mqriskR package

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".

Author(s)

Maintainer: Nii Okine okinean@appstate.edu

Internal: choose a practical upper bound for integrals to infinity

Description

Strategy: find t such that S0(t) is very small, or fall back to a cap.

Usage

.find_upper_t(x, model, ..., eps = 1e-10, t_max = 500)

Internal: safe numerical integration helper

Description

Internal: safe numerical integration helper

Usage

.safe_integrate(f, lower, upper, ..., rel.tol = 1e-10)

Second moment of continuous whole life insurance PV

Description

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

Usage

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

Arguments

Value

Numeric vector of second moments.

Second moment of continuous endowment insurance PV

Description

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

Usage

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

Arguments

Value

Numeric vector of second moments.

Second moment of continuous term insurance PV

Description

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

Usage

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

Arguments

Value

Numeric vector of second moments.

Second moment of continuous deferred insurance PV

Description

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

Usage

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

Arguments

Value

Numeric vector of second moments.

Second moment of deferred insurance PV

Description

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

Usage

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

Arguments

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

Value

Numeric vector of second moments.

Second moment of pure endowment PV

Description

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

Usage

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

Arguments

Value

Numeric vector of second moments.

Second moment of whole life insurance PV

Description

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

Usage

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

Arguments

Value

Numeric vector of second moments.

Second moment of m-thly whole life insurance PV

Description

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

Usage

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

Arguments

Value

Numeric vector of second moments.

Second moment of endowment insurance PV

Description

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

Usage

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

Arguments

Value

Numeric vector of second moments.

Second moment of term insurance PV

Description

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

Usage

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

Arguments

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

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

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

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

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

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

Value

Accrued benefit.

Examples

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

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

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)

APV of gross premiums under a risk discount rate

Description

Computes the actuarial present value of gross premiums:

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

Value

Numeric scalar.

Examples

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

Projected asset share path {}_{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 k.

Usage

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

Arguments

Details

For policy year k,

[{}_{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

{}_{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

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 i^q = i^c.

Usage

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

Arguments

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

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 z for the defined contribution model in Equation (18.1).

Usage

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

Arguments

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)

Continuous whole life insurance APV

Description

Computes \bar{A}_x = \int_0^\infty v^t \, {}_t p_x \mu_{x+t}\,dt.

Usage

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

Arguments

Value

Numeric vector of APVs.

UDD approximation of continuous whole life insurance

Description

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

Usage

Abarx_udd(Ax, i)

Arguments

Value

Continuous whole life insurance APV under UDD.

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

Description

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

Usage

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

Arguments

Details

The integral is evaluated numerically by the trapezoidal rule:

\overline{A}_{x}^{(j)} = \int_0^T v^t {}_{t}p_{x}^{(\tau)} \mu_{x+t}^{(j)} dt

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 \bar{A}_{x:\overline{n}|} = \bar{A}_{x:\overline{n}|}^{1} + v^n\,{}_np_x.

Usage

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

Arguments

Value

Numeric vector of APVs.

Continuous term insurance APV

Description

Computes \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

Value

Numeric vector of APVs.

UDD approximation of continuous term insurance

Description

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

Usage

Abarxn1_udd(Axn1, i)

Arguments

Value

Continuous term insurance APV under UDD.

UDD approximation of continuous endowment insurance

Description

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

Usage

Abarxn_udd(Axn1, nEx, i)

Arguments

Value

Continuous endowment insurance APV under UDD.

Continuous joint-life whole life insurance

Description

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

Usage

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

Arguments

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 \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

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 \overline{A}_{xy}^{2} = \overline{A}_x - \overline{A}_{xy}^{1}.

Usage

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

Arguments

Value

Numeric vector.

Examples

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

Continuous last-survivor whole life insurance

Description

Computes \overline{A}_{\overline{xy}} = \overline{A}_x + \overline{A}_y - \overline{A}_{xy}.

Usage

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

Arguments

Value

Numeric vector.

Examples

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

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

Description

Computes \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

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 \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

Value

Numeric vector.

Examples

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

Whole life insurance APV

Description

Computes A_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

Value

Numeric vector of APVs.

m-thly whole life insurance APV

Description

Computes 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

Value

Numeric vector of APVs.

UDD approximation of m-thly whole life insurance

Description

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

Usage

Ax_m_udd(Ax, i, m)

Arguments

Value

m-thly whole life insurance APV under UDD.

