional life table object.
model

Optional parametric survival model name.
...

Additional arguments passed to survival-model functions.
tol

Numerical tolerance for truncating infinite sums.
k_max

Maximum number of terms in the sum.

Value

Numeric vector of second moments.

Second moment of m-thly deferred insurance PV

Description

Second moment of m-thly deferred insurance PV

Usage

A2nAx_m(x, n, i, m, model, ..., tol = 1e-12, j_max = 100000L)

Arguments

x

Age.
n

Deferral period.
i

Effective annual interest rate.
m

Positive integer payment frequency.
model

Parametric survival model name.
...

Additional model parameters passed to survival-model functions.
tol

Numerical tolerance for truncating the infinite sum.
j_max

Maximum number of m-thly intervals in the sum.

Value

Numeric vector of second moments.

Second moment of pure endowment PV

Description

Computes 2nEx=(v′)nnpx{}^{2}{}_nE_x = (v')^n {}_n p_x.

Usage

A2nEx(x, n, i, tbl = NULL, model = NULL, ...)

Arguments

x

Age.
n

Term.
i

Effective annual interest rate.
tbl

Optional life table object.
model

Optional parametric survival model name.
...

Additional arguments passed to survival-model functions.

Value

Numeric vector of second moments.

Second moment of whole life insurance PV

Description

Computes 2Ax{}^{2}A_x by evaluating AxA_x at doubled force.

Usage

A2x(x, i, tbl = NULL, model = NULL, ..., tol = 1e-12, k_max = 5000)

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.
tol

Numerical tolerance for truncating infinite sums.
k_max

Maximum number of terms in the sum.

Value

Numeric vector of second moments.

Second moment of m-thly whole life insurance PV

Description

Computes 2Ax(m){}^{2}A_x^{(m)} by evaluating Ax(m)A_x^{(m)} at doubled force.

Usage

A2x_m(x, i, m, model, ..., tol = 1e-12, j_max = 100000L)

Arguments

x

Age.
i

Effective annual interest rate.
m

Positive integer payment frequency.
model

Parametric survival model name.
...

Additional model parameters passed to survival-model functions.
tol

Numerical tolerance for truncating the infinite sum.
j_max

Maximum number of m-thly intervals in the sum.

Value

Numeric vector of second moments.

Second moment of endowment insurance PV

Description

Computes 2Ax:n‾∣{}^{2}A_{x:\overline{n}|}.

Usage

A2xn(x, n, i, tbl = NULL, model = NULL, ...)

Arguments

x

Age.
n

Term.
i

Effective annual interest rate.
tbl

Optional life table object.
model

Optional parametric survival model name.
...

Additional arguments passed to survival-model functions.

Value

Numeric vector of second moments.

Second moment of m-thly endowment insurance PV

Description

Second moment of m-thly endowment insurance PV

Usage

A2xn_m(x, n, i, m, model, ...)

Arguments

x

Age.
n

Term.
i

Effective annual interest rate.
m

Positive integer payment frequency.
model

Parametric survival model name.
...

Additional model parameters passed to survival-model functions.

Value

Numeric vector of second moments.

Second moment of term insurance PV

Description

Computes 2Ax:n‾∣1{}^{2}A_{x:\overline{n}|}^{1} by evaluating Ax:n‾∣1A_{x:\overline{n}|}^{1} at doubled force.

Usage

A2xn1(x, n, i, tbl = NULL, model = NULL, ...)

Arguments

x

Age.
n

Term.
i

Effective annual interest rate.
tbl

Optional life table object.
model

Optional parametric survival model name.
...

Additional arguments passed to survival-model functions.

Value

Numeric vector of second moments.

Second moment of m-thly term insurance PV

Description

Second moment of m-thly term insurance PV

Usage

A2xn1_m(x, n, i, m, model, ...)

Arguments

x

Age.
n

Term.
i

Effective annual interest rate.
m

Positive integer payment frequency.
model

Parametric survival model name.
...

Additional model parameters passed to survival-model functions.

Value

Numeric vector of second moments.

Projected Unit Credit accrued liability for a DB plan

Description

Computes the PUC accrued liability as the APV of the portion of the projected benefit attributed to past service.

Usage

AAL_PUC_db(
  projected_benefit,
  past_service,
  total_service,
  v_to_ret,
  p_surv,
  adue_ret
)

Arguments

projected_benefit

Projected benefit at retirement.
past_service

Past service completed.
total_service

Total service at retirement.
v_to_ret

Discount factor to retirement.
p_surv

Active-service survival probability to retirement.
adue_ret

Retirement annuity factor.

Value

PUC accrued liability.

Examples

AAL_PUC_db(projected_benefit = 30000, past_service = 10, total_service = 30,
v_to_ret = 0.5, p_surv = 0.9, adue_ret = 12)

Traditional Unit Credit accrued liability for a DB plan

Description

Computes the TUC accrued liability as the APV of the accrued benefit.

Usage

AAL_TUC_db(accrued_benefit, v_to_ret, p_surv, adue_ret)

Arguments

accrued_benefit

Accrued benefit at the valuation date.
v_to_ret

Discount factor to retirement.
p_surv

Active-service survival probability to retirement.
adue_ret

Retirement annuity factor.

Value

TUC accrued liability.

Examples

AAL_TUC_db(accrued_benefit = 12000, v_to_ret = 0.5, p_surv = 0.9, adue_ret = 12)

Accrued benefit for a career average earnings plan

Description

Computes the accrued benefit using actual salary history only.

Usage

AB_cae(salary_history, p)

Arguments

salary_history

Numeric vector of annual salaries to date.
p

Accrual percentage.

Value

Accrued benefit.

Examples

AB_cae(salary_history = c(100000, 104000, 108160), p = 1)

Accrued benefit for a final average salary plan

Description

Computes the accrued benefit at the current date using service and salary history only.

Usage

AB_fas(salary_history, p, fas_years = 3)

Arguments

salary_history

Numeric vector of annual salaries to date.
p

Accrual percentage, e.g. 2 for 2 percent.
fas_years

Number of years in the final average salary average.

Value

Accrued benefit.

Examples

AB_fas(salary_history = c(150000, 156000), p = 1, fas_years = 2)

Continuous whole life insurance APV

Description

Computes Aˉx=∫0∞vt tpxμx+t dt\bar{A}_x = \int_0^\infty v^t \, {}_t p_x \mu_{x+t}\,dt.

Usage

Abarx(x, i, model, ...)

Arguments

x

Age.
i

Effective annual interest rate.
model

Parametric survival model name.
...

Additional model parameters passed to survival-model functions.

Value

Numeric vector of APVs.

UDD approximation of continuous whole life insurance

Description

Computes Aˉx=(i/δ)Ax\bar{A}_x = (i/\delta)A_x.

Usage

Abarx_udd(Ax, i)

Arguments

Ax

Discrete whole life insurance APV.
i

Effective annual interest rate.

Value

Continuous whole life insurance APV under UDD.

Continuous reversionary annuity to (y) after death of (x)

Description

Computes a‾x∣y=a‾y−a‾xy\overline{a}_{x|y} = \overline{a}_y - \overline{a}_{xy}.

Usage

abarx_y(x, y, i, tbl = NULL, model = NULL, ...)

Arguments

x

Age of first life.
y

Age of second life.
i

Effective annual interest rate.
tbl

Life table.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

abarx_y(40, 50, i = 0.05, model = "uniform", omega = 100)

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

Description

Computes the actuarial present value of a benefit payable at the moment of decrement by Cause jj, 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 tpx(τ){}_{t}p_{x}^{(\tau)}.
muj

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

Force of interest.
benefit

Benefit amount payable on decrement by Cause jj.

Details

The integral is evaluated numerically by the trapezoidal rule:

A‾x(j)=∫0Tvttpx(τ)μx+t(j)dt\overline{A}_{x}^{(j)} = \int_0^T v^t {}_{t}p_{x}^{(\tau)} \mu_{x+t}^{(j)} dt

Value

A numeric scalar.

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)

Continuous endowment insurance APV

Description

Computes Aˉx:n‾∣=Aˉx:n‾∣1+vn npx\bar{A}_{x:\overline{n}|} = \bar{A}_{x:\overline{n}|}^{1} + v^n\,{}_np_x.

Usage

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

Arguments

x

Age.
n

Term.
i

Effective annual interest rate.
model

Parametric survival model name.
...

Additional model parameters passed to survival-model functions.

Value

Numeric vector of APVs.

UDD approximation of continuous endowment insurance

Description

Computes Aˉx:n‾∣=(i/δ)Ax:n‾∣1+nEx\bar{A}_{x:\overline{n}|} = (i/\delta)A_{x:\overline{n}|}^{1} + {}_nE_x.

Usage

Abarxn_udd(Axn1, nEx, i)

Arguments

Axn1

Discrete term insurance APV.
nEx

Pure endowment APV, nEx{}_nE_x.
i

Effective annual interest rate.

Value

Continuous endowment insurance APV under UDD.

Continuous term insurance APV

Description

Computes Aˉx:n‾∣1=∫0nvt tpxμx+t dt\bar{A}_{x:\overline{n}|}^{1} = \int_0^n v^t \, {}_t p_x \mu_{x+t}\,dt.

Usage

Abarxn1(x, n, i, model, ...)

Arguments

x

Age.
n

Term.
i

Effective annual interest rate.
model

Parametric survival model name.
...

Additional model parameters passed to survival-model functions.

Value

Numeric vector of APVs.

UDD approximation of continuous term insurance

Description

Computes Aˉx:n‾∣1=(i/δ)Ax:n‾∣1\bar{A}_{x:\overline{n}|}^{1} = (i/\delta)A_{x:\overline{n}|}^{1}.

Usage

Abarxn1_udd(Axn1, i)

Arguments

Axn1

Discrete term insurance APV.
i

Effective annual interest rate.

Value

Continuous term insurance APV under UDD.

Continuous joint-life whole life insurance

Description

Computes A‾xy=1−δa‾xy\overline{A}_{xy} = 1 - \delta \overline{a}_{xy}.

Usage

Abarxy(x, y, i, tbl = NULL, model = NULL, ...)

Arguments

x

Age of first life.
y

Age of second life.
i

Effective annual interest rate.
tbl

Life table.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

Abarxy(40, 50, i = 0.05, model = "uniform", omega = 100)

Continuous joint-life whole life annuity

Description

Computes a‾xy=∫0∞vt tpxy dt\overline{a}_{xy} = \int_0^\infty v^t\,{}_tp_{xy}\,dt.

Usage

abarxy(x, y, i, tbl = NULL, model = NULL, ...)

Arguments

x

Age of first life.
y

Age of second life.
i

Effective annual interest rate.
tbl

Life table.
model

Survival model.
...

Additional model parameters.

Details

Shared documentation topic used to avoid filename collisions with case-distinct function names on case-insensitive file systems.

Value

Numeric vector.

Examples

abarxy(40, 50, i = 0.05, model = "uniform", omega = 100)

Continuous contingent insurance: benefit on death of (x) if before (y)

Description

Computes A‾xy1=∫0∞vt tpxyμx+t dt\overline{A}_{xy}^{1} = \int_0^\infty v^t\,{}_tp_{xy}\mu_{x+t}\,dt.

Usage

Abarxy1(x, y, i, tbl = NULL, model = NULL, ...)

Arguments

x

Age of first life.
y

Age of second life.
i

Effective annual interest rate.
tbl

Life table.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

Abarxy1(40, 50, i = 0.05, model = "uniform", omega = 100)

Continuous contingent insurance: benefit on death of (x) if after (y)

Description

Computes A‾xy2=A‾x−A‾xy1\overline{A}_{xy}^{2} = \overline{A}_x - \overline{A}_{xy}^{1}.

Usage

Abarxy2(x, y, i, tbl = NULL, model = NULL, ...)

Arguments

x

Age of first life.
y

Age of second life.
i

Effective annual interest rate.
tbl

Life table.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

Abarxy2(40, 50, i = 0.05, model = "uniform", omega = 100)

Continuous last-survivor whole life insurance

Description

Computes A‾xy‾=A‾x+A‾y−A‾xy\overline{A}_{\overline{xy}} = \overline{A}_x + \overline{A}_y - \overline{A}_{xy}.

Usage

Abarxybar(x, y, i, tbl = NULL, model = NULL, ...)

Arguments

x

Age of first life.
y

Age of second life.
i

Effective annual interest rate.
tbl

Life table.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

Abarxybar(40, 50, i = 0.05, model = "uniform", omega = 100)

Continuous last-survivor whole life annuity

Description

Computes a‾xy‾=a‾x+a‾y−a‾xy\overline{a}_{\overline{xy}} = \overline{a}_x + \overline{a}_y - \overline{a}_{xy}.

Usage

abarxybar(x, y, i, tbl = NULL, model = NULL, ...)

Arguments

x

Age of first life.
y

Age of second life.
i

Effective annual interest rate.
tbl

Life table.
model

Survival model.
...

Additional model parameters.

Details

Shared documentation topic used to avoid filename collisions with case-distinct function names on case-insensitive file systems.

Value

Numeric vector.

Examples

abarxybar(40, 50, i = 0.05, model = "uniform", omega = 100)

Continuous reversionary annuity to (x) after death of (y)

Description

Computes a‾y∣x=a‾x−a‾xy\overline{a}_{y|x} = \overline{a}_x - \overline{a}_{xy}.

Usage

abary_x(x, y, i, tbl = NULL, model = NULL, ...)

Arguments

x

Age of first life.
y

Age of second life.
i

Effective annual interest rate.
tbl

Life table.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

abary_x(40, 50, i = 0.05, model = "uniform", omega = 100)

Continuous contingent insurance: benefit on death of (y) if before (x)

Description

Computes A‾xy1=∫0∞vt tpxyμy+t dt\overline{A}_{xy}^{\hspace{1mm}1} = \int_0^\infty v^t\,{}_tp_{xy}\mu_{y+t}\,dt.

Usage

Abaryx1(x, y, i, tbl = NULL, model = NULL, ...)

Arguments

x

Age of first life.
y

Age of second life.
i

Effective annual interest rate.
tbl

Life table.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

Abaryx1(40, 50, i = 0.05, model = "uniform", omega = 100)

Continuous contingent insurance: benefit on death of (y) if after (x)

Description

Computes A‾xy2=A‾y−A‾xy1\overline{A}_{xy}^{\hspace{1mm}2} = \overline{A}_y - \overline{A}_{xy}^{\hspace{1mm}1}.

Usage

Abaryx2(x, y, i, tbl = NULL, model = NULL, ...)

Arguments

x

Age of first life.
y

Age of second life.
i

Effective annual interest rate.
tbl

Life table.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

Abaryx2(40, 50, i = 0.05, model = "uniform", omega = 100)

Joint-life whole life annuity-due

Description

Computes a¨xy\ddot{a}_{xy}.

Usage

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

Arguments

x

Age of first life.
y

Age of second life.
i

Effective annual interest rate.
tbl

Life table.
model

Survival model.
...

Additional model parameters.
k_max

Maximum number of terms.
tol

Convergence tolerance.

Value

Numeric vector.

Examples

adotxy(40, 50, i = 0.05, model = "uniform", omega = 100)

Last-survivor whole life annuity-due

Description

Computes a¨xy‾\ddot{a}_{\overline{xy}}.

Usage

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

Arguments

x

Age of first life.
y

Age of second life.
i

Effective annual interest rate.
tbl

Life table.
model

Survival model.
...

Additional model parameters.
k_max

Maximum number of terms.
tol

Convergence tolerance.

Value

Numeric vector.

Examples

adotxybar(40, 50, i = 0.05, model = "uniform", omega = 100)

Last-survivor temporary annuity-due

Description

Computes a¨xy‾:n‾∣\ddot{a}_{\overline{xy}:\overline{n}|}.

Usage

adotxybarn(x, y, n, i, tbl = NULL, model = NULL, ...)

Arguments

x

Age of first life.
y

Age of second life.
n

Term.
i

Effective annual interest rate.
tbl

Life table.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

adotxybarn(40, 50, n = 10, i = 0.05, model = "uniform", omega = 100)

Joint-life temporary annuity-due

Description

Computes a¨xy:n‾∣\ddot{a}_{xy:\overline{n}|}.

Usage

adotxyn(x, y, n, i, tbl = NULL, model = NULL, ...)

Arguments

x

Age of first life.
y

Age of second life.
n

Term.
i

Effective annual interest rate.
tbl

Life table.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

adotxyn(40, 50, n = 10, i = 0.05, model = "uniform", omega = 100)

AG 38 prefunding ratio

Description

Computes the prefunding ratio in Equation (16.18), capped at 1.

Usage

ag38_prefunding_ratio(excess_payment, nsp_required)

Arguments

excess_payment

Excess payment or shadow-fund amount.
nsp_required

Net single premium required to fully fund the guarantee.

Value

Numeric scalar.

Examples

ag38_prefunding_ratio(60000, 100000)

AG 38 reserve calculation

Description

Computes the main quantities in the Chapter 16 AG 38 reserve calculation, including the prefunding ratio, reduced deficiency reserve, Step (8) reserve, and final increased basic reserve.

Usage

ag38_reserve_ul(
  basic_reserve,
  deficiency_reserve = 0,
  excess_payment,
  nsp_required,
  valuation_nsp,
  surrender_charge = 0
)

Arguments

basic_reserve

Basic reserve.
deficiency_reserve

Deficiency reserve.
excess_payment

Excess payment or shadow-fund amount.
nsp_required

Net single premium required to fully fund the guarantee.
valuation_nsp

Valuation net single premium.
surrender_charge

Applicable surrender charge.

Value

A named list.

Examples

ag38_reserve_ul(
  basic_reserve = 10000,
  deficiency_reserve = 0,
  excess_payment = 60000,
  nsp_required = 100000,
  valuation_nsp = 150000,
  surrender_charge = 5000
)

Full preliminary term first-year modified premium

Description

Computes αF=vqx=Ax:1‾∣1\alpha^F = v q_x = A_{x:\overline{1}|}^{1}.

Computes αF=vqx=Ax:1‾∣1\alpha^F = v q_x = A_{x:\overline{1}|}^{1}.

Usage

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

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

Arguments

x

Issue age.
i

Effective annual interest rate.
tbl

Optional life table object.
model

Optional parametric survival model.
...

Additional model parameters.

Value

Numeric vector.

Numeric vector.

Examples

alphaF(40, i = 0.05, model = "uniform", omega = 100)
alphaF(40, i = 0.05, model = "uniform", omega = 100)

Annual annuity functions (Chapter 8)

Description

Annual whole life, temporary, deferred, and actuarial accumulated value annuity functions in immediate, due, and continuous forms.

Computes the annual whole life annuity-immediate ax=∑t=1∞vt tpxa_x = \sum_{t=1}^{\infty} v^t \, {}_t p_x.

Computes the annual whole life annuity-due a¨x=∑t=0∞vt tpx=1+ax\ddot{a}_x = \sum_{t=0}^{\infty} v^t \, {}_t p_x = 1 + a_x.

Computes the continuous whole life annuity aˉx=∫0∞vt tpx dt\bar{a}_x = \int_0^{\infty} v^t \, {}_t p_x \, dt.

Computes the annual temporary annuity-immediate ax:n‾∣=∑t=1nvt tpxa_{x:\overline{n}|} = \sum_{t=1}^{n} v^t \, {}_t p_x.

Computes the annual temporary annuity-due a¨x:n‾∣=∑t=0n−1vt tpx\ddot{a}_{x:\overline{n}|} = \sum_{t=0}^{n-1} v^t \, {}_t p_x.

Computes the continuous temporary annuity aˉx:n‾∣=∫0nvt tpx dt\bar{a}_{x:\overline{n}|} = \int_0^n v^t \, {}_t p_x \, dt.

Computes the annual deferred whole life annuity-immediate n∣ax=nEx ax+n{}_{n|}a_x = {}_nE_x \, a_{x+n}.

Computes the annual deferred whole life annuity-due n∣a¨x=nEx a¨x+n{}_{n|}\ddot{a}_x = {}_nE_x \, \ddot{a}_{x+n}.

Computes the continuous deferred whole life annuity n∣aˉx=nEx aˉx+n{}_{n|}\bar{a}_x = {}_nE_x \, \bar{a}_{x+n}.

Computes sx:n‾∣=ax:n‾∣/nExs_{x:\overline{n}|} = a_{x:\overline{n}|} / {}_nE_x.

Computes s¨x:n‾∣=a¨x:n‾∣/nEx\ddot{s}_{x:\overline{n}|} = \ddot{a}_{x:\overline{n}|} / {}_nE_x.

Computes sˉx:n‾∣=aˉx:n‾∣/nEx\bar{s}_{x:\overline{n}|} = \bar{a}_{x:\overline{n}|} / {}_nE_x.

Usage

ax(x, i, model, ..., k_max = 5000, tol = 1e-12)

adotx(x, i, model, ..., k_max = 5000, tol = 1e-12)

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

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

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

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

nax(x, n, i, model, ..., k_max = 5000, tol = 1e-12)

nadotx(x, n, i, model, ..., k_max = 5000, tol = 1e-12)

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

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

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

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

Arguments

x

Age.
i

Effective annual interest rate.
model

Survival model.
...

Additional model parameters passed to the survival model.
k_max

Maximum summation horizon for non-terminating models.
tol

Truncation tolerance for non-terminating models.
n

Term in years.

Details

Naming convention follows Chapter 8 notation:

  • ax() = axa_x

  • adotx() = a¨x\ddot{a}_x

  • abarx() = aˉx\bar{a}_x

  • axn() = ax:n‾∣a_{x:\overline{n}|}

  • adotxn() = a¨x:n‾∣\ddot{a}_{x:\overline{n}|}

  • abarxn() = aˉx:n‾∣\bar{a}_{x:\overline{n}|}

  • nax() = n∣ax{}_{n|}a_x

  • nadotx() = n∣a¨x{}_{n|}\ddot{a}_x

  • nabarx() = n∣aˉx{}_{n|}\bar{a}_x

  • sxn() = sx:n‾∣s_{x:\overline{n}|}

  • sdotxn() = s¨x:n‾∣\ddot{s}_{x:\overline{n}|}

  • sbarxn() = sˉx:n‾∣\bar{s}_{x:\overline{n}|}

These functions work directly from the Chapter 5 survival model functions and the Chapter 7 pure endowment function nEx().

Value

Numeric vector.

UDD annuity approximations

Description

UDD-based approximations for Chapter 8 annuity functions.

Computes

a¨x(m)≈α(m)a¨x−β(m).\ddot{a}_x^{(m)} \approx \alpha(m)\ddot{a}_x - \beta(m).

Computes

a¨x:n‾∣(m)≈α(m)a¨x:n‾∣−β(m)(1−nEx).\ddot{a}_{x:\overline{n}|}^{(m)} \approx \alpha(m)\ddot{a}_{x:\overline{n}|} - \beta(m)(1-{}_nE_x).

Computes

n∣a¨x(m)≈α(m) n∣a¨x−β(m) nEx.{}_{n\mid}\ddot{a}_x^{(m)} \approx \alpha(m)\,{}_{n\mid}\ddot{a}_x - \beta(m)\,{}_nE_x.

Computes

ax(m)≈α(m)ax+γ(m).a_x^{(m)} \approx \alpha(m)a_x + \gamma(m).

Computes

ax:n‾∣(m)≈α(m)ax:n‾∣+γ(m)(1−nEx).a_{x:\overline{n}|}^{(m)} \approx \alpha(m)a_{x:\overline{n}|} + \gamma(m)(1-{}_nE_x).

Computes

n∣ax(m)≈α(m) n∣ax+γ(m) nEx.{}_{n\mid}a_x^{(m)} \approx \alpha(m)\,{}_{n\mid}a_x + \gamma(m)\,{}_nE_x.

Computes

s¨x:n‾∣(m)≈α(m)s¨x:n‾∣−β(m)(1nEx−1).\ddot{s}_{x:\overline{n}|}^{(m)} \approx \alpha(m)\ddot{s}_{x:\overline{n}|} - \beta(m)\left(\frac{1}{{}_nE_x}-1\right).

Computes

sx:n‾∣(m)≈α(m)sx:n‾∣+γ(m)(1nEx−1).s_{x:\overline{n}|}^{(m)} \approx \alpha(m)s_{x:\overline{n}|} + \gamma(m)\left(\frac{1}{{}_nE_x}-1\right).

Computes

aˉx≈idδ2a¨x−i−δδ2.\bar{a}_x \approx \frac{id}{\delta^2}\ddot{a}_x - \frac{i-\delta}{\delta^2}.

Uses the identity

aˉx:n‾∣≈1−Aˉx:n‾∣δ\bar{a}_{x:\overline{n}|} \approx \frac{1-\bar{A}_{x:\overline{n}|}}{\delta}

together with the package's existing Chapter 7 UDD insurance approximation for Aˉx:n‾∣\bar{A}_{x:\overline{n}|}.

Computes

n∣aˉx≈nEx aˉx+n.{}_{n\mid}\bar{a}_x \approx {}_nE_x \, \bar{a}_{x+n}.

Usage

adotx_m_udd(x, m, i, model, ...)

adotxn_m_udd(x, n, m, i, model, ...)

nadotx_m_udd(x, n, m, i, model, ...)

ax_m_udd(x, m, i, model, ...)

axn_m_udd(x, n, m, i, model, ...)

nax_m_udd(x, n, m, i, model, ...)

sdotxn_m_udd(x, n, m, i, model, ...)

sxn_m_udd(x, n, m, i, model, ...)

abarx_udd(x, i, model, ...)

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

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

Arguments

x

Age.
m

Number of payments per year.
i

Effective annual interest rate.
model

Survival model.
...

Additional model parameters.
n

Term.

Details

These functions implement the standard Uniform Distribution of Deaths approximations linking annual, m-thly, and continuous annuity values.

The exported functions documented on this page are:

  • ax_m_udd()

  • axn_m_udd()

  • nax_m_udd()

  • adotx_m_udd()

  • adotxn_m_udd()

  • nadotx_m_udd()

  • sxn_m_udd()

  • sdotxn_m_udd()

  • abarx_udd()

  • abarxn_udd()

  • nabarx_udd()

Note that this function relies on the already-existing Abarxn_udd() implementation in the package, so extra survival-model arguments are not used.

Value

Numeric vector.

Woolhouse 2-term annuity approximations

Description

Woolhouse 2-term approximations for Chapter 8 annuity functions.

Usage

ax_m_woolhouse2(x, m, i, model, ...)

adotx_m_woolhouse2(x, m, i, model, ...)

nax_m_woolhouse2(x, n, m, i, model, ...)

nadotx_m_woolhouse2(x, n, m, i, model, ...)

axn_m_woolhouse2(x, n, m, i, model, ...)

adotxn_m_woolhouse2(x, n, m, i, model, ...)

sxn_m_woolhouse2(x, n, m, i, model, ...)

sdotxn_m_woolhouse2(x, n, m, i, model, ...)

abarx_woolhouse2(x, i, model, ...)

Arguments

x

Age.
m

Number of payments per year.
i

Effective annual interest rate.
model

Survival model.
...

Additional model parameters.
n

Term.

Value

Numeric vector.

Woolhouse 3-term annuity approximations

Description

Woolhouse 3-term approximations for Chapter 8 annuity functions.

Usage

ax_m_woolhouse3(x, m, i, model, ...)

adotx_m_woolhouse3(x, m, i, model, ...)

nax_m_woolhouse3(x, n, m, i, model, ...)

nadotx_m_woolhouse3(x, n, m, i, model, ...)

axn_m_woolhouse3(x, n, m, i, model, ...)

adotxn_m_woolhouse3(x, n, m, i, model, ...)

abarx_woolhouse3(x, i, model, ...)

Arguments

x

Age.
m

Number of payments per year.
i

Effective annual interest rate.
model

Survival model.
...

Additional model parameters.
n

Term.

Value

Numeric vector.

Present value of a level annuity-certain

Description

Computes the present value of an n-period annuity certain with level payments of 1 per period.

Usage

annuity_certain(n, i, due = FALSE, m = 1, cont = FALSE)

Arguments

n

Number of payments or periods.
i

Effective interest rate per period.
due

If TRUE, annuity-due; otherwise annuity-immediate.
m

Payment frequency per period (m = 1 means annual).
cont

If TRUE, continuous payment model.

Value

Present value.

Examples

annuity_certain(n = 10, i = 0.05)
annuity_certain(n = 10, i = 0.05, due = TRUE)
annuity_certain(n = 10, i = 0.05, cont = TRUE)

Temporary m-thly annuity-due actuarial accumulated value

Description

Computes

s¨x:n‾∣(m)=a¨x:n‾∣(m)/nEx\ddot{s}_{x:\overline{n}|}^{(m)} = \ddot{a}_{x:\overline{n}|}^{(m)} / {}_nE_x

Usage

sdotxn_m(x, n, m, i, model, ...)

Arguments

x

Age.
n

Term in years.
m

Number of payments per year.
i

Effective annual interest rate.
model

Survival model name.
...

Additional parameters passed to the survival model.

Value

Numeric vector of actuarial accumulated values.

Examples

sdotxn_m(40, n = 10, m = 12, i = 0.05, model = "uniform", omega = 100)

Temporary m-thly annuity-immediate actuarial accumulated value

Description

Computes

sx:n‾∣(m)=ax:n‾∣(m)/nExs_{x:\overline{n}|}^{(m)} = a_{x:\overline{n}|}^{(m)} / {}_nE_x

Usage

sxn_m(x, n, m, i, model, ...)

Arguments

x

Age.
n

Term in years.
m

Number of payments per year.
i

Effective annual interest rate.
model

Survival model name.
...

Additional parameters passed to the survival model.

Value

Numeric vector of actuarial accumulated values.

Examples

sxn_m(40, n = 10, m = 12, i = 0.05, model = "uniform", omega = 100)

Deferred whole life m-thly annuity-due

Description

Computes the deferred whole life m-thly annuity-due using

nEx a¨x+n(m){}_nE_x \, \ddot{a}_{x+n}^{(m)}

Usage

nadotx_m(x, n, m, i, model, ..., k_max = 2e+05, tol = 1e-12)

Arguments

x

Age.
n

Deferral period in years.
m

Number of payments per year.
i

Effective annual interest rate.
model

Survival model name.
...

Additional parameters passed to the survival model.
k_max

Maximum summation horizon for non-terminating models.
tol

Truncation tolerance for non-terminating models.

Value

Numeric vector of annuity values.

Examples

nadotx_m(40, n = 10, m = 12, i = 0.05, model = "uniform", omega = 100)

Deferred whole life m-thly annuity-immediate

Description

Computes the deferred whole life m-thly annuity-immediate using

nEx ax+n(m){}_nE_x \, a_{x+n}^{(m)}

Usage

nax_m(x, n, m, i, model, ..., k_max = 2e+05, tol = 1e-12)

Arguments

x

Age.
n

Deferral period in years.
m

Number of payments per year.
i

Effective annual interest rate.
model

Survival model name.
...

Additional parameters passed to the survival model.
k_max

Maximum summation horizon for non-terminating models.
tol

Truncation tolerance for non-terminating models.

Value

Numeric vector of annuity values.

Examples

nax_m(40, n = 10, m = 12, i = 0.05, model = "uniform", omega = 100)

Temporary m-thly annuity-due

Description

Computes the exact temporary m-thly annuity-due.

Usage

adotxn_m(x, n, m, i, model, ...)

Arguments

x

Age.
n

Term in years.
m

Number of payments per year.
i

Effective annual interest rate.
model

Survival model name.
...

Additional parameters passed to the survival model.

Details

a¨x:n‾∣(m)=1m∑t=0mn−1vt/m t/mpx\ddot{a}_{x:\overline{n}|}^{(m)} = \frac{1}{m}\sum_{t=0}^{mn-1} v^{t/m}\,{}_{t/m}p_x

Value

Numeric vector of annuity values.

Examples

adotxn_m(40, n = 10, m = 12, i = 0.05, model = "uniform", omega = 100)

Temporary m-thly annuity-immediate

Description

Computes the exact temporary m-thly annuity-immediate.

Usage

axn_m(x, n, m, i, model, ...)

Arguments

x

Age.
n

Term in years.
m

Number of payments per year.
i

Effective annual interest rate.
model

Survival model name.
...

Additional parameters passed to the survival model.

Details

ax:n‾∣(m)=1m∑t=1mnvt/m t/mpxa_{x:\overline{n}|}^{(m)} = \frac{1}{m}\sum_{t=1}^{mn} v^{t/m}\,{}_{t/m}p_x

Value

Numeric vector of annuity values.

Examples

axn_m(40, n = 10, m = 12, i = 0.05, model = "uniform", omega = 100)

Whole life m-thly annuity-due

Description

Computes the exact whole life m-thly annuity-due.

Usage

adotx_m(x, m, i, model, ..., k_max = 2e+05, tol = 1e-12)

Arguments

x

Age.
m

Number of payments per year.
i

Effective annual interest rate.
model

Survival model name.
...

Additional parameters passed to the survival model.
k_max

Maximum summation horizon for non-terminating models.
tol

Truncation tolerance for non-terminating models.

Details

a¨x(m)=1m∑t=0∞vt/m t/mpx\ddot{a}_x^{(m)} = \frac{1}{m}\sum_{t=0}^{\infty} v^{t/m}\,{}_{t/m}p_x

Value

Numeric vector of annuity values.

Examples

adotx_m(40, m = 12, i = 0.05, model = "uniform", omega = 100)

Whole life m-thly annuity-immediate

Description

Computes the exact whole life m-thly annuity-immediate, with annual payment rate 1 split into m equal payments of size 1/m.

Usage

ax_m(x, m, i, model, ..., k_max = 2e+05, tol = 1e-12)

Arguments

x

Age.
m

Number of payments per year.
i

Effective annual interest rate.
model

Survival model name.
...

Additional parameters passed to the survival model.
k_max

Maximum summation horizon for non-terminating models.
tol

Truncation tolerance for non-terminating models.

Details

ax(m)=1m∑t=1∞vt/m t/mpxa_x^{(m)} = \frac{1}{m}\sum_{t=1}^{\infty} v^{t/m}\,{}_{t/m}p_x

Value

Numeric vector of annuity values.

Examples

ax_m(40, m = 12, i = 0.05, model = "uniform", omega = 100)

Annuity-insurance relationships (Chapter 8)

Description

This file provides the core Chapter 8 identities linking annual and continuous annuity functions to the corresponding insurance functions.

Computes ax=(v−Ax)/da_x = (v - A_x)/d.

Computes a¨x=(1−Ax)/d\ddot{a}_x = (1 - A_x)/d.

Computes aˉx=(1−Aˉx)/δ\bar{a}_x = (1 - \bar{A}_x)/\delta.

Computes ax:n‾∣=a¨x:n‾∣−1+nExa_{x:\overline{n}|} = \ddot{a}_{x:\overline{n}|} - 1 + {}_nE_x together with a¨x:n‾∣=(1−Ax:n‾∣)/d\ddot{a}_{x:\overline{n}|} = (1 - A_{x:\overline{n}|})/d.

Computes a¨x:n‾∣=(1−Ax:n‾∣)/d\ddot{a}_{x:\overline{n}|} = (1 - A_{x:\overline{n}|})/d.

Computes aˉx:n‾∣=(1−Aˉx:n‾∣)/δ\bar{a}_{x:\overline{n}|} = (1 - \bar{A}_{x:\overline{n}|})/\delta.

Computes n∣ax=nEx ax+n{}_{n|}a_x = {}_nE_x\, a_{x+n}.

Computes n∣a¨x=nEx a¨x+n{}_{n|}\ddot{a}_x = {}_nE_x\, \ddot{a}_{x+n}.

Computes n∣aˉx=nEx aˉx+n{}_{n|}\bar{a}_x = {}_nE_x\, \bar{a}_{x+n}.

Usage

annuity_identity_ax(x, i, model, ...)

annuity_identity_adotx(x, i, model, ...)

annuity_identity_abarx(x, i, model, ...)

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

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

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

annuity_identity_nax(x, n, i, model, ..., k_max = 5000, tol = 1e-12)

annuity_identity_nadotx(x, n, i, model, ..., k_max = 5000, tol = 1e-12)

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

Arguments

x

Age.
i

Effective annual interest rate.
model

Survival model.
...

Additional model parameters passed to the survival model.
n

Term in years.
k_max

Maximum summation horizon for non-terminating models.
tol

Truncation tolerance for non-terminating models.

Details

Included identities:

  • whole life immediate: ax=(v−Ax)/da_x = (v - A_x)/d

  • whole life due: a¨x=(1−Ax)/d\ddot{a}_x = (1 - A_x)/d

  • whole life continuous: aˉx=(1−Aˉx)/δ\bar{a}_x = (1 - \bar{A}_x)/\delta

  • temporary immediate: ax:n‾∣=(1−Ax:n‾∣)/d−1+nExa_{x:\overline{n}|} = (1 - A_{x:\overline{n}|})/d - 1 + {}_nE_x

  • temporary due: a¨x:n‾∣=(1−Ax:n‾∣)/d\ddot{a}_{x:\overline{n}|} = (1 - A_{x:\overline{n}|})/d

  • temporary continuous: aˉx:n‾∣=(1−Aˉx:n‾∣)/δ\bar{a}_{x:\overline{n}|} = (1 - \bar{A}_{x:\overline{n}|})/\delta

  • deferred immediate: n∣ax=nExax+n{}_{n\mid}a_x = {}_nE_x a_{x+n}

  • deferred due: n∣a¨x=nExa¨x+n{}_{n\mid}\ddot{a}_x = {}_nE_x \ddot{a}_{x+n}

  • deferred continuous: n∣aˉx=nExaˉx+n{}_{n\mid}\bar{a}_x = {}_nE_x \bar{a}_{x+n}

These are wrapper functions that evaluate the Chapter 8 relationships using the Chapter 7 insurance functions already implemented in the package.

Hence ax:n‾∣=(1−Ax:n‾∣)/d−1+nExa_{x:\overline{n}|} = (1 - A_{x:\overline{n}|})/d - 1 + {}_nE_x.

Value

Numeric vector.

Varying-payment annuity functions (Chapter 8)

Description

Chapter 8 non-level annuity functions for increasing and decreasing life annuities.

Computes

(Ia)x=∑t=1∞t vt tpx.(Ia)_x = \sum_{t=1}^{\infty} t \, v^t \, {}_tp_x.

Computes

(Ia)x:n‾∣=∑t=1nt vt tpx.(Ia)_{x:\overline{n}|} = \sum_{t=1}^{n} t \, v^t \, {}_tp_x.

Computes

(Da)x:n‾∣=∑t=1n(n+1−t) vt tpx.(Da)_{x:\overline{n}|} = \sum_{t=1}^{n} (n+1-t)\, v^t \, {}_tp_x.

Computes

(Ia¨)x=∑t=0∞(t+1) vt tpx.(I\ddot{a})_x = \sum_{t=0}^{\infty} (t+1)\, v^t \, {}_tp_x.

Computes

(Ia¨)x:n‾∣=∑t=0n−1(t+1) vt tpx.(I\ddot{a})_{x:\overline{n}|} = \sum_{t=0}^{n-1} (t+1)\, v^t \, {}_tp_x.

Computes

(Da¨)x:n‾∣=∑t=0n−1(n−t) vt tpx.(D\ddot{a})_{x:\overline{n}|} = \sum_{t=0}^{n-1} (n-t)\, v^t \, {}_tp_x.

Computes

(Iˉaˉ)x=∫0∞t vt tpx dt.(\bar{I}\bar{a})_x = \int_0^\infty t\,v^t\,{}_tp_x\,dt.

Computes

(Iˉaˉ)x:n‾∣=∫0nt vt tpx dt.(\bar{I}\bar{a})_{x:\overline{n}|} = \int_0^n t\,v^t\,{}_tp_x\,dt.

Computes

(Dˉaˉ)x:n‾∣=∫0n(n−t) vt tpx dt.(\bar{D}\bar{a})_{x:\overline{n}|} = \int_0^n (n-t)\,v^t\,{}_tp_x\,dt.

Usage

Iax(x, i, model, ..., k_max = 5000, tol = 1e-12)

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

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

Iadotx(x, i, model, ..., k_max = 5000, tol = 1e-12)

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

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

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

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

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

Arguments

x

Age.
i

Effective annual interest rate.
model

Survival model.
...

Additional model parameters.
k_max

Maximum summation horizon for non-terminating models.
tol

Truncation tolerance for non-terminating models.
n

Term in years.

Details

The functions implemented here match the notation in Section 8.6:

  • Iax() = (Ia)x(Ia)_x

  • Iaxn() = (Ia)x:n‾∣(Ia)_{x:\overline{n}|}

  • Daxn() = (Da)x:n‾∣(Da)_{x:\overline{n}|}

  • Iadotx() = (Ia¨)x(I\ddot{a})_x

  • Iadotxn() = (Ia¨)x:n‾∣(I\ddot{a})_{x:\overline{n}|}

  • Dadotxn() = (Da¨)x:n‾∣(D\ddot{a})_{x:\overline{n}|}

  • Iabarx() = (Iˉaˉ)x(\bar{I}\bar{a})_x

  • Iabarxn() = (Iˉaˉ)x:n‾∣(\bar{I}\bar{a})_{x:\overline{n}|}

  • Dabarxn() = (Dˉaˉ)x:n‾∣(\bar{D}\bar{a})_{x:\overline{n}|}

Value

Numeric vector.

Numeric vector.

Numeric vector.

APV of gross premiums under a risk discount rate

Description

Computes the actuarial present value of gross premiums:

APVGP=∑t=0n−1Gt+1⋅tpx(τ)(1+r)t.APV_{GP} = \sum_{t=0}^{n-1} \frac{G_{t+1} \cdot {}_tp_x^{(\tau)}}{(1+r)^t}.

Usage

APV_gross_premiums(G, r, p_tau)

Arguments

G

Gross premium vector for policy years 1 through nn.
r

Risk discount rate.
p_tau

One-year in-force probabilities. This may have length n−1n-1 or nn. If length nn, the final entry is ignored.

Value

Numeric scalar.

Examples

APV_gross_premiums(G = rep(95, 3),r = 0.10,p_tau = c(0.99858, 0.99847, 0.99834))

APV of normal retirement benefit for a DB plan

Description

Computes Equation (18.10).

Usage

APV_NR_db(PABz, v_to_ret, p_surv, adue_ret)

Arguments

PABz

Projected annual benefit at retirement.
v_to_ret

Discount factor from current age to retirement.
p_surv

Active-service survival probability to retirement.
adue_ret

Retirement annuity factor.

Value

Actuarial present value of the normal retirement benefit.

Examples

APV_NR_db(PABz = 108008.66, v_to_ret = 1 / 1.06^30, p_surv = 0.8, adue_ret = 12)

Projected asset share path kAS{}_{k}AS

Description

Computes projected asset shares recursively using Equations (14.5b) and (14.6b) of Chapter 14, with optional support for a survival benefit payable at the end of year kk.