Discrete multiple-decrement insurance APV 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 j, matching Equation (14.3b) in Chapter 14.

Usage

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

Arguments

Details

The function evaluates

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 A_{x:\overline{n}|} = A_{x:\overline{n}|}^{1} + {}_nE_x.

Usage

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

Arguments

Value

Numeric vector of APVs.

Term insurance APV

Description

Computes A_{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

Value

Numeric vector of APVs.

m-thly term insurance APV

Description

Computes 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

Value

Numeric vector of APVs.

UDD approximation of m-thly term insurance

Description

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

Usage

Axn1_m_udd(Axn1, i, m)

Arguments

Value

m-thly term insurance APV under UDD.

m-thly endowment insurance APV

Description

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

Usage

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

Arguments

Value

Numeric vector of APVs.

UDD approximation of m-thly endowment insurance

Description

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

Usage

Axn_m_udd(Axn1, nEx, i, m)

Arguments

Value

m-thly endowment insurance APV under UDD.

Joint-life whole life insurance

Description

Computes A_{xy}.

Usage

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

Arguments

Value

Numeric vector.

Examples

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

Last-survivor whole life insurance

Description

Computes A_{\overline{xy}} = A_x + A_y - A_{xy}.

Usage

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

Arguments

Value

Numeric vector.

Examples

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

Last-survivor endowment insurance

Description

Computes A_{\overline{xy}:\overline{n}|}.

Usage

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

Arguments

Value

Numeric vector.

Examples

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

Last-survivor term insurance

Description

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

Usage

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

Arguments

Value

Numeric vector.

Examples

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

Joint-life endowment insurance

Description

Computes A_{xy:\overline{n}|}.

Usage

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

Arguments

Value

Numeric vector.

Examples

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

Joint-life term insurance

Description

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

Usage

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

Arguments

Value

Numeric vector.

Examples

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

Piecewise-continuous decreasing n-year term insurance

Description

Computes

(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

Value

Numeric vector.

Decreasing n-year term insurance

Description

Computes

(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

Value

Numeric vector.

Fully continuous decreasing n-year term insurance

Description

Computes

(\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

Value

Numeric vector.

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

Description

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

Usage

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

Arguments

Details

Under the equivalence-principle premium, this equals the prospective reserve {}_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)

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

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

Value

Numeric vector.

Examples

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

Value

Numeric scalar.

Examples

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

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

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

Value

Numeric vector.

Examples

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

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

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

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

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
)

Piecewise-continuous increasing whole life insurance

Description

Computes

(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

Value

Numeric vector.

Increasing whole life insurance

Description

Computes

(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

Value

Numeric vector.

Increasing n-year term insurance

Description

Computes

(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

Value

Numeric vector.

Internal rate of return of a profit signature

Description

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

Usage

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

Arguments

Value

Numeric scalar.

Examples

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

Fully continuous increasing whole life insurance

Description

Computes

(\bar{I}\bar{A})_x = \int_0^\infty t\, v^t\, {}_tp_x\, \mu_{x+t}\, dt.

Usage

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

Arguments

Value

Numeric vector.

Fully continuous increasing n-year term insurance

Description

Computes

(\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

Value

Numeric vector.

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

Value

Annual retirement income.

Examples

Income_dc(AVz = 824211.35, adue_z = 12)

Entry Age Normal normal cost for a DB plan

Description

Computes Equation (18.13).

Usage

NC_EAN_db(APV_total, adue_active)

Arguments

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

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

Value

TUC normal cost.

Examples

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

Partial net present values

Description

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

Usage

NPV_partial(Pi, r)

Arguments

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

Usage

NPV_profit(Pi, r)

Arguments

Value

Numeric scalar.

Examples

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

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

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

Value

Projected annual benefit.

Examples

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

Continuous premium approximation \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

Details

The numerator is

\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

\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 \mathbf{Pr}=(Pr_0,\dots,Pr_n) into the corresponding profit signature \mathbf{\Pi}=(\Pi_0,\dots,\Pi_n) using Equation (17.3).

Usage

Pi_signature(Pr, p_tau)

Arguments

Value

Numeric vector of length n+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|}\ddot{a}_x) = {}_{n|}\ddot{a}_x / \ddot{a}_{x:\overline{n}|}.

Usage

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

Arguments

Value

Numeric vector.