Usage

AS_path(AS0, G, r, e, b1, b2, q1, q2, p_tau, i, b3 = NULL)

Arguments

AS0

Initial asset share 0AS{}_{0}AS.
G

Level annual premium.
r

Numeric vector of percent-of-premium expense factors.
e

Numeric vector of fixed contract expenses.
b1

Numeric vector of Cause 1 benefit amounts.
b2

Numeric vector of Cause 2 benefit amounts.
q1

Numeric vector of Cause 1 decrement probabilities.
q2

Numeric vector of Cause 2 decrement probabilities.
p_tau

Numeric vector of in-force probabilities.
i

Effective annual interest rate.
b3

Optional numeric vector of survival benefit amounts payable at the end of year kk conditional on survival through year kk. Defaults to a zero vector.

Details

For policy year kk,

[k−1AS+G(1−rk)−ek](1+i)=bk(1)qx+k−1(1)+bk(2)qx+k−1(2)+px+k−1(τ)(bk(3)+kAS)[{}_{k-1}AS + G(1-r_k) - e_k](1+i) = b_k^{(1)} q_{x+k-1}^{(1)} + b_k^{(2)} q_{x+k-1}^{(2)} + p_{x+k-1}^{(\tau)} \left(b_k^{(3)} + {}_{k}AS\right)

so that

kAS=[k−1AS+G(1−rk)−ek](1+i)−bk(1)qx+k−1(1)−bk(2)qx+k−1(2)px+k−1(τ)−bk(3){}_{k}AS = \frac{[{}_{k-1}AS + G(1-r_k) - e_k](1+i) - b_k^{(1)} q_{x+k-1}^{(1)} - b_k^{(2)} q_{x+k-1}^{(2)}}{p_{x+k-1}^{(\tau)}} - b_k^{(3)}

Value

A data frame with columns k and AS.

General projected asset share path (multiple decrements)

Description

General projected asset share path (multiple decrements)

Usage

AS_path_md(AS0, G, r, e, b_mat, q_mat, p_tau, i, b_surv = NULL)

Arguments

AS0

Initial asset share.
G

Premium.
r

Expense percentages.
e

Fixed expenses.
b_mat

Matrix of benefits (rows = years, cols = causes).
q_mat

Matrix of decrement probabilities (same shape as b_mat).
p_tau

In-force probabilities.
i

Interest rate.
b_surv

Optional survival benefits.

Value

A data frame with columns k and AS. Column k gives the policy year from 0 to n, and column AS gives the corresponding projected asset share at each year.

Account-value path for Type A universal life

Description

Computes the year-by-year account value roll-forward for Type A universal life using the explicit form in Equation (16.8), or Equation (16.9) when iq=ici^q = i^c.

Usage

AV_path_ul_typeA(G, r, e, qx, ic, B, iq = ic, AV0 = 0)

Arguments

G

Premium vector GtG_t.
r

Percent-of-premium expense vector rtr_t.
e

Fixed expense vector ete_t.
qx

Mortality vector qx+t−1q_{x+t-1}.
ic

Credited interest rate vector ici^c.
B

Fixed death benefit face amount.
iq

Interest rate vector iqi^q used in cost of insurance. Defaults to ic.
AV0

Initial account value. Defaults to 0.

Value

A data frame with columns t, premium, AV.

Examples

qx <- c(.00076, .00081, .00085, .00090, .00095)
r <- c(.75, .10, .10, .10, .10)
e <- c(100, 20, 20, 20, 20)
G <- rep(5000, 5)

AV_path_ul_typeA(G = G, r = r, e = e, qx = qx, ic = 0.03, B = 100000)

Account-value path for Type B universal life

Description

Computes the year-by-year account value roll-forward for Type B universal life using Equations (16.4) and (16.5a).

Usage

AV_path_ul_typeB(G, r, e, qx, ic, B, iq = ic, AV0 = 0)

Arguments

G

Premium vector GtG_t.
r

Percent-of-premium expense vector rtr_t.
e

Fixed expense vector ete_t.
qx

Mortality vector qx+t−1q_{x+t-1}.
ic

Credited interest rate vector ici^c.
B

Face amount.
iq

Interest rate vector iqi^q used in cost of insurance. Defaults to ic.
AV0

Initial account value. Defaults to 0.

Value

A data frame with columns t, premium, net_contribution, COI, and AV.

Examples

qx <- c(.00076, .00081, .00085, .00090, .00095)
r <- c(.75, .10, .10, .10, .10)
e <- c(100, 20, 20, 20, 20)
G <- rep(5000, 5)

AV_path_ul_typeB(G = G, r = r, e = e, qx = qx, ic = 0.03, B = 100000)

Accumulated value of defined contribution plan contributions

Description

Computes the accumulated value at retirement age zz for the defined contribution model in Equation (18.1).

Usage

AVz_dc(x, z, Sx, c, i, g = NULL, s = NULL)

Arguments

x

Entry age.
z

Retirement age.
Sx

Salary at age x.
c

Contribution rate as a proportion of salary.
i

Annual effective interest rate.
g

Optional constant annual salary growth rate.
s

Optional salary scale vector of length z - x.

Value

The accumulated value of contributions at age z.

Examples

AVz_dc(x = 30, z = 65, Sx = 50000, c = 0.10, i = 0.05, g = 0.04)

Whole life insurance APV

Description

Computes Ax=∑k=0∞vk+1k∣qxA_x = \sum_{k=0}^\infty v^{k+1} {}_{k\mid}q_x.

Usage

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

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.
tol

Numerical tolerance for truncating infinite sums.
k_max

Maximum number of terms in the sum.

Value

Numeric vector of APVs.

Whole life annuity-immediate under mortality improvement

Description

Computes a truncated whole life annuity-immediate under projected mortality.

Usage

ax_improved(x0, i, qx_base_vec, AAx_vec, base_year, issue_year)

Arguments

x0

Issue age.
i

Effective annual interest rate.
qx_base_vec

Base-year one-year death probabilities for successive ages.
AAx_vec

Mortality improvement factors for successive ages.
base_year

Base year.
issue_year

Issue year.

Value

Numeric scalar.

m-thly whole life insurance APV

Description

Computes Ax(m)=∑j=0∞v(j+1)/mPr⁡(j/m<Tx≤(j+1)/m)A_x^{(m)} = \sum_{j=0}^\infty v^{(j+1)/m}\Pr(j/m < T_x \le (j+1)/m).

Usage

Ax_m(x, i, m, model, ..., tol = 1e-12, j_max = 100000L)

Arguments

x

Age.
i

Effective annual interest rate.
m

Positive integer payment frequency.
model

Parametric survival model name.
...

Additional model parameters passed to survival-model functions.
tol

Numerical tolerance for truncating the infinite sum.
j_max

Maximum number of m-thly intervals in the sum.

Value

Numeric vector of APVs.

UDD approximation of m-thly whole life insurance

Description

Computes Ax(m)=(i/i(m))AxA_x^{(m)} = (i/i^{(m)})A_x.

Usage

Ax_m_udd(Ax, i, m)

Arguments

Ax

Discrete whole life insurance APV.
i

Effective annual interest rate.
m

Positive integer payment frequency.

Value

m-thly whole life insurance APV under UDD.

Reversionary annuity to (y) after death of (x)

Description

Computes ax∣y=ay−axya_{x|y} = a_y - a_{xy}.

Usage

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

Arguments

x

Age of first life.
y

Age of second life.
i

Effective annual interest rate.
tbl

Life table.
model

Survival model.
...

Additional model parameters.
k_max

Maximum number of terms.
tol

Convergence tolerance.

Value

Numeric vector.

Examples

ax_y(40, 50, i = 0.05, model = "uniform", omega = 100)

Discrete multiple-decrement insurance APV Ax(j)A_{x}^{(j)}

Description

Computes the actuarial present value of a benefit payable at the end of the year of decrement if decrement occurs by Cause jj, matching Equation (14.3b) in Chapter 14.

Usage

Axj_md(qj, ptau, i, benefit = 1)

Arguments

qj

Numeric vector of conditional probabilities qx+k(j)q_{x+k}^{(j)} for Cause jj.
ptau

Numeric vector of survival probabilities kpx(τ){}_{k}p_{x}^{(\tau)} of remaining in force to duration kk.
i

Effective annual interest rate.
benefit

Benefit amount payable on decrement by Cause jj.

Details

The function evaluates

Ax(j)=∑k=0n−1vk+1kpx(τ)qx+k(j)A_{x}^{(j)} = \sum_{k=0}^{n-1} v^{k+1} {}_{k}p_{x}^{(\tau)} q_{x+k}^{(j)}

with an optional benefit amount multiplier.

Value

A numeric scalar.

Examples

q1 <- c(.02, .02, .02, .02, .02)
q2 <- c(.03, .04, .05, .06, .00)
q3 <- c(.00, .00, .00, .00, .98)
qtau <- q1 + q2 + q3

ptau <- numeric(length(qtau))
ptau[1] <- 1
for (k in 2:length(qtau)) {
  ptau[k] <- prod(1 - qtau[1:(k - 1)])
}

Axj_md(qj = q1, ptau = ptau, i = 0.06, benefit = 1000)

Endowment insurance APV

Description

Computes Ax:n‾∣=Ax:n‾∣1+nExA_{x:\overline{n}|} = A_{x:\overline{n}|}^{1} + {}_nE_x.

Usage

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

Arguments

x

Age.
n

Term.
i

Effective annual interest rate.
tbl

Optional life table object.
model

Optional parametric survival model name.
...

Additional arguments passed to survival-model functions.

Value

Numeric vector of APVs.

Temporary annuity-immediate under mortality improvement

Description

Computes

ax:n‾∣=∑t=1nvt tpx(improved)a_{x:\overline{n}|} = \sum_{t=1}^n v^t \, {}_tp_x^{(\mathrm{improved})}

using projected mortality rates.

Usage

axn_improved(x0, n, i, qx_base_vec, AAx_vec, base_year, issue_year)

Arguments

x0

Issue age.
n

Term in years.
i

Effective annual interest rate.
qx_base_vec

Base-year one-year death probabilities for ages x0, x0+1, ..., x0+n-1.
AAx_vec

Mortality improvement factors for ages x0, x0+1, ..., x0+n-1.
base_year

Base year.
issue_year

Issue year.

Value

Numeric scalar.

m-thly endowment insurance APV

Description

Computes Ax:n‾∣(m)=Ax:n‾∣1(m)+vn npxA_{x:\overline{n}|}^{(m)} = A_{x:\overline{n}|}^{1(m)} + v^n\,{}_np_x.

Usage

Axn_m(x, n, i, m, model, ...)

Arguments

x

Age.
n

Term.
i

Effective annual interest rate.
m

Positive integer payment frequency.
model

Parametric survival model name.
...

Additional model parameters passed to survival-model functions.

Value

Numeric vector of APVs.

UDD approximation of m-thly endowment insurance

Description

Computes Ax:n‾∣(m)=(i/i(m))Ax:n‾∣1+nExA_{x:\overline{n}|}^{(m)} = (i/i^{(m)})A_{x:\overline{n}|}^{1} + {}_nE_x.

Usage

Axn_m_udd(Axn1, nEx, i, m)

Arguments

Axn1

Discrete term insurance APV.
nEx

Pure endowment APV.
i

Effective annual interest rate.
m

Positive integer payment frequency.

Value

m-thly endowment insurance APV under UDD.

Term insurance APV

Description

Computes Ax:n‾∣1=∑k=0n−1vk+1k∣qxA_{x:\overline{n}|}^{1} = \sum_{k=0}^{n-1} v^{k+1} {}_{k\mid}q_x.

Usage

Axn1(x, n, i, tbl = NULL, model = NULL, ...)

Arguments

x

Age.
n

Term.
i

Effective annual interest rate.
tbl

Optional life table object.
model

Optional parametric survival model name.
...

Additional arguments passed to survival-model functions.

Value

Numeric vector of APVs.

m-thly term insurance APV

Description

Computes Ax:n‾∣1(m)=∑j=0mn−1v(j+1)/mPr⁡(j/m<Tx≤(j+1)/m)A_{x:\overline{n}|}^{1(m)} = \sum_{j=0}^{mn-1} v^{(j+1)/m}\Pr(j/m < T_x \le (j+1)/m).

Usage

Axn1_m(x, n, i, m, model, ...)

Arguments

x

Age.
n

Term.
i

Effective annual interest rate.
m

Positive integer payment frequency.
model

Parametric survival model name.
...

Additional model parameters passed to survival-model functions.

Value

Numeric vector of APVs.

UDD approximation of m-thly term insurance

Description

Computes Ax:n‾∣1(m)=(i/i(m))Ax:n‾∣1A_{x:\overline{n}|}^{1(m)} = (i/i^{(m)})A_{x:\overline{n}|}^{1}.

Usage

Axn1_m_udd(Axn1, i, m)

Arguments

Axn1

Discrete term insurance APV.
i

Effective annual interest rate.
m

Positive integer payment frequency.

Value

m-thly term insurance APV under UDD.

Joint-life whole life insurance

Description

Computes AxyA_{xy}.

Usage

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

Arguments

x

Age of first life.
y

Age of second life.
i

Effective annual interest rate.
tbl

Life table.
model

Survival model.
...

Additional model parameters.
k_max

Maximum number of terms.
tol

Convergence tolerance.

Value

Numeric vector.

Examples

Axy(40, 50, i = 0.05, model = "uniform", omega = 100)

Joint-life whole life annuity-immediate

Description

Computes axya_{xy}.

Usage

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

Arguments

x

Age of first life.
y

Age of second life.
i

Effective annual interest rate.
tbl

Life table.
model

Survival model.
...

Additional model parameters.
k_max

Maximum number of terms.
tol

Convergence tolerance.

Details

Shared documentation topic used to avoid filename collisions with case-distinct function names on case-insensitive file systems.

Value

Numeric vector.

Examples

axy(40, 50, i = 0.05, model = "uniform", omega = 100)

Last-survivor whole life insurance

Description

Computes Axy‾=Ax+Ay−AxyA_{\overline{xy}} = A_x + A_y - A_{xy}.

Usage

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

Arguments

x

Age of first life.
y

Age of second life.
i

Effective annual interest rate.
tbl

Life table.
model

Survival model.
...

Additional model parameters.
k_max

Maximum number of terms.
tol

Convergence tolerance.

Value

Numeric vector.

Examples

Axybar(40, 50, i = 0.05, model = "uniform", omega = 100)

Last-survivor whole life annuity-immediate

Description

Computes axy‾a_{\overline{xy}}.

Usage

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

Arguments

x

Age of first life.
y

Age of second life.
i

Effective annual interest rate.
tbl

Life table.
model

Survival model.
...

Additional model parameters.
k_max

Maximum number of terms.
tol

Convergence tolerance.

Details

Shared documentation topic used to avoid filename collisions with case-distinct function names on case-insensitive file systems.

Value

Numeric vector.

Examples

axybar(40, 50, i = 0.05, model = "uniform", omega = 100)

Last-survivor endowment insurance

Description

Computes Axy‾:n‾∣A_{\overline{xy}:\overline{n}|}.

Usage

Axybarn(x, y, n, i, tbl = NULL, model = NULL, ...)

Arguments

x

Age of first life.
y

Age of second life.
n

Term.
i

Effective annual interest rate.
tbl

Life table.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

Axybarn(40, 50, n = 10, i = 0.05, model = "uniform", omega = 100)

Last-survivor temporary annuity-immediate

Description

Computes axy‾:n‾∣a_{\overline{xy}:\overline{n}|}.

Usage

axybarn(x, y, n, i, tbl = NULL, model = NULL, ...)

Arguments

x

Age of first life.
y

Age of second life.
n

Term.
i

Effective annual interest rate.
tbl

Life table.
model

Survival model.
...

Additional model parameters.

Details

Shared documentation topic used to avoid filename collisions with case-distinct function names on case-insensitive file systems.

Value

Numeric vector.

Examples

axybarn(40, 50, n = 10, i = 0.05, model = "uniform", omega = 100)

Last-survivor term insurance

Description

Computes Axy‾:n‾∣1A^{1}_{\overline{xy}:\overline{n}|}.

Usage

Axybarn1(x, y, n, i, tbl = NULL, model = NULL, ...)

Arguments

x

Age of first life.
y

Age of second life.
n

Term.
i

Effective annual interest rate.
tbl

Life table.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

Axybarn1(40, 50, n = 10, i = 0.05, model = "uniform", omega = 100)

Joint-life endowment insurance

Description

Computes Axy:n‾∣A_{xy:\overline{n}|}.

Usage

Axyn(x, y, n, i, tbl = NULL, model = NULL, ...)

Arguments

x

Age of first life.
y

Age of second life.
n

Term.
i

Effective annual interest rate.
tbl

Life table.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

Axyn(40, 50, n = 10, i = 0.05, model = "uniform", omega = 100)

Joint-life temporary annuity-immediate

Description

Computes axy:n‾∣a_{xy:\overline{n}|}.

Usage

axyn(x, y, n, i, tbl = NULL, model = NULL, ...)

Arguments

x

Age of first life.
y

Age of second life.
n

Term.
i

Effective annual interest rate.
tbl

Life table.
model

Survival model.
...

Additional model parameters.

Details

Shared documentation topic used to avoid filename collisions with case-distinct function names on case-insensitive file systems.

Value

Numeric vector.

Examples

axyn(40, 50, n = 10, i = 0.05, model = "uniform", omega = 100)

Joint-life term insurance

Description

Computes Axy:n‾∣1A^{1}_{xy:\overline{n}|}.

Usage

Axyn1(x, y, n, i, tbl = NULL, model = NULL, ...)

Arguments

x

Age of first life.
y

Age of second life.
n

Term.
i

Effective annual interest rate.
tbl

Life table.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

Axyn1(40, 50, n = 10, i = 0.05, model = "uniform", omega = 100)

Reversionary annuity to (x) after death of (y)

Description

Computes ay∣x=ax−axya_{y|x} = a_x - a_{xy}.

Usage

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

Arguments

x

Age of first life.
y

Age of second life.
i

Effective annual interest rate.
tbl

Life table.
model

Survival model.
...

Additional model parameters.
k_max

Maximum number of terms.
tol

Convergence tolerance.

Value

Numeric vector.

Examples

ay_x(40, 50, i = 0.05, model = "uniform", omega = 100)

Full preliminary term renewal modified premium

Description

Computes the FPT renewal premium βF=Px+1\beta^F = P_{x+1} for whole life insurance.

Computes the FPT renewal premium βF=Px+1\beta^F = P_{x+1} for whole life insurance.

Usage

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

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

Arguments

x

Issue age.
i

Effective annual interest rate.
tbl

Optional life table object.
model

Optional parametric survival model.
...

Additional model parameters.

Value

Numeric vector.

Numeric vector.

Examples

betaF(40, i = 0.05, model = "uniform", omega = 100)
betaF(40, i = 0.05, model = "uniform", omega = 100)

Spot-rate actuarial present value functions

Description

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

Computes

nEx=(1+zn)−n⋅npx.{}_nE_x = (1+z_n)^{-n}\cdot {}_np_x.

Computes

Ax:n‾∣1=∑t=1n(1+zt)−t⋅t−1px⋅qx+t−1.A_{x:\overline{n}|}^1 = \sum_{t=1}^{n}(1+z_t)^{-t}\cdot {}_{t-1}p_x \cdot q_{x+t-1}.

Computes

Ax:n‾∣=Ax:n‾∣1+nExA_{x:\overline{n}|} = A_{x:\overline{n}|}^1 + {}_nE_x

using spot-rate discount factors.

For an immediate annuity,

ax:n‾∣=∑t=1n(1+zt)−t⋅tpx.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,…,n1,\dots,n.
benefit

Amount of each annuity payment.
type

Either "immediate" or "due".

Details

For an annuity-due,

a¨x:n‾∣=∑t=0n−1(1+zt)−t⋅tpx,\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.

Value

A numeric scalar.

A numeric scalar.

A numeric scalar.

A numeric scalar.

Examples

qx <- c(.02, .03, .04, .05, .06)
z  <- c(.03, .04, .05, .06, .07)
nEx_spot(qx, z, benefit = 1000000)

qx <- c(.02, .03, .04, .05, .06)
z  <- c(.03, .04, .05, .06, .07)
Axn1_spot(qx, z)

qx <- c(.02, .03, .04, .05, .06)
z  <- c(.03, .04, .05, .06, .07)
Axn_spot(qx, z)

qx <- c(.02, .03, .04, .05, .06)
z  <- c(.03, .04, .05, .06, .07)
axn_spot(qx, z, type = "due")

Variable-interest actuarial present value functions

Description

Chapter 15 functions for actuarial present values under variable annual effective interest rates interpreted as a yearly scenario i1,i2,…,ini_1, i_2, \dots, i_n.

Computes the APV of an nn-year pure endowment under a variable annual interest scenario:

nEx=vn⋅npx.{}_nE_x = v_n \cdot {}_np_x.

Computes the APV of an nn-year term insurance with benefit paid at the end of the year of death under a variable annual interest scenario:

Ax:n‾∣1=∑t=1nvt⋅t−1px⋅qx+t−1.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 nn-year endowment insurance under a variable annual interest scenario:

Ax:n‾∣=Ax:n‾∣1+nEx.A_{x:\overline{n}|} = A_{x:\overline{n}|}^1 + {}_nE_x.

Computes the APV of an nn-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 qx,qx+1,…,qx+n−1q_x, q_{x+1}, \dots, q_{x+n-1}.
i

Numeric vector of annual effective interest rates i1,i2,…,ini_1, i_2, \dots, i_n.
benefit

Amount of each annuity payment.
type

Either "immediate" or "due".

Details

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

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

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

For an immediate annuity,

ax:n‾∣=∑t=1nvt⋅tpx.a_{x:\overline{n}|} = \sum_{t=1}^{n} v_t \cdot {}_tp_x.

For an annuity-due,

a¨x:n‾∣=∑t=0n−1vt⋅tpx,\ddot{a}_{x:\overline{n}|} = \sum_{t=0}^{n-1} v_t \cdot {}_tp_x,

with v0=1v_0 = 1.

Value

A numeric scalar.

A numeric scalar.

A numeric scalar.

A numeric scalar.

Examples

qx <- c(.03, .04, .05, .06, .07)
nEx_var(qx = qx, i = c(.06, .07, .08, .09, .10), benefit = 1000)

qx <- c(.03, .04, .05, .06, .07)
Axn1_var(qx = qx, i = c(.06, .07, .08, .09, .10))

qx <- c(.03, .04, .05, .06, .07)
Axn_var(qx = qx, i = c(.06, .07, .08, .09, .10))

qx <- rep(.02, 5)
axn_var(qx = qx, i = c(.06, .05, .04, .03, .03), type = "immediate")
axn_var(qx = qx, i = c(.03, .04, .05, .06, .07), type = "due")

Cost of insurance for Type B universal life

Description

Computes the one-period cost of insurance for a Type B universal life policy using Equation (16.5a):

COIt=Bqx+t−11+iq.COI_t = \frac{B q_{x+t-1}}{1+i^q}.

Usage

coi_ul_typeB(B, qx, iq)

Arguments

B

Face amount.
qx

Mortality rate for the period.
iq

Interest rate used in the cost-of-insurance calculation.

Value

Numeric vector.

Examples

coi_ul_typeB(B = 100000, qx = 0.00076, iq = 0.03)

Target contribution rate for a defined contribution plan

Description

Solves Equation (18.5) for the contribution rate required to achieve a target replacement ratio.

Usage

contribution_rate_target(x, z, Sx, RR_target, i, adue_z, g = NULL, s = NULL)

Arguments

x

Entry age.
z

Retirement age.
Sx

Salary at age x.
RR_target

Target replacement ratio.
i

Annual effective interest rate.
adue_z

Whole life annuity-due factor at age z.
g

Optional constant annual salary growth rate.
s

Optional salary scale vector of length z - x.

Value

Required contribution rate.

Examples

contribution_rate_target(
  x = 30, z = 65, Sx = 60000, RR_target = 0.50,
  i = 0.06, adue_z = 11, g = 0.04
)

Covariance of term and deferred insurance PVs

Description

Computes Cov(Zx:n‾∣1,n∣Zx)=−Ax:n‾∣1⋅n∣Ax\mathrm{Cov}(Z_{x:\overline{n}|}^{1}, {}_{n\mid}Z_x) = -A_{x:\overline{n}|}^{1} \cdot {}_{n\mid}A_x.

Usage

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

Arguments

x

Age.
n

Term.
i

Effective annual interest rate.
tbl

Optional life table object.
model

Optional parametric survival model name.
...

Additional arguments passed to survival-model functions.

Value

Numeric vector of covariances.

Covariance of term insurance and pure endowment PVs

Description

Computes Cov(Zx:n‾∣1,Zx:n‾∣1(pure endow))=−Ax:n‾∣1⋅nEx\mathrm{Cov}(Z_{x:\overline{n}|}^{1}, Z_{x:\overline{n}|}^{1\text{(pure endow)}}) = -A_{x:\overline{n}|}^{1} \cdot {}_nE_x.

Usage

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

Arguments

x

Age.
n

Term.
i

Effective annual interest rate.
tbl

Optional life table object.
model

Optional parametric survival model name.
...

Additional arguments passed to survival-model functions.

Value

Numeric vector of covariances.

Cumulative hazard for age-at-failure T0

Description

Computes Λ0(t)=∫0tλ0(y) dy\Lambda_0(t)=\int_0^t \lambda_0(y)\,dy.

Usage

cumhaz0(t, model, ...)

Arguments

t

Numeric vector of times (t≥0t \ge 0).
model

One of "uniform", "exponential", "gompertz", "makeham", "weibull".
...

Model parameters:

  • uniform: omega

  • exponential: lambda

  • gompertz: B, c

  • makeham: A, B, c

  • weibull: shape, scale

Value

Numeric vector of cumulative hazard values.

Piecewise-continuous decreasing n-year term insurance

Description

Computes

(DAˉ)x:n‾∣1=∫0n⌊n+1−t⌋ vt tpx μx+t dt.(D\bar{A})_{x:\overline{n}|}^{1} = \int_0^n \lfloor n+1-t \rfloor\, v^t\, {}_tp_x\, \mu_{x+t}\, dt.

Usage

DAbarxn1(x, n, i, model, ...)

Arguments

x

Age.
n

Term.
i

Effective annual interest rate.
model

Parametric survival model.
...

Additional model parameters.

Value

Numeric vector.

Decreasing n-year term insurance

Description

Computes

(DA)x:n‾∣1=∑k=0n−1(n−k)vk+1Pr⁡(Kx=k).(DA)_{x:\overline{n}|}^{1} = \sum_{k=0}^{n-1} (n-k) v^{k+1} \Pr(K_x = k).

Usage

DAxn1(x, n, i, tbl = NULL, model = NULL, ...)

Arguments

x

Age.
n

Term.
i

Effective annual interest rate.
tbl

Optional life table object.
model

Optional parametric survival model.
...

Additional model parameters.

Value

Numeric vector.

Fully continuous decreasing n-year term insurance

Description

Computes

(DˉAˉ)x:n‾∣1=∫0n(n−t) vt tpx μx+t dt.(\bar{D}\bar{A})_{x:\overline{n}|}^{1} = \int_0^n (n-t)\, v^t\, {}_tp_x\, \mu_{x+t}\, dt.

Usage

DbarAbarxn1(x, n, i, model, ...)

Arguments

x

Age.
n

Term.
i

Effective annual interest rate.
model

Parametric survival model.
...

Additional model parameters.

Value

Numeric vector.

Ordered gross gain decomposition

Description

Decomposes total gross gain into interest, mortality, and expense components in a user-specified order.

Decomposes total gross gain into interest, mortality, and expense components in a user-specified order. The first two components are computed sequentially, and the last component is taken as the balancing item so that the components sum exactly to total gain.

Usage

decompGg_disc(
  VtG,
  Vt1G,
  G,
  i_assumed,
  q_assumed,
  r_assumed = 0,
  e_assumed = 0,
  s_assumed = 0,
  i_actual,
  q_actual,
  r_actual = 0,
  e_actual = 0,
  s_actual = 0,
  b = 1,
  order = c("interest", "mortality", "expense")
)

decompGg_disc(
  VtG,
  Vt1G,
  G,
  i_assumed,
  q_assumed,
  r_assumed = 0,
  e_assumed = 0,
  s_assumed = 0,
  i_actual,
  q_actual,
  r_actual = 0,
  e_actual = 0,
  s_actual = 0,
  b = 1,
  order = c("interest", "mortality", "expense")
)

Arguments

VtG

Gross reserve at duration t.
Vt1G

Gross reserve at duration t+1.
G

Gross premium.
i_assumed

Assumed annual effective interest rate.
q_assumed

Assumed mortality rate.
r_assumed

Assumed percent-of-premium expense rate.
e_assumed

Assumed per-policy expense.
s_assumed

Assumed settlement expense.
i_actual

Actual annual effective interest rate.
q_actual

Actual mortality rate.
r_actual

Actual percent-of-premium expense rate.
e_actual

Actual per-policy expense.
s_actual

Actual settlement expense.
b

Benefit amount. Default 1.
order

Character vector giving the order of decomposition.

Value

Named numeric vector.

Named numeric vector.

Examples

decompGg_disc(
  VtG = 3950.73, Vt1G = 4607.07, G = 685,
  i_assumed = 0.06, q_assumed = 0.00592,
  r_assumed = 0.05, e_assumed = 0, s_assumed = 300,
  i_actual = 0.065, q_actual = 0.005,
  r_actual = 0.06, e_actual = 0, s_actual = 100,
  b = 50000,
  order = c("interest", "mortality", "expense")
)
decompGg_disc(
  VtG = 3950.73, Vt1G = 4607.07, G = 685,
  i_assumed = 0.06, q_assumed = 0.00592,
  r_assumed = 0.05, e_assumed = 0, s_assumed = 300,
  i_actual = 0.065, q_actual = 0.005,
  r_actual = 0.06, e_actual = 0, s_actual = 100,
  b = 50000,
  order = c("interest", "mortality", "expense")
)

Discount factor for compound interest

Description

Discount factor for compound interest

Usage

discount(i, t)

Arguments

i

Effective interest rate.
t

Time (can be vector).

Value

Discount factor at time t.

Examples

discount(0.05, 0:5)

Discounted payback period

Description

Returns the first duration tt for which the partial net present value NPV(t)NPV(t) is nonnegative.

Usage

discounted_payback_period(Pi, r)

Arguments

Pi

Profit signature vector.
r

Risk discount rate.

Value

Integer scalar, or NA_integer_ if the payback period is not reached.

Examples

Pi <- c(-15.00, 8.42, 8.39, 8.58)
discounted_payback_period(Pi, r = 0.10)

Distribution functions for age-at-failure T0

Description

Convenience functions for the CDF and density of T0T_0.

Usage

F0(t, model, ...)

f0(t, model, ...)

Arguments

t

Numeric vector of times (t≥0t \ge 0).
model

One of "uniform", "exponential", "gompertz", "makeham", "weibull".
...

Model parameters:

  • uniform: omega

  • exponential: lambda

  • gompertz: B, c

  • makeham: A, B, c

  • weibull: shape, scale

Details

  • F0(t) computes F0(t)=Pr(T0≤t)=1−S0(t)F_0(t)=Pr(T_0 \le t)=1-S_0(t)

  • f0(t) computes f0(t)=ddtF0(t)f_0(t)=\frac{d}{dt}F_0(t)

Value

Numeric vector. For F0: CDF values in [0,1][0,1]. For f0: density values (≥0\ge 0).

Doubled force of interest

Description

Computes δ′=2δ\delta' = 2\delta.

Usage

double_force_delta(delta)

Arguments

delta

Numeric vector of forces of interest.

Value

Numeric vector of doubled forces of interest.

Effective annual interest at doubled force

Description

If δ′=2δ\delta' = 2\delta, then i′=(1+i)2−1i' = (1+i)^2 - 1.

Usage

double_force_i(i)

Arguments

i

Numeric vector of effective annual interest rates.

Value

Numeric vector of effective annual rates corresponding to doubled force.

Compute deaths between ages x and x+1

Description

Compute deaths between ages x and x+1

Usage

dx(tbl, x)

Arguments

tbl

A life_table object.
x

Ages.

Value

Numeric vector of d_x values.

Cause-specific decrements dx(j)d_x^{(j)}

Description

Cause-specific decrements dx(j)d_x^{(j)}

Usage

dxj(lxtau, qxj)

Arguments

lxtau

Number alive at age x in the multiple-decrement table.
qxj

Numeric vector of cause-specific decrement probabilities.

Value

Numeric vector.

Examples

dxj(1000, c(0.011, 0.100))

Total decrements dx(τ)d_x^{(\tau)}

Description

Total decrements dx(τ)d_x^{(\tau)}

Usage

dxtau(lxtau, qxj)

Arguments

lxtau

Number alive at age x in the multiple-decrement table.
qxj

Numeric vector of cause-specific decrement probabilities.

Value

Numeric scalar.

Examples

dxtau(1000, c(0.011, 0.100))

Mean present value of loss at duration t for whole life insurance

Description

Computes the Chapter 10 conditional mean E[tLx∣Kx≥t]E[{}_tL_x \mid K_x \ge t] for a fully discrete whole life insurance.

Usage

ELtx(x, t, i, P, model, ...)

Arguments

x

Issue age.
t

Duration.
i

Effective annual interest rate.
P

Annual premium.
model

Survival model.
...

Additional model parameters.

Details

Under the equivalence-principle premium, this equals the prospective reserve tVx{}_tV_x.

Value

Numeric vector.

Examples

prem <- Px(40, i = 0.05, model = "uniform", omega = 100)
ELtx(40, t = 10, i = 0.05, P = prem, model = "uniform", omega = 100)

Complete expectation of life at age x

Description

Computes e∘x=∫0∞tpx dt\overset{\circ}{e}_x=\int_0^\infty {}_t p_x \, dt. Uses closed form for uniform and exponential; numeric integration otherwise.

Usage

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

Arguments

x

Numeric vector of ages (x≥0x \ge 0).
model

One of "uniform", "exponential", "gompertz", "makeham", "weibull".
...

Model parameters:

  • uniform: omega

  • exponential: lambda

  • gompertz: B, c

  • makeham: A, B, c

  • weibull: shape, scale

tol

Tolerance used to choose a finite integration bound (numeric).

Value

Numeric vector of complete expectations.

Complete expectation of life from a life table

Description

Computes e∘x=∫0∞tpx dt\overset{\circ}{e}_x = \int_0^\infty {}_t p_x \, dt using a within-year assumption: "udd", "cf", or "balducci".

Usage

ex_complete_tab(tbl, x, assumption = c("udd", "cf", "balducci"))

Arguments

tbl

A life_table object.
x

Numeric vector of integer ages.
assumption

One of "udd", "cf", "balducci".

Value

Numeric vector of complete expectations e∘x\overset{\circ}{e}_x.

Curtate expectation of life at age x

Description

Computes ex=E[Kx]=∑k=1∞kpxe_x = E[K_x] = \sum_{k=1}^\infty {}_k p_x with truncation.

Usage

ex_curtate(x, model, ..., k_max = 5000, tol = 1e-12)

Arguments

x

Numeric vector of ages (x≥0x \ge 0).
model

One of "uniform", "exponential", "gompertz", "makeham", "weibull".
...

Model parameters:

  • uniform: omega

  • exponential: lambda

  • gompertz: B, c

  • makeham: A, B, c

  • weibull: shape, scale

k_max

Maximum integer duration to sum to.
tol

Stop early if the summand is < tol for several steps.

Value

Numeric vector of curtate expectations.

Curtate expectation of life from a life table

Description

Computes the curtate expectation of life ex=∑k=1∞kpxe_x = \sum_{k=1}^\infty {}_k p_x in the discrete tabular setting.

Usage

ex_curtate_tab(tbl, x)

Arguments

tbl

A life_table object.
x

Numeric vector of integer ages.

Value

Numeric vector of curtate expectations exe_x.

Temporary complete expectation of life from a life table

Description

Computes e∘x:n‾∣=∫0ntpx dt\overset{\circ}{e}_{x:\overline{n}|} = \int_0^n {}_t p_x \, dt using a within-year assumption: "udd", "cf", or "balducci".

Usage

ex_temp_complete_tab(tbl, x, n, assumption = c("udd", "cf", "balducci"))

Arguments

tbl

A life_table object.
x

Numeric vector of integer ages.
n

Numeric vector of nonnegative numbers.
assumption

One of "udd", "cf", "balducci".

Value

Numeric vector of temporary complete expectations.

Temporary curtate expectation of life from a life table

Description

Computes ex:n‾∣=∑k=1nkpxe_{x:\overline{n}|} = \sum_{k=1}^{n} {}_k p_x for integer n in the discrete tabular setting.

Usage

ex_temp_curtate_tab(tbl, x, n)

Arguments

tbl

A life_table object.
x

Numeric vector of integer ages.
n

Numeric vector of nonnegative integers.

Value

Numeric vector of temporary curtate expectations.

Forward rate fn,kf_{n,k} from spot rates

Description

Computes the nn-year forward kk-year annual effective rate implied by annual effective spot rates:

(1+zn+k)n+k=(1+zn)n(1+fn,k)k.(1+z_{n+k})^{n+k} = (1+z_n)^n (1+f_{n,k})^k.

Usage

fnk_from_z(z, n, k)

Arguments

z

Numeric vector of annual effective spot rates.
n

Forward start in years.
k

Forward maturity in years.

Details

The input vector z should contain annual effective spot rates for maturities 1, 2, ..., length(z).

Value

A numeric scalar.

Examples

z <- c(0.03, 0.04, 0.05, 0.06, 0.07)
fnk_from_z(z, n = 1, k = 4)
fnk_from_z(z, n = 2, k = 2)

Matrix of all determinable forward rates from spot rates

Description

Constructs an upper-left triangular matrix of annual effective forward rates fn,kf_{n,k} implied by annual effective spot rates z1,…,zmz_1,\dots,z_m.

Usage

forward_matrix_from_z(z)

Arguments

z

Numeric vector of annual effective spot rates.

Details

Rows correspond to n=1,…,m−1n = 1,\dots,m-1 and columns correspond to k=1,…,m−1k = 1,\dots,m-1. Entries that are not determinable are returned as NA.

Value

A numeric matrix.

Examples

z <- c(0.03, 0.04, 0.05, 0.06, 0.07)
forward_matrix_from_z(z)

Conditional density for Tx

Description

Computes fx(t)=f0(x+t)/S0(x)f_x(t)=f_0(x+t)/S_0(x).

Usage

fx(t, x, model, ...)

Arguments

t

Numeric vector of durations (t≥0t \ge 0).
x

Numeric vector of ages (x≥0x \ge 0).
model

One of "uniform", "exponential", "gompertz", "makeham", "weibull".
...

Model parameters:

  • uniform: omega

  • exponential: lambda

  • gompertz: B, c

  • makeham: A, B, c

  • weibull: shape, scale

Details

Vectorization rule: - If t and x are the same length, values are computed elementwise. - If one of t or x has length 1, it is recycled to match the other.

Value

Numeric vector of densities (≥0\ge 0).

Fractional conditional density from a life table

Description

Computes the conditional density fx(t∣T0>x)=tpxμx+tf_x(t \mid T_0 > x) = {}_t p_x \mu_{x+t} for 0<t<10 < t < 1.

Usage

fx_tab(tbl, x, t, assumption = c("udd", "cf", "balducci"))

Arguments

tbl

A life_table object.
x

Numeric vector of integer ages.
t

Numeric vector of fractional durations with 0≤t≤10 \le t \le 1.
assumption

One of "udd", "cf", "balducci".

Value

Numeric vector of conditional density values.

Gain or loss in a multiple-decrement model

Description

Computes the gain or loss expression from Section 14.6.

Usage

gain_loss_md(
  Vt,
  G,
  r,
  e,
  i,
  b1,
  b2,
  s1 = 0,
  s2 = 0,
  q1,
  q2,
  Vt1,
  year_end_cause2 = FALSE,
  q1prime = NULL,
  q2prime = NULL
)

Arguments

Vt

Gross premium reserve at time tt.
G

Gross premium for the year.
r

Percent-of-premium expense factor.
e

Fixed expense at the beginning of the year.
i

Earned interest rate.
b1

Cause 1 benefit.
b2

Cause 2 benefit.
s1

Claim settlement expense for Cause 1.
s2

Claim settlement expense for Cause 2.
q1

Cause 1 decrement probability.
q2

Cause 2 decrement probability.
Vt1

Gross premium reserve at time t+1t+1.
year_end_cause2

Logical; if TRUE, use the year-end Cause 2 form.
q1prime

Single-decrement Cause 1 probability for the year-end Cause 2 case.
q2prime

Single-decrement Cause 2 probability for the year-end Cause 2 case.

Details

With within-year decrement probabilities, the function evaluates

[tVG+G(1−r)−e](1+i)−[(b(1)+s(1))q(1)+(b(2)+s(2))q(2)+p(τ)t+1VG][{}_{t}V^G + G(1-r) - e](1+i) - \left[(b^{(1)}+s^{(1)})q^{(1)} + (b^{(2)}+s^{(2)})q^{(2)} + p^{(\tau)} {}_{t+1}V^G \right]

If year_end_cause2 = TRUE, the Cause 2 decrement is treated as occurring only at year end, matching Equation (14.30).

Value

A numeric scalar.

Examples

gain_loss_md(
  Vt = 115.00, G = 16, r = 0, e = 3, i = 0.06,
  b1 = 1000, b2 = 110, s1 = 0, s2 = 0,
  q1 = 0.01, q2 = 0.10, Vt1 = 128.83
)

Interest gain helper for continuous-style recursion

Description

Interest gain helper for continuous-style recursion

Usage

GI_cont(Vt, Vt1, P, delta_actual, p_assumed, benefit = 0, h = 1)

Arguments

Vt

Reserve at time t.
Vt1

Reserve at time t+h.
P

Premium rate.
delta_actual

Actual force of interest.
p_assumed

Assumed survival probability over the step.
benefit

Benefit paid at start of step. Default 0.
h

Step length. Default 1.

Value

Numeric vector.

Examples

GI_cont(Vt = 10, Vt1 = 11, P = 1, delta_actual = 0.05, p_assumed = 0.99)

Interest gain for a discrete insurance contract

Description

Interest gain for a discrete insurance contract

Usage

GI_disc(Vt, Vt1, P, i_actual, q_assumed, B = 1)

Arguments

Vt

Reserve at duration t.
Vt1

Reserve at duration t+1.
P

Net premium for the year.
i_actual

Actual annual effective interest rate.
q_assumed

Assumed mortality rate for the year.
B

Benefit amount. Defaults to 1.

Value

Numeric vector.

Examples

GI_disc(Vt = 0.1, Vt1 = 0.11, P = 0.02, i_actual = 0.05, q_assumed = 0.01)

Mortality gain helper for continuous-style recursion

Description

Mortality gain helper for continuous-style recursion

Usage

GM_cont(Vt, Vt1, P, delta_assumed, p_actual, benefit = 0, h = 1)

Arguments

Vt

Reserve at time t.
Vt1

Reserve at time t+h.
P

Premium rate.
delta_assumed

Assumed force of interest.
p_actual

Actual survival probability over the step.
benefit

Benefit paid at start of step. Default 0.
h

Step length. Default 1.

Value

Numeric vector.