Examples

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

Deferred annuity-immediate premium

Description

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

Usage

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

Arguments

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

\mathbf{Pr} = (Pr_0, Pr_1, \dots, Pr_n)

where Pr_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

Details

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

Value

Numeric vector of length n+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
)

Survival function for age-at-failure T0

Description

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

Usage

S0(t, model, ...)

Arguments

Details

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

Value

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

Convert survival probabilities to life-table values

Description

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

Usage

S0_to_lx(S0, radix = 1e+05)

Arguments

Value

Numeric vector of l_x values.

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

Details

For a death-only contract with no settlement expense and no second decrement, the recursion sets

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 {}_tV^Z.

Value

Numeric vector of zeroized reserves of length n+1.

Examples

V_zeroized(
  qx = c(.015, .017, .019, .021, .024),
  i = 0.06,
  G = 19279,
  benefit = 1000000,
  e = 240
)

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

Value

Numeric scalar.

Examples

Vprefloor_crvm_ul(r = 0.33506, pvfb_minus_pvfp = 70)

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

Description

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

Usage

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

Arguments

Value

Numeric vector.

Examples

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

Continuous joint-life whole life annuity

Description

Computes \overline{a}_{xy} = \int_0^\infty v^t\,{}_tp_{xy}\,dt.

Usage

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

Arguments

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 last-survivor whole life annuity

Description

Computes \overline{a}_{\overline{xy}} = \overline{a}_x + \overline{a}_y - \overline{a}_{xy}.

Usage

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

Arguments

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 \overline{a}_{y|x} = \overline{a}_x - \overline{a}_{xy}.

Usage

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

Arguments

Value

Numeric vector.

Examples

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

Joint-life whole life annuity-due

Description

Computes \ddot{a}_{xy}.

Usage

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

Arguments

Value

Numeric vector.

Examples

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

Last-survivor whole life annuity-due

Description

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

Usage

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

Arguments

Value

Numeric vector.

Examples

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

Last-survivor temporary annuity-due

Description

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

Usage

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

Arguments

Value

Numeric vector.

Examples

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

Joint-life temporary annuity-due

Description

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

Usage

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

Arguments

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

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

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 \alpha^F = v q_x = A_{x:\overline{1}|}^{1}.

Computes \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

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 a_x = \sum_{t=1}^{\infty} v^t \, {}_t p_x.

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

Computes the continuous whole life annuity \bar{a}_x = \int_0^{\infty} v^t \, {}_t p_x \, dt.

Computes the annual temporary annuity-immediate a_{x:\overline{n}|} = \sum_{t=1}^{n} v^t \, {}_t p_x.

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

Computes the continuous temporary annuity \bar{a}_{x:\overline{n}|} = \int_0^n v^t \, {}_t p_x \, dt.

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

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

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

Computes s_{x:\overline{n}|} = a_{x:\overline{n}|} / {}_nE_x.

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

Computes \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

Details

Naming convention follows Chapter 8 notation:

• ax() = a_x

• adotx() = \ddot{a}_x

• abarx() = \bar{a}_x

• axn() = a_{x:\overline{n}|}

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

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

• nax() = {}_{n|}a_x

• nadotx() = {}_{n|}\ddot{a}_x

• nabarx() = {}_{n|}\bar{a}_x

• sxn() = s_{x:\overline{n}|}

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

• sbarxn() = \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.

Annuity approximations (Chapter 8)

Description

Chapter 8 approximation formulas for m-thly and continuous life annuities.

Details

This file implements:

• UDD approximations for m-thly annuities,

• UDD approximations for continuous annuities,

• Woolhouse 2-term approximations,

• Woolhouse 3-term approximations.

UDD annuity approximations

Description

UDD-based approximations for Chapter 8 annuity functions.

Computes

\ddot{a}_x^{(m)} \approx \alpha(m)\ddot{a}_x - \beta(m).

Computes

\ddot{a}_{x:\overline{n}|}^{(m)} \approx \alpha(m)\ddot{a}_{x:\overline{n}|} - \beta(m)(1-{}_nE_x).

Computes

{}_{n\mid}\ddot{a}_x^{(m)} \approx \alpha(m)\,{}_{n\mid}\ddot{a}_x - \beta(m)\,{}_nE_x.

Computes

a_x^{(m)} \approx \alpha(m)a_x + \gamma(m).

Computes

a_{x:\overline{n}|}^{(m)} \approx \alpha(m)a_{x:\overline{n}|} + \gamma(m)(1-{}_nE_x).

Computes

{}_{n\mid}a_x^{(m)} \approx \alpha(m)\,{}_{n\mid}a_x + \gamma(m)\,{}_nE_x.

Computes

\ddot{s}_{x:\overline{n}|}^{(m)} \approx \alpha(m)\ddot{s}_{x:\overline{n}|} - \beta(m)\left(\frac{1}{{}_nE_x}-1\right).

Computes

s_{x:\overline{n}|}^{(m)} \approx \alpha(m)s_{x:\overline{n}|} + \gamma(m)\left(\frac{1}{{}_nE_x}-1\right).

Computes

\bar{a}_x \approx \frac{id}{\delta^2}\ddot{a}_x - \frac{i-\delta}{\delta^2}.

Uses the identity

\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 \bar{A}_{x:\overline{n}|}.

Computes

{}_{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

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

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

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

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)

m-thly contingent annuity functions (Chapter 8)

Description

Chapter 8 functions for life annuities payable m-thly.

Details

These functions implement the exact m-thly formulas from Section 8.5 using the survival model functions already available in the package.

In all cases, the annuity is one unit per year payable in m equal installments, so each payment is of size 1/m.

Temporary m-thly annuity-due actuarial accumulated value

Description

Computes

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

Usage

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

Arguments

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

s_{x:\overline{n}|}^{(m)} = a_{x:\overline{n}|}^{(m)} / {}_nE_x

Usage

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

Arguments

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

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

Usage

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

Arguments

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

{}_nE_x \, a_{x+n}^{(m)}

Usage

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

Arguments

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

Details

\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

Details

a_{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

Details

\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

Details

a_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 a_x = (v - A_x)/d.

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

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

Computes a_{x:\overline{n}|} = \ddot{a}_{x:\overline{n}|} - 1 + {}_nE_x together with \ddot{a}_{x:\overline{n}|} = (1 - A_{x:\overline{n}|})/d.

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

Computes \bar{a}_{x:\overline{n}|} = (1 - \bar{A}_{x:\overline{n}|})/\delta.

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

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

Computes {}_{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

Details

Included identities:

• whole life immediate: a_x = (v - A_x)/d

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

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

• temporary immediate: a_{x:\overline{n}|} = (1 - A_{x:\overline{n}|})/d - 1 + {}_nE_x

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

• temporary continuous: \bar{a}_{x:\overline{n}|} = (1 - \bar{A}_{x:\overline{n}|})/\delta

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

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

• deferred continuous: {}_{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 a_{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 = \sum_{t=1}^{\infty} t \, v^t \, {}_tp_x.

Computes

(Ia)_{x:\overline{n}|} = \sum_{t=1}^{n} t \, v^t \, {}_tp_x.

Computes

(Da)_{x:\overline{n}|} = \sum_{t=1}^{n} (n+1-t)\, v^t \, {}_tp_x.

Computes

(I\ddot{a})_x = \sum_{t=0}^{\infty} (t+1)\, v^t \, {}_tp_x.

Computes

(I\ddot{a})_{x:\overline{n}|} = \sum_{t=0}^{n-1} (t+1)\, v^t \, {}_tp_x.

Computes

(D\ddot{a})_{x:\overline{n}|} = \sum_{t=0}^{n-1} (n-t)\, v^t \, {}_tp_x.

Computes

(\bar{I}\bar{a})_x = \int_0^\infty t\,v^t\,{}_tp_x\,dt.

Computes

(\bar{I}\bar{a})_{x:\overline{n}|} = \int_0^n t\,v^t\,{}_tp_x\,dt.

Computes

(\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

Details

The functions implemented here match the notation in Section 8.6:

• Iax() = (Ia)_x

• Iaxn() = (Ia)_{x:\overline{n}|}

• Daxn() = (Da)_{x:\overline{n}|}

• Iadotx() = (I\ddot{a})_x

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

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

• Iabarx() = (\bar{I}\bar{a})_x

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

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

Value

Numeric vector.

Numeric vector.

Numeric vector.

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

Value

Numeric scalar.

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

Description

Computes a_{x|y} = a_y - a_{xy}.

Usage

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

Arguments

Value

Numeric vector.

Examples

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

Temporary annuity-immediate under mortality improvement

Description

Computes

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

Value

Numeric scalar.