Examples

GM_cont(Vt = 10, Vt1 = 11, P = 1, delta_assumed = 0.05, p_actual = 0.99)

Mortality gain for a discrete insurance contract

Description

Mortality gain for a discrete insurance contract

Usage

GM_disc(Vt, Vt1, P, i_assumed, q_actual, B = 1)

Arguments

Vt

Reserve at duration t.
Vt1

Reserve at duration t+1.
P

Net premium for the year.
i_assumed

Assumed annual effective interest rate.
q_actual

Actual mortality rate for the year.
B

Benefit amount. Defaults to 1.

Value

Numeric vector.

Examples

GM_disc(Vt = 0.1, Vt1 = 0.11, P = 0.02, i_assumed = 0.04, q_actual = 0.01)

Guaranteed maturity fund roll-forward

Description

Computes the one-period guaranteed maturity fund roll-forward used in Example 16.9.

Usage

GMF_rollforward_ul(GMF_prev, GMP, r, policy_charge, i)

Arguments

GMF_prev

Prior guaranteed maturity fund.
GMP

Guaranteed maturity premium.
r

Expense factor applied to GMP.
policy_charge

Guaranteed policy charge.
i

Guaranteed interest rate.

Value

Numeric scalar.

Examples

GMF_rollforward_ul(140.40, 14.49, 0.04, 11.80, 0.03)

Total gain for a continuous-style one-step recursion

Description

Total gain for a continuous-style one-step recursion

Usage

GT_cont(Vt, Vt1, P, delta_actual, p_actual, benefit = 0, h = 1)

Arguments

Vt

Reserve at time t.
Vt1

Reserve at time t+h.
P

Premium rate.
delta_actual

Actual force of interest.
p_actual

Actual survival probability over the step.
benefit

Benefit paid at start of step. Default 0.
h

Step length. Default 1.

Value

Numeric vector.

Examples

GT_cont(Vt = 10, Vt1 = 11, P = 1, delta_actual = 0.05, p_actual = 0.99)

Total gain for a discrete insurance contract

Description

Computes the Chapter 10 total gain: amount on hand at year-end minus amount required.

Usage

GT_disc(Vt, Vt1, P, i_actual, q_actual, B = 1)

Arguments

Vt

Reserve at duration t.
Vt1

Reserve at duration t+1.
P

Net premium for the year.
i_actual

Actual annual effective interest rate.
q_actual

Actual mortality rate for the year.
B

Benefit amount. Defaults to 1.

Value

Numeric vector.

Examples

GT_disc(Vt = 0.1, Vt1 = 0.11, P = 0.02, i_actual = 0.05, q_actual = 0.01)

Total gross gain for a discrete insurance contract

Description

Computes the Chapter 11 total gain under gross premiums and gross reserves.

Computes the Chapter 11 total gain under gross premiums and gross reserves.

Usage

GTg_disc(
  VtG,
  Vt1G,
  G,
  i_actual,
  q_actual,
  r_actual = 0,
  e_actual = 0,
  s_actual = 0,
  b = 1
)

GTg_disc(
  VtG,
  Vt1G,
  G,
  i_actual,
  q_actual,
  r_actual = 0,
  e_actual = 0,
  s_actual = 0,
  b = 1
)

Arguments

VtG

Gross reserve at duration t.
Vt1G

Gross reserve at duration t+1.
G

Gross premium.
i_actual

Actual annual effective interest rate.
q_actual

Actual mortality rate.
r_actual

Actual percent-of-premium expense rate.
e_actual

Actual per-policy expense.
s_actual

Actual settlement expense.
b

Benefit amount. Default 1.

Value

Numeric vector.

Numeric vector.

Examples

GTg_disc(
  VtG = 0.10, Vt1G = 0.12, G = 0.02,
  i_actual = 0.05, q_actual = 0.01,
  r_actual = 0.03, e_actual = 0, s_actual = 0.01, b = 1
)
GTg_disc(
  VtG = 0.10, Vt1G = 0.12, G = 0.02,
  i_actual = 0.05, q_actual = 0.01,
  r_actual = 0.03, e_actual = 0, s_actual = 0.01, b = 1
)

Hazard / force for age-at-failure T0

Description

Computes λ0(t)=f0(t)/S0(t)\lambda_0(t)=f_0(t)/S_0(t).

Usage

hazard0(t, model, ...)

Arguments

t

Numeric vector of times (t≥0t \ge 0).
model

One of "uniform", "exponential", "gompertz", "makeham", "weibull".
...

Model parameters:

  • uniform: omega

  • exponential: lambda

  • gompertz: B, c

  • makeham: A, B, c

  • weibull: shape, scale

Value

Numeric vector of hazard/force values.

h-pay whole life net level premium reserve

Description

Computes the Chapter 10 prospective reserve for an h-pay whole life policy.

Usage

htVx(x, h, t, i, model, ...)

Arguments

x

Issue age.
h

Premium-paying period in years.
t

Duration.
i

Effective annual interest rate.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

htVx(40, h = 10, t = 5, i = 0.05, model = "uniform", omega = 100)

Credited rates from raw index growth rates

Description

Applies participation, floor, cap, and optional margin to raw index growth rates for an indexed universal life contract.

Usage

i_credit_eiul(
  i_raw,
  part = 1,
  floor = 0,
  cap = Inf,
  margin = 0,
  margin_after_participation = TRUE
)

Arguments

i_raw

Numeric vector of raw index growth rates.
part

Participation rate.
floor

Minimum credited rate.
cap

Maximum credited rate.
margin

Index margin. Defaults to 0.
margin_after_participation

Logical; if TRUE, subtract the margin after applying the participation rate.

Value

Numeric vector of credited rates.

Examples

raw <- iP_eiul(c(1000, 1050, 1200, 1100))
i_credit_eiul(raw, part = 1.10, floor = 0.01, cap = 0.10)

Piecewise-continuous increasing whole life insurance

Description

Computes

(IAˉ)x=∫0∞⌊t+1⌋ vt tpx μx+t dt.(I\bar{A})_x = \int_0^\infty \lfloor t+1 \rfloor\, v^t\, {}_tp_x\, \mu_{x+t}\, dt.

Usage

IAbarx(x, i, model, ...)

Arguments

x

Age.
i

Effective annual interest rate.
model

Parametric survival model.
...

Additional model parameters.

Value

Numeric vector.

Increasing whole life insurance

Description

Computes

(IA)x=∑k=0∞(k+1)vk+1Pr⁡(Kx=k).(IA)_x = \sum_{k=0}^{\infty} (k+1) v^{k+1} \Pr(K_x = k).

Usage

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

Arguments

x

Age.
i

Effective annual interest rate.
tbl

Optional life table object.
model

Optional parametric survival model.
...

Additional model parameters.
tol

Numerical tolerance for truncation.
k_max

Maximum number of terms.

Value

Numeric vector.

Increasing n-year term insurance

Description

Computes

(IA)x:n‾∣1=∑k=0n−1(k+1)vk+1Pr⁡(Kx=k).(IA)_{x:\overline{n}|}^{1} = \sum_{k=0}^{n-1} (k+1) v^{k+1} \Pr(K_x = k).

Usage

IAxn1(x, n, i, tbl = NULL, model = NULL, ...)

Arguments

x

Age.
n

Term.
i

Effective annual interest rate.
tbl

Optional life table object.
model

Optional parametric survival model.
...

Additional model parameters.

Value

Numeric vector.

Fully continuous increasing whole life insurance

Description

Computes

(IˉAˉ)x=∫0∞t vt tpx μx+t dt.(\bar{I}\bar{A})_x = \int_0^\infty t\, v^t\, {}_tp_x\, \mu_{x+t}\, dt.

Usage

IbarAbarx(x, i, model, ...)

Arguments

x

Age.
i

Effective annual interest rate.
model

Parametric survival model.
...

Additional model parameters.

Value

Numeric vector.

Fully continuous increasing n-year term insurance

Description

Computes

(IˉAˉ)x:n‾∣1=∫0nt vt tpx μx+t dt.(\bar{I}\bar{A})_{x:\overline{n}|}^{1} = \int_0^n t\, v^t\, {}_tp_x\, \mu_{x+t}\, dt.

Usage

IbarAbarxn1(x, n, i, model, ...)

Arguments

x

Age.
n

Term.
i

Effective annual interest rate.
model

Parametric survival model.
...

Additional model parameters.

Value

Numeric vector.

Monthly-average index growth rate

Description

Computes the monthly-average index growth rate from an initial index value and 12 monthly closing values, matching Equation (16.13).

Usage

iMA_eiul(index)

Arguments

index

Numeric vector of length 13 containing the initial index value followed by the 12 monthly closing values.

Value

Numeric scalar.

Examples

idx <- c(1000, 1020, 1100, 1150, 1080, 1040, 960, 1030, 1000, 1070, 1150, 1200, 1150)
iMA_eiul(idx)

Retirement income from a defined contribution accumulation

Description

Converts a defined contribution accumulation to annual annuity-due income.

Usage

Income_dc(AVz, adue_z)

Arguments

AVz

Accumulated value at retirement.
adue_z

Whole life annuity-due factor at retirement age.

Value

Annual retirement income.

Examples

Income_dc(AVz = 824211.35, adue_z = 12)

Convert between compound-interest quantities

Description

Provides consistent conversions between: - effective interest rate i - effective discount rate d - force of interest delta

Usage

interest_convert(i = NULL, d = NULL, delta = NULL, m = NULL)

Arguments

i

Effective interest rate.
d

Effective discount rate.
delta

Force of interest.
m

Optional compounding frequency for the nominal rate convertible m-thly.

Details

Exactly one of i, d, or delta must be provided.

Value

A list with elements i, d, delta and, if m is supplied, im (the nominal rate convertible m-thly).

Examples

interest_convert(i = 0.05)
interest_convert(d = 0.04761905)
interest_convert(delta = log(1.05))

Point-to-point index growth rates

Description

Computes annual point-to-point index growth rates from successive index values, matching Equation (16.12).

Usage

iP_eiul(index)

Arguments

index

Numeric vector of index closing values.

Value

Numeric vector of growth rates.

Examples

iP_eiul(c(1000, 1050, 1200, 1100))

Internal rate of return of a profit signature

Description

Computes the internal rate of return (IRR), defined as the rate rr for which the net present value is zero.

Usage

IRR_profit(Pi, interval = c(0, 1), tol = .Machine$double.eps^0.5)

Arguments

Pi

Profit signature vector.
interval

Numeric vector of length 2 giving the search interval for uniroot().
tol

Tolerance passed to uniroot().

Value

Numeric scalar.

Examples

Pi <- c(-15.00, 8.42, 8.39, 8.58)
IRR_profit(Pi)

Construct a life table

Description

Build a discrete life table from one of lx, qx, px, or S0.

Usage

life_table(x, lx = NULL, qx = NULL, px = NULL, S0 = NULL, radix = 1e+05)

Arguments

x

Numeric vector of ages.
lx

Numeric vector of l_x values.
qx

Numeric vector of q_x values.
px

Numeric vector of p_x values.
S0

Numeric vector of S_0(x) values.
radix

Radix used when converting S0 to lx, or when building from qx/px.

Value

A data.frame with class "life_table".

Extract life-table survivor values

Description

Extract life-table survivor values

Usage

lx(tbl, x)

Arguments

tbl

A life_table object.
x

Ages.

Value

Numeric vector of l_x values.

Extract select-table survivor value

Description

Returns l[x]+tl_{[x]+t} from a select life table.

Usage

lx_select(tbl, x_sel, t)

Arguments

tbl

A select_life_table object.
x_sel

Numeric vector of ages at selection.
t

Numeric vector of durations since selection.

Value

Numeric vector of survivor values.

Convert life-table values to survival probabilities

Description

Converts Chapter 6 life-table values lxl_x into Chapter 5 survival probabilities S0(x)=lx/l0S_0(x) = l_x / l_0.

Usage

lx_to_S0(lx)

Arguments

lx

Numeric vector of life-table survivor values.

Value

Numeric vector of S0(x)S_0(x) values.

n-step transition probability for a discrete-time Markov chain

Description

Computes the (i,j)(i,j) entry of PnP^n, useful for Chapter 14 examples involving discrete-time multi-state models such as CCRC and risk-class models.

Usage

markov_nstep_prob(P, n, i, j)

Arguments

P

Transition probability matrix.
n

Nonnegative integer number of steps.
i

Starting state index.
j

Ending state index.

Value

A numeric scalar.

Examples

P <- matrix(
  c(0.94, 0.03, 0.02, 0.01,
    0.50, 0.30, 0.18, 0.02,
    0.00, 0.00, 0.93, 0.07,
    0.00, 0.00, 0.00, 1.00),
  nrow = 4, byrow = TRUE
)

markov_nstep_prob(P, n = 3, i = 1, j = 1)
markov_nstep_prob(P, n = 3, i = 1, j = 3)

Build a multiple-decrement table

Description

Build a multiple-decrement table

Usage

md_table(x, qxj, radix = 1e+05)

Arguments

x

Integer vector of ages or durations.
qxj

Matrix/data.frame of cause-specific decrement probabilities. Rows correspond to ages in x, columns correspond to causes.
radix

Starting lx(τ)l_x^{(\tau)}.

Value

Data frame containing q(j)q^{(j)}, q(τ)q^{(\tau)}, p(τ)p^{(\tau)}, l(τ)l^{(\tau)}, d(j)d^{(j)}, and d(τ)d^{(\tau)}.

Examples

x <- 45:50
qmat <- cbind(
  q1 = c(.011, .012, .013, .014, .015, .016),
  q2 = c(.100, .100, .100, .100, .100, .100)
)
md_table(x, qmat, radix = 1000)

Mean reserve for whole life insurance

Description

Computes the mean reserve t+1/2V{}_{t+1/2}V.

Computes the mean reserve t+1/2V{}_{t+1/2}V.

Usage

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

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

Arguments

x

Issue age.
t

Integer contract duration.
i

Effective annual interest rate.
tbl

Optional life table object.
model

Optional parametric survival model.
...

Additional model parameters.

Value

Numeric vector.

Numeric vector.

Examples

meanVx(40, t = 10, i = 0.05, model = "uniform", omega = 100)
meanVx(40, t = 10, i = 0.05, model = "uniform", omega = 100)

Fractional force of mortality from a life table

Description

Computes μx+t\mu_{x+t} under UDD, constant force, or Balducci.

Usage

mux_tab(tbl, x, t, assumption = c("udd", "cf", "balducci"))

Arguments

tbl

A life_table object.
x

Numeric vector of integer ages.
t

Numeric vector of fractional durations with 0≤t≤10 \le t \le 1.
assumption

One of "udd", "cf", "balducci".

Value

Numeric vector of μx+t\mu_{x+t} values.

Continuous deferred insurance APV

Description

Computes n∣Aˉx=vn npx Aˉx+n{}_{n\mid}\bar{A}_x = v^n\,{}_np_x\,\bar{A}_{x+n}.

Usage

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

Arguments

x

Age.
n

Deferral period.
i

Effective annual interest rate.
model

Parametric survival model name.
...

Additional model parameters passed to survival-model functions.

Value

Numeric vector of APVs.

UDD approximation of continuous deferred insurance

Description

Computes n∣Aˉx=(i/δ) n∣Ax{}_{n\mid}\bar{A}_x = (i/\delta)\,{}_{n\mid}A_x.

Usage

nAbarx_udd(nAx, i)

Arguments

nAx

Discrete deferred insurance APV.
i

Effective annual interest rate.

Value

Continuous deferred insurance APV under UDD.

Deferred insurance APV

Description

Computes n∣Ax=nEx⋅Ax+n{}_{n\mid}A_x = {}_nE_x \cdot A_{x+n}.

Usage

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

Arguments

x

Age.
n

Deferral period.
i

Effective annual interest rate.
tbl

Optional life table object.
model

Optional parametric survival model name.
...

Additional arguments passed to survival-model functions.
tol

Numerical tolerance for truncating infinite sums.
k_max

Maximum number of terms in the sum.

Value

Numeric vector of APVs.

m-thly deferred insurance APV

Description

Computes n∣Ax(m)=vn npx Ax+n(m){}_{n\mid}A_x^{(m)} = v^n\,{}_np_x\,A_{x+n}^{(m)}.

Usage

nAx_m(x, n, i, m, model, ..., tol = 1e-12, j_max = 100000L)

Arguments

x

Age.
n

Deferral period.
i

Effective annual interest rate.
m

Positive integer payment frequency.
model

Parametric survival model name.
...

Additional model parameters passed to survival-model functions.
tol

Numerical tolerance for truncating the infinite sum.
j_max

Maximum number of m-thly intervals in the sum.

Value

Numeric vector of APVs.

UDD approximation of m-thly deferred insurance

Description

Computes n∣Ax(m)=(i/i(m)) n∣Ax{}_{n\mid}A_x^{(m)} = (i/i^{(m)})\,{}_{n\mid}A_x.

Usage

nAx_m_udd(nAx, i, m)

Arguments

nAx

Discrete deferred insurance APV.
i

Effective annual interest rate.
m

Positive integer payment frequency.

Value

m-thly deferred insurance APV under UDD.

Deferred temporary annuity-immediate under mortality improvement

Description

Computes

u∣ax:n‾∣=∑t=u+1u+nvt tpx(improved){}_{u\mid}a_{x:\overline{n}|} = \sum_{t=u+1}^{u+n} v^t \, {}_tp_x^{(\mathrm{improved})}

Usage

naxn_improved(x0, u, n, i, qx_base_vec, AAx_vec, base_year, issue_year)

Arguments

x0

Issue age.
u

Deferral period in years.
n

Temporary payment period in years.
i

Effective annual interest rate.
qx_base_vec

Base-year one-year death probabilities for ages x0, x0+1, ..., x0+u+n-1.
AAx_vec

Mortality improvement factors for ages x0, x0+1, ..., x0+u+n-1.
base_year

Base year.
issue_year

Issue year.

Value

Numeric scalar.

Entry Age Normal normal cost for a DB plan

Description

Computes Equation (18.13).

Usage

NC_EAN_db(APV_total, adue_active)

Arguments

APV_total

Total actuarial present value of benefits.
adue_active

Active-service annuity-due factor.

Value

Entry Age Normal normal cost.

Examples

NC_EAN_db(APV_total = 25000, adue_active = 15)

Projected Unit Credit normal cost for a DB plan

Description

Computes the PUC normal cost as the APV of the portion of the projected benefit attributed to the current year of service.

Usage

NC_PUC_db(projected_benefit, total_service, v_to_ret, p_surv, adue_ret)

Arguments

projected_benefit

Projected benefit at retirement.
total_service

Total service at retirement.
v_to_ret

Discount factor to retirement.
p_surv

Active-service survival probability to retirement.
adue_ret

Retirement annuity factor.

Value

PUC normal cost.

Examples

NC_PUC_db(
  projected_benefit = 30000,
  total_service = 30,
  v_to_ret = 0.5,
  p_surv = 0.9,
  adue_ret = 12
)

Traditional Unit Credit normal cost for a DB plan

Description

Computes the TUC normal cost as the APV of the current year's accrual.

Usage

NC_TUC_db(accrual_benefit, v_to_ret, p_surv, adue_ret)

Arguments

accrual_benefit

Benefit accrued in the current year.
v_to_ret

Discount factor to retirement.
p_surv

Active-service survival probability to retirement.
adue_ret

Retirement annuity factor.

Value

TUC normal cost.

Examples

NC_TUC_db(accrual_benefit = 1560, v_to_ret = 0.5, p_surv = 0.9, adue_ret = 12)

Compute deaths over an n-year interval from a life table

Description

Compute deaths over an n-year interval from a life table

Usage

ndx(tbl, x, n)

Arguments

tbl

A life_table object.
x

Ages.
n

Nonnegative integer durations.

Value

Numeric vector of _n d_x values.

Pure endowment APV

Description

Computes nEx=vn npx{}_nE_x = v^n \, {}_n p_x.

Usage

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

Arguments

x

Age.
n

Term.
i

Effective annual interest rate.
tbl

Optional life table object.
model

Optional parametric survival model name.
...

Additional arguments passed to survival-model functions.

Value

Numeric vector of APVs.

Joint-life pure endowment

Description

Computes nExy=vn npxy{}_nE_{xy} = v^n\,{}_np_{xy}.

Usage

nExy(x, y, n, i, tbl = NULL, model = NULL, ...)

Arguments

x

Age of first life.
y

Age of second life.
n

Term.
i

Effective annual interest rate.
tbl

Life table.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

nExy(40, 50, n = 10, i = 0.05, model = "uniform", omega = 100)

Last-survivor pure endowment

Description

Computes nExy‾=vn npxy‾{}_nE_{\overline{xy}} = v^n\,{}_np_{\overline{xy}}.

Usage

nExybar(x, y, n, i, tbl = NULL, model = NULL, ...)

Arguments

x

Age of first life.
y

Age of second life.
n

Term.
i

Effective annual interest rate.
tbl

Life table.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

nExybar(40, 50, n = 10, i = 0.05, model = "uniform", omega = 100)

Curtate death probability from a life table

Description

Computes k∣qx=kpx−k+1px{}_{k|} q_x = {}_k p_x - {}_{k+1} p_x

Usage

nkqx(tbl, x, k)

Arguments

tbl

A life_table object.
x

Numeric vector of ages.
k

Nonnegative integer.

Value

Numeric vector of k∣qx{}_{k|} q_x values.

Deferred death probability from a life table

Description

Computes the probability that a life aged x survives n years and then dies within the following m years: n∣mqx=npx⋅mqx+n{}_{n|m} q_x = {}_n p_x \cdot {}_m q_{x+n}

Usage

nmxq(tbl, x, n, m)

Arguments

tbl

A life_table object.
x

Numeric vector of ages.
n

Nonnegative integer deferred period.
m

Nonnegative integer subsequent period.