Joint-life whole life annuity-immediate

Description

Computes a_{xy}.

Usage

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

Arguments

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 annuity-immediate

Description

Computes a_{\overline{xy}}.

Usage

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

Arguments

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 temporary annuity-immediate

Description

Computes a_{\overline{xy}:\overline{n}|}.

Usage

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

Arguments

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)

Joint-life temporary annuity-immediate

Description

Computes a_{xy:\overline{n}|}.

Usage

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

Arguments

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)

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

Description

Computes a_{y|x} = a_x - a_{xy}.

Usage

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

Arguments

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 \beta^F = P_{x+1} for whole life insurance.

Computes the FPT renewal premium \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

Value

Numeric vector.

Numeric vector.

Examples

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

Shared parameters for Chapter 12 continuous multi-life functions

Description

Shared parameters for Chapter 12 continuous multi-life functions

Arguments

Spot-rate actuarial present value functions

Description

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

Computes

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

Computes

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

Computes

A_{x:\overline{n}|} = A_{x:\overline{n}|}^1 + {}_nE_x

using spot-rate discount factors.

For an immediate annuity,

a_{x:\overline{n}|} = \sum_{t=1}^{n}(1+z_t)^{-t}\cdot {}_tp_x.

Usage

nEx_spot(qx, z, benefit = 1)

Axn1_spot(qx, z, benefit = 1)

Axn_spot(qx, z, benefit = 1)

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

Arguments

Details

For an annuity-due,

\ddot{a}_{x:\overline{n}|} = \sum_{t=0}^{n-1}(1+z_t)^{-t}\cdot {}_tp_x,

where the time-0 discount factor is 1.

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 i_1, i_2, \dots, i_n.

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

{}_nE_x = v_n \cdot {}_np_x.

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

A_{x:\overline{n}|}^1 = \sum_{t=1}^{n} v_t \cdot {}_{t-1}p_x \cdot q_{x+t-1}.

Computes the APV of an n-year endowment insurance under a variable annual interest scenario:

A_{x:\overline{n}|} = A_{x:\overline{n}|}^1 + {}_nE_x.

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

Usage

nEx_var(qx, i, benefit = 1)

Axn1_var(qx, i, benefit = 1)

Axn_var(qx, i, benefit = 1)

axn_var(qx, i, type = c("immediate", "due"), benefit = 1)

Arguments

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,

a_{x:\overline{n}|} = \sum_{t=1}^{n} v_t \cdot {}_tp_x.

For an annuity-due,

\ddot{a}_{x:\overline{n}|} = \sum_{t=0}^{n-1} v_t \cdot {}_tp_x,

with v_0 = 1.

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):

COI_t = \frac{B q_{x+t-1}}{1+i^q}.

Usage

coi_ul_typeB(B, qx, iq)

Arguments

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

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 \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

Value

Numeric vector of covariances.

Covariance of term insurance and pure endowment PVs

Description

Computes \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

Value

Numeric vector of covariances.

Cumulative hazard for age-at-failure T0

Description

Computes \Lambda_0(t)=\int_0^t \lambda_0(y)\,dy.

Usage

cumhaz0(t, model, ...)

Arguments

Value

Numeric vector of cumulative hazard values.

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

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

Value

Discount factor at time t.

Examples

discount(0.05, 0:5)

Discounted payback period

Description

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

Usage

discounted_payback_period(Pi, r)

Arguments

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 T_0.

Usage

F0(t, model, ...)

f0(t, model, ...)

Arguments

Details

• F0(t) computes F_0(t)=Pr(T_0 \le t)=1-S_0(t)

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

Value

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

Doubled force of interest

Description

Computes \delta' = 2\delta.

Usage

double_force_delta(delta)

Arguments

Value

Numeric vector of doubled forces of interest.

Effective annual interest at doubled force

Description

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

Usage

double_force_i(i)

Arguments

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

Value

Numeric vector of d_x values.

Cause-specific decrements d_x^{(j)}

Description

Cause-specific decrements d_x^{(j)}

Usage

dxj(lxtau, qxj)

Arguments

Value

Numeric vector.

Examples

dxj(1000, c(0.011, 0.100))

Total decrements d_x^{(\tau)}

Description

Total decrements d_x^{(\tau)}

Usage

dxtau(lxtau, qxj)

Arguments

Value

Numeric scalar.