Details

This function is for integer n and m in the discrete tabular setting.

Value

Numeric vector of n∣mqx{}_{n|m} q_x values.

Deferred select-life death probability

Description

Computes n∣mq[x]+t=np[x]+t⋅mq[x]+t+n{}_{n|m} q_{[x]+t} = {}_n p_{[x]+t} \cdot {}_m q_{[x]+t+n}.

Usage

nmxq_select(tbl, x_sel, t, n, m)

Arguments

tbl

A select_life_table object.
x_sel

Numeric vector of ages at selection.
t

Numeric vector of current durations since selection.
n

Numeric vector of nonnegative integer deferred periods.
m

Numeric vector of nonnegative integer death windows.

Value

Numeric vector of deferred death probabilities.

Partial net present values

Description

Computes the sequence NPV(0),NPV(1),…,NPV(n)NPV(0), NPV(1), \dots, NPV(n) of partial net present values from a profit signature.

Usage

NPV_partial(Pi, r)

Arguments

Pi

Profit signature vector.
r

Risk discount rate.

Value

Numeric vector.

Examples

Pi <- c(-15.00, 8.42, 8.39, 8.58)
NPV_partial(Pi, r = 0.10)

Net present value of a profit signature

Description

Computes the Chapter 17 net present value:

NPV=∑t=0nΠt(1+r)t.NPV = \sum_{t=0}^{n}\frac{\Pi_t}{(1+r)^t}.

Usage

NPV_profit(Pi, r)

Arguments

Pi

Profit signature vector.
r

Risk discount rate.

Value

Numeric scalar.

Examples

Pi <- c(-15.00, 8.42, 8.39, 8.58)
NPV_profit(Pi, r = 0.10)

Compute n-year survival probability from a life table

Description

Compute n-year survival probability from a life table

Usage

npx(tbl, x, n)

Arguments

tbl

A life_table object.
x

Ages.
n

Nonnegative integer durations.

Value

Numeric vector of _n p_x values.

Select-life survival probability

Description

Computes np[x]+t=l[x]+t+n/l[x]+t{}_n p_{[x]+t} = l_{[x]+t+n} / l_{[x]+t}

Usage

npx_select(tbl, x_sel, t, n)

Arguments

tbl

A select_life_table object.
x_sel

Numeric vector of ages at selection.
t

Numeric vector of current durations since selection.
n

Numeric vector of nonnegative integer future durations.

Details

in the discrete select-table setting.

Value

Numeric vector of survival probabilities.

npx(τ){}_n p_x^{(\tau)} from a multiple-decrement table

Description

npx(τ){}_n p_x^{(\tau)} from a multiple-decrement table

Usage

npxtau_md(tbl, x, n)

Arguments

tbl

Output from md_table().
x

Starting age.
n

Number of years.

Value

Numeric scalar.

Examples

x <- 45:50
qmat <- cbind(q1 = c(.011, .012, .013, .014, .015, .016), q2 = rep(.1, 6))
tbl <- md_table(x, qmat, radix = 1000)
npxtau_md(tbl, x = 46, n = 3)

Compute n-year death probability from a life table

Description

Compute n-year death probability from a life table

Usage

nqx(tbl, x, n)

Arguments

tbl

A life_table object.
x

Ages.
n

Nonnegative integer durations.

Value

Numeric vector of _n q_x values.

Select-life death probability

Description

Computes nq[x]+t=1−np[x]+t{}_n q_{[x]+t} = 1 - {}_n p_{[x]+t}.

Usage

nqx_select(tbl, x_sel, t, n)

Arguments

tbl

A select_life_table object.
x_sel

Numeric vector of ages at selection.
t

Numeric vector of current durations since selection.
n

Numeric vector of nonnegative integer future durations.

Value

Numeric vector of death probabilities.

nqx(j){}_n q_x^{(j)} from a multiple-decrement table

Description

nqx(j){}_n q_x^{(j)} from a multiple-decrement table

Usage

nqxj_md(tbl, x, n, j)

Arguments

tbl

Output from md_table().
x

Starting age.
n

Number of years.
j

Cause index.

Value

Numeric scalar.

Examples

x <- 45:50
qmat <- cbind(q1 = c(.011, .012, .013, .014, .015, .016), q2 = rep(.1, 6))
tbl <- md_table(x, qmat, radix = 1000)
nqxj_md(tbl, x = 46, n = 2, j = 1)

nqx(τ){}_n q_x^{(\tau)} from a multiple-decrement table

Description

nqx(τ){}_n q_x^{(\tau)} from a multiple-decrement table

Usage

nqxtau_md(tbl, x, n)

Arguments

tbl

Output from md_table().
x

Starting age.
n

Number of years.

Value

Numeric scalar.

Examples

x <- 45:50
qmat <- cbind(q1 = c(.011, .012, .013, .014, .015, .016), q2 = rep(.1, 6))
tbl <- md_table(x, qmat, radix = 1000)
nqxtau_md(tbl, x = 46, n = 2)

Projected annual benefit under a career average earnings DB plan

Description

Computes the projected annual benefit for a CAE plan using Equation (18.9).

Usage

PAB_cae(x, z, CASx, p, past_salary_total = 0, g = NULL, s = NULL)

Arguments

x

Current or entry age.
z

Retirement age.
CASx

Current annual salary at age x.
p

Accrual percentage, e.g. 1 for 1 percent.
past_salary_total

Optional total of actual past salaries.
g

Optional constant annual salary growth rate.
s

Optional salary scale vector of length z - x.

Value

Projected annual benefit.

Examples

PAB_cae(x = 30, z = 65, CASx = 100000, p = 1, g = 0.04)

Projected annual benefit under a final average salary DB plan

Description

Computes the projected annual benefit for a final average salary plan using Equation (18.6).

Usage

PAB_fas(x, z, CASx, p, fas_years = 3, past_service = 0, g = NULL, s = NULL)

Arguments

x

Current or entry age.
z

Retirement age.
CASx

Current annual salary at age x.
p

Accrual percentage, e.g. 2 for 2 percent.
fas_years

Number of years in the final average salary period.
past_service

Past years of service already completed at age x.
g

Optional constant annual salary growth rate.
s

Optional salary scale vector of length z - x.

Value

Projected annual benefit.

Examples

PAB_fas(x = 35, z = 65, CASx = 60000, p = 2, fas_years = 3, g = 0.04)

Continuous premium approximation P‾\overline{P} by trapezoidal rule

Description

Approximates the annual continuous premium in the disability model allowing for recovery, as in Example 14.18.

Usage

Pbar_trapz_ms(t, tp00, tp01, delta, mu02, mu12, B02 = 1, B12 = 1, R = 0)

Arguments

t

Numeric vector of time points.
tp00

Numeric vector of values tpx00{}_{t}p_{x}^{00}.
tp01

Numeric vector of values tpx01{}_{t}p_{x}^{01}.
delta

Force of interest.
mu02

Function of time returning μx+t02\mu_{x+t}^{02}.
mu12

Function of time returning μx+t12\mu_{x+t}^{12}.
B02

Benefit payable on death while healthy.
B12

Benefit payable on death while disabled.
R

Continuous income rate while disabled.

Details

The numerator is

∫vt[tpx00μx+t02B02+tpx01μx+t12B12+tpx01R]dt\int v^t \left[{}_{t}p_{x}^{00}\mu_{x+t}^{02}B^{02} + {}_{t}p_{x}^{01}\mu_{x+t}^{12}B^{12} + {}_{t}p_{x}^{01}R \right] dt

and the denominator is

∫vttpx00dt\int v^t {}_{t}p_{x}^{00} dt

Value

A numeric scalar.

Examples

mu01 <- function(t) 0.10 * t + 0.20
mu02 <- function(t) 0.20
mu10 <- function(t) 0.50
mu12 <- function(t) 0.125 * t + 0.20

ex1410 <- tp00_tp01_euler(
  h = 0.10, n = 2.0,
  mu01 = mu01, mu02 = mu02, mu10 = mu10, mu12 = mu12
)

Pbar_trapz_ms(
  t = ex1410$t,
  tp00 = ex1410$tp00,
  tp01 = ex1410$tp01,
  delta = 0.04,
  mu02 = mu02,
  mu12 = mu12,
  B02 = 1000,
  B12 = 1000,
  R = 1000
)

Profit signature from a profit vector

Description

Converts a Chapter 17 profit vector Pr=(Pr0,…,Prn)\mathbf{Pr}=(Pr_0,\dots,Pr_n) into the corresponding profit signature Π=(Π0,…,Πn)\mathbf{\Pi}=(\Pi_0,\dots,\Pi_n) using Equation (17.3).

Usage

Pi_signature(Pr, p_tau)

Arguments

Pr

Profit vector of length n+1n+1.
p_tau

One-year in-force probabilities. This may have length n−1n-1 or nn. If length nn, the final entry is ignored for the profit-signature calculation.

Value

Numeric vector of length n+1n+1.

Examples

Pr <- c(-15.00, 8.42, 8.40, 8.61)
Pi_signature(Pr, p_tau = c(0.99858, 0.99847, 0.99834))

Deferred annuity-due premium

Description

Computes P(n∣a¨x)=n∣a¨x/a¨x:n‾∣P({}_{n|}\ddot{a}_x) = {}_{n|}\ddot{a}_x / \ddot{a}_{x:\overline{n}|}.

Usage

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

Arguments

x

Issue age.
n

Deferral period.
i

Effective annual interest rate.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

PnAdotx(40, n = 20, i = 0.05, model = "uniform", omega = 100)

Deferred annuity-immediate premium

Description

Computes P(n∣ax)=n∣ax/a¨x:n‾∣P({}_{n|}a_x) = {}_{n|}a_x / \ddot{a}_{x:\overline{n}|}.

Usage

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

Arguments

x

Issue age.
n

Deferral period.
i

Effective annual interest rate.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

Pnax(40, n = 20, i = 0.05, model = "uniform", omega = 100)

Profit vector for a discrete profit-analysis model

Description

Computes the Chapter 17 profit vector

Pr=(Pr0,Pr1,…,Prn)\mathbf{Pr} = (Pr_0, Pr_1, \dots, Pr_n)

where Pr0Pr_0 is the negative pre-contract expense and the yearly expected profit values are calculated from the general discrete expression in Equation (17.1).

Usage

Pr_vector_disc(
  V,
  G,
  i,
  r = 0,
  e = 0,
  q1,
  q2 = 0,
  b1,
  b2 = 0,
  s1 = 0,
  s2 = 0,
  p_tau = NULL,
  pre_contract_expense = 0
)

Arguments

V

Vector of gross premium reserves tVG{}_tV^G of length n+1n+1, including the issue-time reserve and the terminal reserve.
G

Gross premium vector for policy years 1 through nn.
i

Interest-rate vector for policy years 1 through nn.
r

Percent-of-premium expense vector.
e

Fixed expense vector.
q1

First decrement probabilities, typically death.
q2

Second decrement probabilities, typically surrender or lapse. Defaults to 0.
b1

Benefit vector for decrement 1.
b2

Benefit vector for decrement 2. Defaults to 0.
s1

Settlement-expense vector for decrement 1. Defaults to 0.
s2

Settlement-expense vector for decrement 2. Defaults to 0.
p_tau

Optional vector of in-force probabilities px+t(τ)p_{x+t}^{(\tau)}. If omitted, it is computed as 1−q(1)−q(2)1-q^{(1)}-q^{(2)}.
pre_contract_expense

Positive pre-contract expense amount. The returned first element is Pr0=−pre_contract_expensePr_0 = -\text{pre\_contract\_expense}.

Details

This implementation allows for two decrements, typically death and withdrawal/surrender.

Value

Numeric vector of length n+1n+1.

Examples

V <- c(0, 5.66, 6.17, 0)
qx <- c(0.00142, 0.00153, 0.00166)
Pr_vector_disc(
  V = V, G = 95, i = 0.06, r = 0.05, e = 10,
  q1 = qx, b1 = 50000, pre_contract_expense = 15
)

Chapter 9 premium, loss, and expense functions

Description

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.

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.

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.

Profit margin

Description

Computes the Chapter 17 profit margin:

Profit Margin=NPVAPVGP.\text{Profit Margin} = \frac{NPV}{APV_{GP}}.

Usage

profit_margin(NPV, APV_GP)

Arguments

NPV

Net present value of profits.
APV_GP

Actuarial present value of gross premiums.

Value

Numeric scalar.

Examples

profit_margin(6.03, 259.52)

Present value of cash flows at time 0

Description

Present value of cash flows at time 0

Usage

pv_cashflows(cf, t, i)

Arguments

cf

Cash flow amounts (positive = inflow, negative = outflow).
t

Times of cash flows (same length as cf).
i

Effective interest rate.

Value

Present value at time 0.

Examples

pv_cashflows(c(-100, 60, 60), c(0, 1, 2), i = 0.10)

Present value of cash flows using spot rates

Description

Discounts deterministic cash flows using spot rates matched to their maturities.

Usage

pv_spot_cashflows(
  amounts,
  times,
  spot,
  compounding = c("annual", "semiannual")
)

Arguments

amounts

Numeric vector of cash flow amounts.
times

Numeric vector of payment times in years.
spot

Numeric vector of spot rates matched elementwise to times. Use 0 for any time-0 entry.
compounding

Either "annual" or "semiannual".

Details

For annual compounding, each positive-time cash flow at time tt is discounted by (1+zt)−t(1+z_t)^{-t}.

For semiannual nominal compounding, each positive-time cash flow at time tt is discounted by (1+zt/2)−2t(1+z_t/2)^{-2t}.

Time-0 cash flows are left undiscounted.

Value

A numeric scalar.

Examples

pv_spot_cashflows(
  amounts = c(200000, 50000, 50000, 100000),
  times   = c(0, 0.5, 1, 2),
  spot    = c(0, 0.02440, 0.02601, 0.02936),
  compounding = "semiannual"
)

Project one-year survival probability under mortality improvement

Description

Computes projected one-year survival probability

px[Y]=1−qx[Y]p_x^{[Y]} = 1 - q_x^{[Y]}

.

Usage

px_proj(qx_base, AAx, base_year, proj_year)

Arguments

qx_base

Base-year one-year death probability qx[B]q_x^{[B]}.
AAx

Mortality improvement factor AAxAA_x.
base_year

Base year BB.
proj_year

Projection year YY. May be scalar or vector.

Value

Numeric vector of projected one-year survival probabilities.

Construct life-table values from p_x values

Description

Builds life-table survivor values recursively from lx+1=lxpxl_{x+1} = l_x p_x, starting from a chosen radix.

Usage

px_to_lx(px, radix = 1e+05)

Arguments

px

Numeric vector of one-year survival probabilities pxp_x.
radix

Positive radix l0l_0.

Value

Numeric vector of lxl_x values of length length(px)+1.

Survival probability px(τ)p_x^{(\tau)}

Description

Survival probability px(τ)p_x^{(\tau)}

Usage

pxtau(qxj)

Arguments

qxj

Numeric vector of cause-specific decrement probabilities.

Value

Numeric scalar.

Examples

pxtau(c(0.011, 0.100))

One-year persistency rates for universal life

Description

Computes the one-year persistency rates px(τ)p_x^{(\tau)} under either Equation (16.14) or Equation (16.15).

Usage

pxtau_ul(qd, qw, year_end_withdrawal = TRUE)

Arguments

qd

Mortality probabilities.
qw

Withdrawal probabilities.
year_end_withdrawal

Logical; if TRUE, use (1−q(d))(1−q(w))(1-q^{(d)})(1-q^{(w)}). Otherwise use 1−q(d)−q(w)1-q^{(d)}-q^{(w)}.

Value

Numeric vector.

Examples

qd <- c(.001, .002, .003)
qw <- c(.02, .02, .03)
pxtau_ul(qd, qw)

Dependent probabilities qx(j)q_x^{(j)} from independent probabilities qx′(j)q_x^{\prime(j)} under constant force

Description

Dependent probabilities qx(j)q_x^{(j)} from independent probabilities qx′(j)q_x^{\prime(j)} under constant force

Usage

qx_dep_cf(qxprime)

Arguments

qxprime

Numeric vector of independent probabilities.

Value

Numeric vector of dependent probabilities.

Examples

qx_dep_cf(c(0.20, 0.10))

Dependent probabilities qx(j)q_x^{(j)} from independent probabilities qx′(j)q_x^{\prime(j)} under SUDD

Description

Two-decrement case only.

Usage

qx_dep_sudd(q1prime, q2prime)

Arguments

q1prime

Independent probability for decrement 1.
q2prime

Independent probability for decrement 2.

Value

Numeric vector c(q1, q2).

Examples

qx_dep_sudd(0.20, 0.10)

Project one-year death probability under mortality improvement

Description

Computes projected one-year death probability

qx[Y]=qx[B](1−AAx)Y−Bq_x^{[Y]} = q_x^{[B]} (1-AA_x)^{Y-B}

.

Usage

qx_proj(qx_base, AAx, base_year, proj_year)

Arguments

qx_base

Base-year one-year death probability qx[B]q_x^{[B]}.
AAx

Mortality improvement factor AAxAA_x.
base_year

Base year BB.
proj_year

Projection year YY. May be scalar or vector.

Value

Numeric vector of projected one-year death probabilities.

Compute one-year death probability from a life table

Description

Compute one-year death probability from a life table

Usage

qx_tab(tbl, x)

Arguments

tbl

A life_table object.
x

Ages.

Value

Numeric vector of q_x values.

Construct life-table values from q_x values

Description

Builds life-table survivor values recursively from lx+1=lx(1−qx)l_{x+1} = l_x (1-q_x), starting from a chosen radix.

Usage

qx_to_lx(qx, radix = 1e+05)

Arguments

qx

Numeric vector of one-year death probabilities qxq_x.
radix

Positive radix l0l_0.

Value

Numeric vector of lxl_x values of length length(qx)+1.

Independent probabilities qx′(j)q_x^{\prime(j)} from dependent probabilities qx(j)q_x^{(j)} under MUDD

Description

Independent probabilities qx′(j)q_x^{\prime(j)} from dependent probabilities qx(j)q_x^{(j)} under MUDD

Usage

qxprime_mudd(qxj)

Arguments

qxj

Numeric vector of dependent probabilities qx(j)q_x^{(j)}.

Value

Numeric vector of independent probabilities qx′(j)q_x^{\prime(j)}.

Examples

qxprime_mudd(c(.20, .10))

Independent probabilities qx′(j)q_x^{\prime(j)} from dependent probabilities qx(j)q_x^{(j)} under SUDD

Description

Two-decrement case only.

Usage

qxprime_sudd(q1, q2)

Arguments

q1

Dependent probability for decrement 1.
q2

Dependent probability for decrement 2.

Value

Numeric vector c(q1prime, q2prime).

Examples

qxprime_sudd(0.20, 0.10)

Total probability of decrement qx(τ)q_x^{(\tau)}

Description

Total probability of decrement qx(τ)q_x^{(\tau)}

Usage

qxtau(qxj)

Arguments

qxj

Numeric vector of cause-specific decrement probabilities.

Value

Numeric scalar.

Examples

qxtau(c(0.011, 0.100))

Replacement ratio for a defined benefit plan

Description

Computes a DB replacement ratio as benefit divided by a chosen salary measure.

Usage

replacement_ratio_db(benefit, salary)

Arguments

benefit

Annual retirement benefit.
salary

Salary measure used in the denominator.

Value

Replacement ratio.

Examples

replacement_ratio_db(benefit = 108008.66, salary = 187119.09)

Replacement ratio for a defined contribution plan

Description

Computes the replacement ratio defined in Equation (18.4).

Usage

replacement_ratio_dc(x, z, Sx, c, i, adue_z, g = NULL, s = NULL)

Arguments

x

Entry age.
z

Retirement age.
Sx

Salary at age x.
c

Contribution rate.
i

Annual effective interest rate.
adue_z

Whole life annuity-due factor at age z.
g

Optional constant annual salary growth rate.
s

Optional salary scale vector of length z - x.

Value

Replacement ratio.

Examples

replacement_ratio_dc(
  x = 30, z = 65, Sx = 50000, c = 0.10, i = 0.05,
  adue_z = 12, g = 0.04
)

Deferred insurance reserve functions

Description

Chapter 10 reserve functions for deferred insurance contracts.

Usage

tVnAx(x, n, t, i, model, ...)

htVnAx(x, n, h, t, i, model, ...)

Arguments

x

Issue age.
n

Deferral period.
t

Duration.
i

Effective annual interest rate.
model

Survival model.
...

Additional model parameters.
h

Premium-paying period.

Details

tVnAx() computes the reserve for an nn-year deferred insurance funded over the deferred period.

htVnAx() computes the reserve when premiums are limited to the first hh years, with h≤nh \le n.

Value

Numeric vector.

Numeric vector.

Examples

tVnAx(40, n = 20, t = 10, i = 0.05, model = "uniform", omega = 100)
htVnAx(40, n = 20, h = 10, t = 5, i = 0.05, model = "uniform", omega = 100)

Ratio rt=AVt/GMFtr_t = AV_t / GMF_t capped at 1

Description

Computes the ratio in Equation (16.16).

Usage

rt_ul(AV, GMF)

Arguments

AV

Account value.
GMF

Guaranteed maturity fund.

Value

Numeric scalar.

Examples

rt_ul(AV = 4918.20, GMF = 14678.57)

Survival function for age-at-failure T0

Description

Computes the survival distribution function S0(t)=Pr(T0>t)S_0(t)=Pr(T_0>t).

Usage

S0(t, model, ...)

Arguments

t

Numeric vector of times (t≥0t \ge 0).
model

One of "uniform", "exponential", "gompertz", "makeham", "weibull".
...

Model parameters:

  • uniform: omega

  • exponential: lambda

  • gompertz: B, c

  • makeham: A, B, c

  • weibull: shape, scale

Details

Supported models (Chapter 5): uniform (de Moivre), exponential, Gompertz, Makeham, Weibull.

Value

Numeric vector of survival probabilities in [0,1][0,1].

Convert survival probabilities to life-table values

Description

Converts Chapter 5 survival function values S0(x)S_0(x) into Chapter 6 life-table values lx=l0S0(x)l_x = l_0 S_0(x) using a chosen radix.

Usage

S0_to_lx(S0, radix = 1e+05)

Arguments

S0

Numeric vector of survival probabilities.
radix

Positive radix l0l_0.

Value

Numeric vector of lxl_x values.

Salary scale under constant annual growth

Description

Constructs salary scale factors sks_k under a constant annual growth rate.

Usage

salary_scale(k, g, base_age = min(k), s_base = 1)

Arguments

k

Numeric vector of ages.
g

Annual salary growth rate.
base_age

Age at which the scale is normalized.
s_base

Salary scale value at base_age.

Value

Numeric vector of salary scale factors.

Examples

salary_scale(k = 30:34, g = 0.04, base_age = 30)

Construct a select life table

Description

Builds a select-life-table object from vectors of selection age, duration since selection, attained age, and survivor values.

Usage

select_life_table(x_sel, duration, attained_age, lx)

Arguments

x_sel

Numeric vector of ages at selection.
duration

Numeric vector of durations since selection.
attained_age

Numeric vector of attained ages.
lx

Numeric vector of select-table survivor values.

Value

A data.frame with class "select_life_table".

Solve the yield rate by the equation of value

Description

Finds the interest rate i such that the present value of the cash flows is 0.

Usage

solve_yield(cf, t, interval = c(-0.99, 1), tol = 1e-10)

Arguments

cf

Cash flows.
t

Times.
interval

Two-length numeric vector bracketing the root.
tol

Tolerance passed to uniroot.

Value

Yield rate i.

Examples

solve_yield(c(-100, 60, 60), c(0, 1, 2), interval = c(-0.5, 1))

Backward Euler reserve path from maturity

Description

Starting from a terminal reserve value at time T, computes reserves backward on a grid using the backward Euler-style Thiele step.

Usage

thiele_backward_path(times, V_terminal, P, delta, mu, benefit = 1)

Arguments

times

Vector of times in increasing order.
V_terminal

Reserve at the final time.
P

Premium rate, scalar or vector of length length(times)-1.
delta

Force of interest, scalar or vector of length length(times)-1.
mu

Force of mortality, scalar or vector of length length(times)-1.
benefit

Benefit amount, scalar or vector of length length(times)-1.

Value

Numeric vector of reserve values on the grid.

Examples

times <- seq(19, 20, by = 0.25)
thiele_backward_path(times, V_terminal = 1000, P = 26.96, delta = 0.058, mu = 0.002, benefit = 1000)

One backward Euler-style Thiele step

Description

Approximates the reserve at time tt from a known reserve at time t+ht+h.

Usage

thiele_backward_step(V_next, P, delta, mu, benefit = 1, h = 1)

Arguments

V_next

Reserve at time t+h.
P

Premium rate.
delta

Force of interest.
mu

Force of mortality at time t.
benefit

Benefit amount. Defaults to 1.
h

Step size.

Value

Numeric vector.

Examples

thiele_backward_step(V_next = 1000, P = 26.96, delta = 0.058, mu = 0.002, benefit = 1000, h = 1)

Reserve derivative from Thiele's equation

Description

Computes dV/dt=P+δV−μ(B−V)dV/dt = P + \delta V - \mu(B - V).

Usage

thiele_dVdt(V, P, delta, mu, benefit = 1)

Arguments

V

Reserve at time t.
P

Premium rate.
delta

Force of interest.
mu

Force of mortality.
benefit

Benefit amount. Defaults to 1.

Value

Numeric vector.

Examples

thiele_dVdt(V = 900, P = 25, delta = 0.05, mu = 0.002, benefit = 1000)

Reserve derivatives for the disability model with recovery

Description

Computes the right-hand sides of the coupled Thiele differential equations in Equations (14.25) and (14.26) for the healthy-life reserve tV‾(0){}_{t}\overline{V}^{(0)} and the disabled-life reserve tV‾(1){}_{t}\overline{V}^{(1)}.

Usage

thiele_dVdt_01(t, V0, V1, delta, Pbar, B, R, mu01, mu02, mu10, mu12)

Arguments

t

Time.
V0

Value of tV‾(0){}_{t}\overline{V}^{(0)}.
V1

Value of tV‾(1){}_{t}\overline{V}^{(1)}.
delta

Force of interest.
Pbar

Continuous premium rate.
B

Death benefit.
R

Continuous disability income rate.
mu01

Function of time returning μx+t01\mu_{x+t}^{01}.
mu02

Function of time returning μx+t02\mu_{x+t}^{02}.
mu10

Function of time returning μx+t10\mu_{x+t}^{10}.
mu12

Function of time returning μx+t12\mu_{x+t}^{12}.

Details

The equations are

ddttV‾(0)=P‾+δtV‾(0)−μx+t02(B−tV‾(0))−μx+t01(tV‾(1)−tV‾(0))\frac{d}{dt}{}_{t}\overline{V}^{(0)} = \overline{P} + \delta {}_{t}\overline{V}^{(0)} - \mu_{x+t}^{02}(B-{}_{t}\overline{V}^{(0)}) - \mu_{x+t}^{01}({}_{t}\overline{V}^{(1)}-{}_{t}\overline{V}^{(0)})

and

ddttV‾(1)=δtV‾(1)−R−μx+t12(B−tV‾(1))−μx+t10(tV‾(0)−tV‾(1))\frac{d}{dt}{}_{t}\overline{V}^{(1)} = \delta {}_{t}\overline{V}^{(1)} - R - \mu_{x+t}^{12}(B-{}_{t}\overline{V}^{(1)}) - \mu_{x+t}^{10}({}_{t}\overline{V}^{(0)}-{}_{t}\overline{V}^{(1)})

Value

A named numeric vector with components dV0 and dV1.

Examples

mu01 <- function(t) 0.10 * t + 0.20
mu02 <- function(t) 0.20
mu10 <- function(t) 0.50
mu12 <- function(t) 0.125 * t + 0.20

thiele_dVdt_01(
  t = 2.0, V0 = 0, V1 = 0,
  delta = 0.04, Pbar = 446.95,
  B = 1000, R = 1000,
  mu01 = mu01, mu02 = mu02, mu10 = mu10, mu12 = mu12
)

Backward reserve path for the disability model with recovery

Description

Computes the backward Euler reserve path for the healthy-life reserve tV‾(0){}_{t}\overline{V}^{(0)} and disabled-life reserve tV‾(1){}_{t}\overline{V}^{(1)} using Equations (14.27) and (14.28).

Usage

thiele_path_01(
  h,
  n,
  delta,
  Pbar,
  B,
  R,
  mu01,
  mu02,
  mu10,
  mu12,
  V0_n = 0,
  V1_n = 0
)

Arguments

h

Step size.
n

Final time.
delta

Force of interest.
Pbar

Continuous premium rate.
B

Death benefit.
R

Continuous disability income rate.
mu01

Function of time returning μx+t01\mu_{x+t}^{01}.
mu02

Function of time returning μx+t02\mu_{x+t}^{02}.
mu10

Function of time returning μx+t10\mu_{x+t}^{10}.
mu12

Function of time returning μx+t12\mu_{x+t}^{12}.
V0_n

Terminal value of nV‾(0){}_{n}\overline{V}^{(0)}.
V1_n

Terminal value of nV‾(1){}_{n}\overline{V}^{(1)}.

Value

A data frame with columns t, tV0, and tV1.

Examples

mu01 <- function(t) 0.10 * t + 0.20
mu02 <- function(t) 0.20
mu10 <- function(t) 0.50
mu12 <- function(t) 0.125 * t + 0.20

thiele_path_01(
  h = 0.10, n = 2.0, delta = 0.04, Pbar = 446.95,
  B = 1000, R = 1000,
  mu01 = mu01, mu02 = mu02, mu10 = mu10, mu12 = mu12
)

Euler approximation for tpx00{}_{t}p_{x}^{00} and tpx01{}_{t}p_{x}^{01}

Description

Computes the Euler approximations in the disability model allowing for recovery, as in Equations (14.20) and (14.21).

Usage

tp00_tp01_euler(h, n, mu01, mu02, mu10, mu12, p00_0 = 1, p01_0 = 0)

Arguments

h

Step size.
n

Final time.
mu01

Function of time returning μx+t01\mu_{x+t}^{01}.
mu02

Function of time returning μx+t02\mu_{x+t}^{02}.
mu10

Function of time returning μx+t10\mu_{x+t}^{10}.
mu12

Function of time returning μx+t12\mu_{x+t}^{12}.
p00_0

Initial value of 0px00{}_{0}p_{x}^{00}.
p01_0

Initial value of 0px01{}_{0}p_{x}^{01}.

Details

The model uses three states:

  • State 0: healthy

  • State 1: disabled

  • State 2: deceased

Value

A data frame with columns t, tp00, tp01, and tp02.

Examples

mu01 <- function(t) 0.10 * t + 0.20
mu02 <- function(t) 0.20
mu10 <- function(t) 0.50
mu12 <- function(t) 0.125 * t + 0.20

tp00_tp01_euler(
  h = 0.10, n = 2.0,
  mu01 = mu01, mu02 = mu02, mu10 = mu10, mu12 = mu12
)

Conditional survival probability for Tx

Description

Computes tpx=S0(x+t)/S0(x){}_t p_x = S_0(x+t)/S_0(x).

Usage

tpx(t, x, model, ...)

Arguments

t

Numeric vector of durations (t≥0t \ge 0).
x

Numeric vector of ages (x≥0x \ge 0).
model

One of "uniform", "exponential", "gompertz", "makeham", "weibull".
...

Model parameters:

  • uniform: omega

  • exponential: lambda

  • gompertz: B, c

  • makeham: A, B, c

  • weibull: shape, scale

Details

Vectorization rule: - If t and x are the same length, values are computed elementwise. - If one of t or x has length 1, it is recycled to match the other.

Value

Numeric vector in [0,1][0,1].

Multi-year survival probability under mortality improvement

Description

Computes survival over n years starting at age x0 in year issue_year, using base-year mortality qx_base_vec and improvement factors AAx_vec.

Usage

tpx_improved(x0, n, qx_base_vec, AAx_vec, base_year, issue_year)

Arguments

x0

Issue age.
n

Number of years.
qx_base_vec

Base-year one-year death probabilities for successive ages.
AAx_vec

Mortality improvement factors for successive ages.
base_year

Base year.
issue_year

Issue year.

Details

The vectors should correspond to ages x0, x0+1, ..., x0+n-1.

The survival probability is

∏j=0n−1(1−qx+j[issue_year+j])\prod_{j=0}^{n-1} \left(1 - q_{x+j}^{[issue\_year+j]}\right)

.

Value

Numeric scalar.

Fractional survival probability from a life table

Description

Computes tpx{}_t p_x for 0≤t≤10 \le t \le 1 from a discrete life table under one of the standard Chapter 6 assumptions: UDD, constant force, or Balducci.

Usage

tpx_tab(tbl, x, t, assumption = c("udd", "cf", "balducci"))

Arguments

tbl

A life_table object.
x

Numeric vector of integer ages.
t

Numeric vector of fractional durations with 0≤t≤10 \le t \le 1.
assumption

One of "udd", "cf", "balducci".

Value

Numeric vector of tpx{}_t p_x values.

Total survival probability tpx(τ){}_t p_x^{(\tau)} under constant forces

Description

Total survival probability tpx(τ){}_t p_x^{(\tau)} under constant forces

Usage

tpx_tau_cf(mu, t)

Arguments

mu

Numeric vector of cause-specific forces.
t

Time.

Value

Numeric scalar/vector.

Examples

tpx_tau_cf(c(0.10, 0.20), 5)

Single-decrement survival probability tpx′(j){}_t p_x^{\prime(j)} under constant force

Description

Single-decrement survival probability tpx′(j){}_t p_x^{\prime(j)} under constant force

Usage

tpxprimej_cf(mu, t)

Arguments

mu

Force of decrement for cause j.
t

Time.

Value

Numeric scalar/vector.

Examples

tpxprimej_cf(0.10, 5)

Cumulative persistency to the end of each policy year

Description

Computes tpx(τ){}_tp_x^{(\tau)} from the one-year persistency rates.

Usage

tpxtau_ul(qd, qw, year_end_withdrawal = TRUE)

Arguments

qd

Mortality probabilities.
qw

Withdrawal probabilities.
year_end_withdrawal

Logical; if TRUE, use Equation (16.15).

Value

Numeric vector.

Examples

qd <- c(.001, .002, .003)
qw <- c(.02, .02, .03)
tpxtau_ul(qd, qw)

Joint-life survival probability

Description

Computes tpxy=tpx tpy{}_tp_{xy} = {}_tp_x\,{}_tp_y under independence.

Usage

tpxy(x, y, t, tbl = NULL, model = NULL, ...)

Arguments

x

Age of first life.
y

Age of second life.
t

Time.
tbl

Life table.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

tpxy(40, 50, t = 10, model = "uniform", omega = 100)

Last-survivor survival probability

Description

Computes tpxy‾=tpx+tpy−tpxy{}_tp_{\overline{xy}} = {}_tp_x + {}_tp_y - {}_tp_{xy}.

Usage

tpxybar(x, y, t, tbl = NULL, model = NULL, ...)

Arguments

x

Age of first life.
y

Age of second life.
t

Time.
tbl

Life table.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

tpxybar(40, 50, t = 10, model = "uniform", omega = 100)

Conditional failure probability for Tx

Description

Computes tqx=1−tpx{}_t q_x = 1 - {}_t p_x.

Usage

tqx(t, x, model, ...)

Arguments

t

Numeric vector of durations (t≥0t \ge 0).
x

Numeric vector of ages (x≥0x \ge 0).
model

One of "uniform", "exponential", "gompertz", "makeham", "weibull".
...

Model parameters:

  • uniform: omega

  • exponential: lambda

  • gompertz: B, c

  • makeham: A, B, c

  • weibull: shape, scale

Value

Numeric vector in [0,1][0,1].

Fractional failure probability from a life table

Description

Computes tqx=1−tpx{}_t q_x = 1 - {}_t p_x for 0≤t≤10 \le t \le 1.

Usage

tqx_tab(tbl, x, t, assumption = c("udd", "cf", "balducci"))

Arguments

tbl

A life_table object.
x

Numeric vector of integer ages.
t

Numeric vector of fractional durations with 0≤t≤10 \le t \le 1.
assumption

One of "udd", "cf", "balducci".

Value

Numeric vector of tqx{}_t q_x values.

Cause-specific probability tqx(j){}_t q_x^{(j)} under constant forces

Description

Cause-specific probability tqx(j){}_t q_x^{(j)} under constant forces

Usage

tqxj_cf(mu, j, t)

Arguments

mu

Numeric vector of cause-specific forces.
j

Cause index.
t

Time.

Value

Numeric scalar/vector.

Examples

tqxj_cf(c(0.10, 0.20), j = 1, t = 5)

Independent probabilities tqx′(j){}_t q_x^{\prime(j)} under MUDD

Description

Independent probabilities tqx′(j){}_t q_x^{\prime(j)} under MUDD

Usage

tqxprime_mudd(qxj, t)

Arguments

qxj

Numeric vector of dependent probabilities qx(j)q_x^{(j)}.
t

Time in [0,1].

Value

Numeric vector.

Examples

tqxprime_mudd(c(.20, .10), t = 0.5)

Single-decrement failure probability tqx′(j){}_t q_x^{\prime(j)} under constant force

Description

Single-decrement failure probability tqx′(j){}_t q_x^{\prime(j)} under constant force

Usage

tqxprimej_cf(mu, t)

Arguments

mu

Force of decrement for cause j.
t

Time.

Value

Numeric scalar/vector.

Examples

tqxprimej_cf(0.10, 5)

Joint-life failure probability

Description

Computes tqxy=1−tpxy{}_tq_{xy} = 1 - {}_tp_{xy}.

Usage

tqxy(x, y, t, tbl = NULL, model = NULL, ...)

Arguments

x

Age of first life.
y

Age of second life.
t

Time.
tbl

Life table.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

tqxy(40, 50, t = 10, model = "uniform", omega = 100)

Probability that (x) fails before (y) within n years

Description

Computes nqxy1=∫0ntpxyμx+t dt{}_n q_{xy}^{1} = \int_0^n {}_tp_{xy}\mu_{x+t}\,dt under independence.

Usage

tqxy1(x, y, n, tbl = NULL, model = NULL, ...)

Arguments

x

Age of first life.
y

Age of second life.
n

Term in years.
tbl

Life table.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

tqxy1(40, 50, n = 10, model = "uniform", omega = 100)

Probability that (x) fails after (y) within n years

Description

Computes nqxy2=nqx−nqxy1{}_n q_{xy}^{2} = {}_n q_x - {}_n q_{xy}^{1}.

Usage

tqxy2(x, y, n, tbl = NULL, model = NULL, ...)

Arguments

x

Age of first life.
y

Age of second life.
n

Term in years.
tbl

Life table.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

tqxy2(40, 50, n = 10, model = "uniform", omega = 100)

Last-survivor failure probability

Description

Computes tqxy‾=1−tpxy‾{}_tq_{\overline{xy}} = 1 - {}_tp_{\overline{xy}}.

Usage

tqxybar(x, y, t, tbl = NULL, model = NULL, ...)

Arguments

x

Age of first life.
y

Age of second life.
t

Time.
tbl

Life table.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

tqxybar(40, 50, t = 10, model = "uniform", omega = 100)

Probability that (y) fails before (x) within n years

Description

Computes nqxy1=∫0ntpxyμy+t dt{}_n q_{xy}^{\hspace{1mm}1} = \int_0^n {}_tp_{xy}\mu_{y+t}\,dt under independence.

Usage

tqyx1(x, y, n, tbl = NULL, model = NULL, ...)

Arguments

x

Age of first life.
y

Age of second life.
n

Term in years.
tbl

Life table.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

tqyx1(40, 50, n = 10, model = "uniform", omega = 100)

Probability that (y) fails after (x) within n years

Description

Computes nqxy2=nqy−nqxy1{}_n q_{xy}^{\hspace{1mm}2} = {}_n q_y - {}_n q_{xy}^{\hspace{1mm}1}.

Usage

tqyx2(x, y, n, tbl = NULL, model = NULL, ...)

Arguments

x

Age of first life.
y

Age of second life.
n

Term in years.
tbl

Life table.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

tqyx2(40, 50, n = 10, model = "uniform", omega = 100)

Fractional-duration whole life reserve

Description

Computes the interpolated reserve t+sV=(tV+Px)(1−s)+t+1V⋅s{}_{t+s}V = ({}_tV + P_x)(1-s) + {}_{t+1}V \cdot s for 0≤s≤10 \le s \le 1.

Computes the interpolated reserve t+sV=(tV+Px)(1−s)+t+1V⋅s{}_{t+s}V = ({}_tV + P_x)(1-s) + {}_{t+1}V \cdot s for 0≤s≤10 \le s \le 1.

Usage

tsVx(x, t, s, i, tbl = NULL, model = NULL, ...)

tsVx(x, t, s, i, tbl = NULL, model = NULL, ...)

Arguments

x

Issue age.
t

Integer contract duration.
s

Fractional part of duration in [0, 1].
i

Effective annual interest rate.
tbl

Optional life table object.
model

Optional parametric survival model.
...

Additional model parameters.

Value

Numeric vector.

Numeric vector.

Examples

tsVx(40, t = 10, s = 0.5, i = 0.05, model = "uniform", omega = 100)
tsVx(40, t = 10, s = 0.5, i = 0.05, model = "uniform", omega = 100)

Fractional-duration endowment reserve

Description

Computes the interpolated reserve t+sV=(tV+P)(1−s)+t+1V⋅s{}_{t+s}V = ({}_tV + P)(1-s) + {}_{t+1}V \cdot s for an n-year endowment insurance.

Computes the interpolated reserve t+sV=(tV+P)(1−s)+t+1V⋅s{}_{t+s}V = ({}_tV + P)(1-s) + {}_{t+1}V \cdot s for an n-year endowment insurance.

Usage

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

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

Arguments

x

Issue age.
n

Term in years.
t

Integer duration with t < n.
s

Fractional part in [0, 1].
i

Effective annual interest rate.
tbl

Optional life table object.
model

Optional parametric survival model.
...

Additional model parameters.

Value

Numeric vector.

Numeric vector.

Examples

tsVxn(40, n = 20, t = 10, s = 0.5, i = 0.05, model = "uniform", omega = 100)
tsVxn(40, n = 20, t = 10, s = 0.5, i = 0.05, model = "uniform", omega = 100)

Fractional-duration term reserve

Description

Computes the interpolated reserve t+sV=(tV+P)(1−s)+t+1V⋅s{}_{t+s}V = ({}_tV + P)(1-s) + {}_{t+1}V \cdot s for an n-year term insurance.

Computes the interpolated reserve t+sV=(tV+P)(1−s)+t+1V⋅s{}_{t+s}V = ({}_tV + P)(1-s) + {}_{t+1}V \cdot s for an n-year term insurance.

Usage

tsVxn1(x, n, t, s, i, tbl = NULL, model = NULL, ...)

tsVxn1(x, n, t, s, i, tbl = NULL, model = NULL, ...)

Arguments

x

Issue age.
n

Term in years.
t

Integer duration with t < n.
s

Fractional part in [0, 1].
i

Effective annual interest rate.
tbl

Optional life table object.
model

Optional parametric survival model.
...

Additional model parameters.

Value

Numeric vector.

Numeric vector.

Examples

tsVxn1(40, n = 20, t = 10, s = 0.5, i = 0.05, model = "uniform", omega = 100)
tsVxn1(40, n = 20, t = 10, s = 0.5, i = 0.05, model = "uniform", omega = 100)

Fully continuous whole life reserve

Description

Computes the Chapter 10 reserve for a whole life insurance with continuous premiums and immediate payment of claims.

Usage

tVbarAbarx(x, t, i, model, ...)

Arguments

x

Issue age.
t

Duration, allowed to be any nonnegative numeric value.
i

Effective annual interest rate.
model

Survival model.
...

Additional model parameters.

Details

In the fully continuous setting, reserve time tt may be any nonnegative real value.

Value

Numeric vector.

Examples

tVbarAbarx(40, t = 10, i = 0.05, model = "uniform", omega = 100)
tVbarAbarx(40, t = c(19, 19.25, 19.5, 19.75, 20), i = 0.06,
           model = "uniform", omega = 100)

Whole life reserve with continuous premiums

Description

Computes the Chapter 10 reserve for a discrete whole life insurance funded by continuous premiums.

Usage

tVbarx(x, t, i, model, ...)

Arguments

x

Issue age.
t

Duration.
i

Effective annual interest rate.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

tVbarx(40, t = 10, i = 0.05, model = "uniform", omega = 100)

Whole life expense reserve

Description

Computes the Chapter 11 expense reserve for a fully discrete whole life insurance with renewal expenses.

Computes the Chapter 11 expense reserve for a fully discrete whole life insurance with renewal expenses.

Usage

tVEx(
  x,
  t,
  i,
  G,
  benefit = 1,
  renewal_premium_pct = 0,
  renewal_policy_exp = 0,
  settlement_exp = 0,
  tbl = NULL,
  model = NULL,
  ...
)

tVEx(
  x,
  t,
  i,
  G,
  benefit = 1,
  renewal_premium_pct = 0,
  renewal_policy_exp = 0,
  settlement_exp = 0,
  tbl = NULL,
  model = NULL,
  ...
)

Arguments

x

Issue age.
t

Duration.
i

Effective annual interest rate.
G

Gross annual premium.
benefit

Insurance amount. Default 1.
renewal_premium_pct

Renewal percent-of-premium expense.
renewal_policy_exp

Renewal per-policy expense.
settlement_exp

Settlement expense at death.
tbl

Optional life table object.
model

Optional parametric survival model.
...

Additional model parameters.

Value

Numeric vector.

Numeric vector.

Examples

tVEx(
  x = 40, t = 10, i = 0.05, G = 0.03,
  benefit = 1, renewal_premium_pct = 0.10,
  renewal_policy_exp = 0.002, settlement_exp = 0.02,
  model = "uniform", omega = 100
)
tVEx(
  x = 40, t = 10, i = 0.05, G = 0.03,
  benefit = 1, renewal_premium_pct = 0.10,
  renewal_policy_exp = 0.002, settlement_exp = 0.02,
  model = "uniform", omega = 100
)

Full preliminary term reserve for whole life insurance

Description

Computes the FPT reserve for a whole life insurance. For whole life insurance, 1VF=0{}_1V^F = 0 and for t≥1t \ge 1, tVF=t−1Vx+1NLP{}_tV^F = {}_{t-1}V_{x+1}^{NLP}.

Computes the FPT reserve for a whole life insurance. For whole life insurance, 1VF=0{}_1V^F = 0 and for t≥1t \ge 1, tVF=t−1Vx+1NLP{}_tV^F = {}_{t-1}V_{x+1}^{NLP}.

Usage

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

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

Arguments

x

Issue age.
t

Duration.
i

Effective annual interest rate.
tbl

Optional life table object.
model

Optional parametric survival model.
...

Additional model parameters.

Value

Numeric vector.

Numeric vector.

Examples

tVFx(40, t = 5, i = 0.05, model = "uniform", omega = 100)
tVFx(40, t = 5, i = 0.05, model = "uniform", omega = 100)

Whole life gross premium reserve

Description

Computes the Chapter 11 prospective gross premium reserve for a fully discrete whole life insurance with annual premiums and renewal expenses.

Computes the Chapter 11 prospective gross premium reserve for a fully discrete whole life insurance with annual premiums and renewal expenses.

Usage

tVGx(
  x,
  t,
  i,
  G,
  benefit = 1,
  renewal_premium_pct = 0,
  renewal_policy_exp = 0,
  settlement_exp = 0,
  tbl = NULL,
  model = NULL,
  ...
)

tVGx(
  x,
  t,
  i,
  G,
  benefit = 1,
  renewal_premium_pct = 0,
  renewal_policy_exp = 0,
  settlement_exp = 0,
  tbl = NULL,
  model = NULL,
  ...
)

Arguments

x

Issue age.
t

Duration.
i

Effective annual interest rate.
G

Gross annual premium.
benefit

Insurance amount. Default 1.
renewal_premium_pct

Renewal percent-of-premium expense.
renewal_policy_exp

Renewal per-policy expense.
settlement_exp

Settlement expense at death.
tbl

Optional life table object.
model

Optional parametric survival model.
...

Additional model parameters.

Details

This function is intended for durations after issue, where future expenses are modeled through renewal premium expenses, renewal per-policy expenses, and settlement expense.

Value

Numeric vector.

Numeric vector.

Examples

tVGx(
  x = 40, t = 10, i = 0.05, G = 0.03,
  benefit = 1, renewal_premium_pct = 0.10,
  renewal_policy_exp = 0.002, settlement_exp = 0.02,
  model = "uniform", omega = 100
)
tVGx(
  x = 40, t = 10, i = 0.05, G = 0.03,
  benefit = 1, renewal_premium_pct = 0.10,
  renewal_policy_exp = 0.002, settlement_exp = 0.02,
  model = "uniform", omega = 100
)

Deferred annuity-due reserve

Description

Computes the Chapter 10 reserve tV(n∣a¨x){}_tV({}_{n|}\ddot{a}_x) for t<nt < n.

Usage

tVnAdotx(x, n, t, i, model, ...)

Arguments

x

Issue age.
n

Deferral period.
t

Duration.
i

Effective annual interest rate.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

tVnAdotx(40, n = 20, t = 10, i = 0.05, model = "uniform", omega = 100)

Deferred annuity-immediate reserve

Description

Computes the reserve for an n-year deferred annuity-immediate for t<nt < n.

Usage

tVnax(x, n, t, i, model, ...)

Arguments

x

Issue age.
n

Deferral period.
t

Duration.
i

Effective annual interest rate.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

tVnax(40, n = 20, t = 10, i = 0.05, model = "uniform", omega = 100)

Pure endowment net level premium reserve

Description

Computes the Chapter 10 prospective reserve for an n-year pure endowment.

Usage

tVnEx(x, n, t, i, model, ...)

Arguments

x

Issue age.
n

Term in years.
t

Duration.
i

Effective annual interest rate.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

tVnEx(40, n = 20, t = 10, i = 0.05, model = "uniform", omega = 100)

Whole life net level premium reserve

Description

Computes the Chapter 10 prospective reserve for a whole life insurance with annual premiums: reserve at duration t equals future APV of benefits minus future APV of net premiums.

Usage

tVx(x, t, i, model, ...)

Arguments

x

Issue age.
t

Duration.
i

Effective annual interest rate.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

tVx(40, t = 10, i = 0.05, model = "uniform", omega = 100)

Whole life reserve with m-thly premiums

Description

Computes the Chapter 10 reserve for a whole life insurance funded by true m-thly premiums.

Usage

tVx_m(x, t, m, i, model, ...)

Arguments

x

Issue age.
t

Duration.
m

Number of premium payments per year.
i

Effective annual interest rate.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

tVx_m(40, t = 10, m = 12, i = 0.05, model = "uniform", omega = 100)

Whole life net level premium reserve by retrospective method

Description

Computes the Chapter 10 retrospective reserve tVx=Pxs¨x:t‾∣−tkx{}_tV_x = P_x \ddot{s}_{x:\overline{t}|} - {}_tk_x.

Usage

tVx_ret(x, t, i, model, ...)

Arguments

x

Issue age.
t

Duration.
i

Effective annual interest rate.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

tVx_ret(40, t = 10, i = 0.05, model = "uniform", omega = 100)

Endowment insurance net level premium reserve

Description

Computes the Chapter 10 prospective reserve for an n-year endowment insurance.

Usage

tVxn(x, n, t, i, model, ...)

Arguments

x

Issue age.
n

Term in years.
t

Duration.
i

Effective annual interest rate.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

tVxn(40, n = 20, t = 10, i = 0.05, model = "uniform", omega = 100)

Endowment insurance reserve by retrospective method

Description

Computes the Chapter 10 retrospective reserve tVx:n‾∣=Px:n‾∣s¨x:t‾∣−tkx{}_tV_{x:\overline{n}|} = P_{x:\overline{n}|} \ddot{s}_{x:\overline{t}|} - {}_tk_x for t≤nt \le n.

Usage

tVxn_ret(x, n, t, i, model, ...)

Arguments

x

Issue age.
n

Term in years.
t

Duration.
i

Effective annual interest rate.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

tVxn_ret(40, n = 20, t = 10, i = 0.05, model = "uniform", omega = 100)

Term insurance net level premium reserve

Description

Computes the Chapter 10 prospective reserve for an n-year term insurance.

Usage

tVxn1(x, n, t, i, model, ...)

Arguments

x

Issue age.
n

Term in years.
t

Duration.
i

Effective annual interest rate.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

tVxn1(40, n = 20, t = 10, i = 0.05, model = "uniform", omega = 100)

Term insurance reserve by retrospective method

Description

Computes the retrospective term reserve for t≤nt \le n.

Usage

tVxn1_ret(x, n, t, i, model, ...)

Arguments

x

Issue age.
n

Term in years.
t

Duration.
i

Effective annual interest rate.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

tVxn1_ret(40, n = 20, t = 10, i = 0.05, model = "uniform", omega = 100)

UDD multiplier for continuous insurance approximations

Description

Under UDD, Aˉx=(i/δ)Ax\bar{A}_x = (i/\delta) A_x and similarly for term and deferred insurance.

Usage

udd_continuous_multiplier(i)

Arguments

i

Numeric vector of effective annual interest rates.

Value

Numeric vector equal to i/δi/\delta.

UDD multiplier for m-thly insurance approximations

Description

Under UDD, Ax(m)=(i/i(m))AxA_x^{(m)} = (i / i^{(m)}) A_x and similarly for term and deferred insurance.

Usage

udd_mthly_multiplier(i, m)

Arguments

i

Numeric vector of effective annual interest rates.
m

Positive integer payment frequency.

Value

Numeric vector equal to i/i(m)i / i^{(m)}.

Zeroized reserves for a discrete death-only contract

Description

Computes the zeroized reserve sequence by backward recursion, setting negative reserves equal to zero.

Usage

V_zeroized(qx, i, G, benefit, r = 0, e = 0, V_terminal = 0, floor_zero = TRUE)

Arguments

qx

Mortality vector.
i

Interest-rate vector.
G

Gross premium vector.
benefit

Death-benefit vector.
r

Percent-of-premium expense vector.
e

Fixed-expense vector.
V_terminal

Terminal reserve. Defaults to 0.
floor_zero

Logical; if TRUE, negative reserves are reset to 0.

Details

For a death-only contract with no settlement expense and no second decrement, the recursion sets

Prt+1=(tVZ+Gt+1(1−rt+1)−et+1)(1+it+1)−[Bqx+t+t+1VZpx+t]Pr_{t+1} = ({}_tV^Z + G_{t+1}(1-r_{t+1}) - e_{t+1})(1+i_{t+1}) - [Bq_{x+t} + {}_{t+1}V^Z p_{x+t}]

equal to zero, solving backward for tVZ{}_tV^Z.

Value

Numeric vector of zeroized reserves of length n+1n+1.

Examples

V_zeroized(
  qx = c(.015, .017, .019, .021, .024),
  i = 0.06,
  G = 19279,
  benefit = 1000000,
  e = 240
)

Variance of continuous whole life insurance PV

Description

Computes Var(Zˉx)=2Aˉx−Aˉx2\mathrm{Var}(\bar{Z}_x) = {}^{2}\bar{A}_x - \bar{A}_x^2.

Usage

var_Abarx(x, i, model, ...)

Arguments

x

Age.
i

Effective annual interest rate.
model

Parametric survival model name.
...

Additional model parameters passed to survival-model functions.

Value

Numeric vector of variances.

Variance of continuous endowment insurance PV

Description

Variance of continuous endowment insurance PV

Usage

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

Arguments

x

Age.
n

Term.
i

Effective annual interest rate.
model

Parametric survival model name.
...

Additional model parameters passed to survival-model functions.

Value

Numeric vector of variances.

Variance of continuous term insurance PV

Description

Computes Var(Zˉx:n‾∣1)=2Aˉx:n‾∣1−(Aˉx:n‾∣1)2\mathrm{Var}(\bar{Z}_{x:\overline{n}|}^{1}) = {}^{2}\bar{A}_{x:\overline{n}|}^{1} - (\bar{A}_{x:\overline{n}|}^{1})^2.

Usage

var_Abarxn1(x, n, i, model, ...)

Arguments

x

Age.
n

Term.
i

Effective annual interest rate.
model

Parametric survival model name.
...

Additional model parameters passed to survival-model functions.

Value

Numeric vector of variances.

Variance of whole life insurance PV

Description

Computes Var(Zx)=2Ax−Ax2\mathrm{Var}(Z_x) = {}^{2}A_x - A_x^2.

Usage

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

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.
tol

Numerical tolerance for truncating infinite sums.
k_max

Maximum number of terms in the sum.

Value

Numeric vector of variances.

Variance of m-thly whole life insurance PV

Description

Computes Var(Zx(m))=2Ax(m)−(Ax(m))2\mathrm{Var}(Z_x^{(m)}) = {}^{2}A_x^{(m)} - (A_x^{(m)})^2.

Usage

var_Ax_m(x, i, m, model, ..., tol = 1e-12, j_max = 100000L)

Arguments

x

Age.
i

Effective annual interest rate.
m

Positive integer payment frequency.
model

Parametric survival model name.
...

Additional model parameters passed to survival-model functions.
tol

Numerical tolerance for truncating the infinite sum.
j_max

Maximum number of m-thly intervals in the sum.

Value

Numeric vector of variances.

Variance of endowment insurance PV

Description

Variance of endowment insurance PV

Usage

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

Arguments

x

Age.
n

Term.
i

Effective annual interest rate.
tbl

Optional life table object.
model

Optional parametric survival model name.
...

Additional arguments passed to survival-model functions.

Value

Numeric vector of variances.

Variance of m-thly endowment insurance PV

Description

Variance of m-thly endowment insurance PV

Usage

var_Axn_m(x, n, i, m, model, ...)

Arguments

x

Age.
n

Term.
i

Effective annual interest rate.
m

Positive integer payment frequency.
model

Parametric survival model name.
...

Additional model parameters passed to survival-model functions.

Value

Numeric vector of variances.

Variance of term insurance PV

Description

Computes Var(Zx:n‾∣1)=2Ax:n‾∣1−(Ax:n‾∣1)2\mathrm{Var}(Z_{x:\overline{n}|}^{1}) = {}^{2}A_{x:\overline{n}|}^{1} - (A_{x:\overline{n}|}^{1})^2.

Usage

var_Axn1(x, n, i, tbl = NULL, model = NULL, ...)

Arguments

x

Age.
n

Term.
i

Effective annual interest rate.
tbl

Optional life table object.
model

Optional parametric survival model name.
...

Additional arguments passed to survival-model functions.

Value

Numeric vector of variances.

Variance of m-thly term insurance PV

Description

Variance of m-thly term insurance PV

Usage

var_Axn1_m(x, n, i, m, model, ...)

Arguments

x

Age.
n

Term.
i

Effective annual interest rate.
m

Positive integer payment frequency.
model

Parametric survival model name.
...

Additional model parameters passed to survival-model functions.

Value

Numeric vector of variances.

Variance of continuous deferred insurance PV

Description

Variance of continuous deferred insurance PV

Usage

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

Arguments

x

Age.
n

Deferral period.
i

Effective annual interest rate.
model

Parametric survival model name.
...

Additional model parameters passed to survival-model functions.

Value

Numeric vector of variances.

Variance of deferred insurance PV

Description

Variance of deferred insurance PV

Usage

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

Arguments

x

Age.
n

Deferral period.
i

Effective annual interest rate.
tbl

Optional life table object.
model

Optional parametric survival model name.
...

Additional arguments passed to survival-model functions.
tol

Numerical tolerance for truncating infinite sums.
k_max

Maximum number of terms in the sum.

Value

Numeric vector of variances.

Variance of m-thly deferred insurance PV

Description

Variance of m-thly deferred insurance PV

Usage

var_nAx_m(x, n, i, m, model, ..., tol = 1e-12, j_max = 100000L)

Arguments

x

Age.
n

Deferral period.
i

Effective annual interest rate.
m

Positive integer payment frequency.
model

Parametric survival model name.
...

Additional model parameters passed to survival-model functions.
tol

Numerical tolerance for truncating the infinite sum.
j_max

Maximum number of m-thly intervals in the sum.

Value

Numeric vector of variances.

Variance of pure endowment PV

Description

Variance of pure endowment PV

Usage

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

Arguments

x

Age.
n

Term.
i

Effective annual interest rate.
tbl

Optional life table object.
model

Optional parametric survival model name.
...

Additional arguments passed to survival-model functions.

Value

Numeric vector of variances.

Variance of present value of loss at duration t for whole life insurance

Description

Computes the Chapter 10 conditional variance Var(tLx∣Kx≥t)\mathrm{Var}({}_tL_x \mid K_x \ge t) for a fully discrete whole life insurance.

Usage

varLtx(x, t, i, P, model, ...)

Arguments

x

Issue age.
t

Duration.
i

Effective annual interest rate.
P

Annual premium.
model

Survival model.
...

Additional model parameters.

Value

Numeric vector.

Examples

prem <- Px(40, i = 0.05, model = "uniform", omega = 100)
varLtx(40, t = 10, i = 0.05, P = prem, model = "uniform", omega = 100)

Pre-floor CRVM reserve for universal life

Description

Computes the pre-floor CRVM reserve from Equation (16.17).

Usage

Vprefloor_crvm_ul(r, pvfb_minus_pvfp)

Arguments

r

Ratio rtr_t.
pvfb_minus_pvfp

Difference (PVFB)t−(PVFP)t(PVFB)_t - (PVFP)_t.

Value

Numeric scalar.

Examples

Vprefloor_crvm_ul(r = 0.33506, pvfb_minus_pvfp = 70)

Discount factors under a variable annual interest scenario

Description

Computes the sequence of discount factors

v1,  v2,  …,  vnv_1,\; v_2,\; \dots,\; v_n

where

vt=∏k=1t(1+ik)−1.v_t = \prod_{k=1}^{t}(1+i_k)^{-1}.

Usage

vt_var(i)

Arguments

i

Numeric vector of annual effective interest rates i1,i2,…,ini_1, i_2, \dots, i_n.

Details

This corresponds to the Chapter 15 notation jvt{}_j v^t for a fixed scenario jj.

Value

Numeric vector of discount factors of the same length as i.

Examples

vt_var(c(0.06, 0.07, 0.08))

Bootstrap annual spot rates from annual coupon-bond yields

Description

Bootstraps annual effective zero-coupon yields from par annual coupon-bearing bond yields of the same maturities.

Usage

z_from_coupon_annual(maturity, coupon_yield, par = 1000)

Arguments

maturity

Integer vector of maturities in years, in increasing order.
coupon_yield

Numeric vector of annual coupon yields.
par

Par value of each bond.

Value

Numeric vector of annual effective spot rates.

Examples

maturity <- 1:4
coupon_yield <- c(0.02, 0.04, 0.06, 0.08)
z_from_coupon_annual(maturity, coupon_yield)

Bootstrap semiannual nominal spot rates from coupon-bond yields

Description

Bootstraps the semiannual nominal annual zero-coupon yields from par coupon-bearing bond yields of the same maturities.

Usage

z_from_coupon_semi(maturity, coupon_yield, par = 1000)

Arguments

maturity

Numeric vector of maturities in years, typically 0.5, 1.0, 1.5, ..., in increasing order.
coupon_yield

Numeric vector of nominal annual coupon yields convertible semiannually.
par

Par value of each bond.

Details

Both coupon yields and spot yields are interpreted as nominal annual rates convertible semiannually.

Value

Numeric vector of semiannual nominal annual spot rates.

Examples

maturity <- c(0.5, 1.0, 1.5, 2.0)
coupon_yield <- c(0.0244, 0.0260, 0.0276, 0.0293)
z_from_coupon_semi(maturity, coupon_yield)

Spot rates from forward one-year rates

Description

Converts annual effective forward one-year rates f0,1,f1,1,…,fn−1,1f_{0,1}, f_{1,1}, \dots, f_{n-1,1} into annual effective spot rates z1,z2,…,znz_1, z_2, \dots, z_n.

Usage

z_from_fn1(fn1)

Arguments

fn1

Numeric vector of annual effective forward one-year rates.

Details

Since

(1+zn)n=∏j=0n−1(1+fj,1),(1+z_n)^n = \prod_{j=0}^{n-1}(1+f_{j,1}),

the spot rates are recovered directly.

Value

Numeric vector of annual effective spot rates.

Examples

z_from_fn1(c(0.04, 0.05, 0.06, 0.07, 0.08))