Examples

dxtau(1000, c(0.011, 0.100))

Complete expectation of life at age x

Description

Computes \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

Value

Numeric vector of complete expectations.

Complete expectation of life from a life table

Description

Computes \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

Value

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

Curtate expectation of life at age x

Description

Computes e_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

Value

Numeric vector of curtate expectations.

Curtate expectation of life from a life table

Description

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

Usage

ex_curtate_tab(tbl, x)

Arguments

Value

Numeric vector of curtate expectations e_x.

Temporary complete expectation of life from a life table

Description

Computes \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

Value

Numeric vector of temporary complete expectations.

Temporary curtate expectation of life from a life table

Description

Computes e_{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

Value

Numeric vector of temporary curtate expectations.

Forward rate f_{n,k} from spot rates

Description

Computes the n-year forward k-year annual effective rate implied by annual effective spot rates:

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

Usage

fnk_from_z(z, n, k)

Arguments

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 f_{n,k} implied by annual effective spot rates z_1,\dots,z_m.

Usage

forward_matrix_from_z(z)

Arguments

Details

Rows correspond to n = 1,\dots,m-1 and columns correspond to k = 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 f_x(t)=f_0(x+t)/S_0(x).

Usage

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

Arguments

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 (\ge 0).

Fractional conditional density from a life table

Description

Computes the conditional density f_x(t \mid T_0 > x) = {}_t p_x \mu_{x+t} for 0 < t < 1.

Usage

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

Arguments

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

Details

With within-year decrement probabilities, the function evaluates

[{}_{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
)

Hazard / force for age-at-failure T0

Description

Computes \lambda_0(t)=f_0(t)/S_0(t).

Usage

hazard0(t, model, ...)

Arguments

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

Value

Numeric vector.

Examples

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

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

Value

Numeric scalar.

Examples

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

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

Value

Numeric vector of growth rates.

Examples

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

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

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)

Continuous insurance models (Chapter 7)

Description

Continuous contingent payment / insurance functions matching Chapter 7 notation.

Details

These functions handle:

• continuous whole life insurance: \bar{A}_x,

• continuous term insurance: \bar{A}_{x:\overline{n}|}^{1},

• continuous deferred insurance: {}_{n\mid}\bar{A}_x,

• continuous endowment insurance: \bar{A}_{x:\overline{n}|},

• second moments and variances.

These functions are evaluated from the parametric survival-model framework developed earlier in the package.

Discrete insurance models (Chapter 7)

Description

Discrete contingent payment / insurance functions matching Chapter 7 notation.

Details

These functions handle:

• whole life insurance: A_x,

• term insurance: A_{x:\overline{n}|}^{1},

• deferred insurance: {}_{n\mid}A_x,

• pure endowment: {}_nE_x,

• endowment insurance: A_{x:\overline{n}|},

• second moments and variances.

The functions may be evaluated from either:

• a life table object via tbl = ..., or

• a parametric survival model via model = ... and additional parameters.

m-thly insurance models (Chapter 7)

Description

Exact m-thly contingent payment / insurance functions matching Chapter 7 notation.

Details

These functions handle:

• m-thly whole life insurance: A_x^{(m)},

• m-thly term insurance: A_{x:\overline{n}|}^{1(m)},

• m-thly deferred insurance: {}_{n\mid}A_x^{(m)},

• m-thly endowment insurance: A_{x:\overline{n}|}^{(m)},

• second moments and variances.

These functions are evaluated exactly from the parametric survival-model framework. For UDD approximations from annual life tables, use the corresponding *_udd helpers in insurance_utils.R.

Insurance utilities (Chapter 7)

Description

Helper functions for Chapter 7 insurance models.

Details

These utilities focus on:

• doubled-force calculations for second moments,

• UDD approximation multipliers,

• UDD approximations for continuous and m-thly insurance functions.

General interest conversion functions such as interest_convert() and discount() are defined elsewhere in the package and are reused here.

Insurance models with varying benefits (Chapter 7)

Description

Functions for discrete and continuous contingent payment models with varying benefits, matching Chapter 7 notation.

Details

Included functions:

• (IA)_x

• (IA)_{x:\overline{n}|}^{1}

• (DA)_{x:\overline{n}|}^{1}

• (\bar{I}\bar{A})_x

• (I\bar{A})_x

• (\bar{I}\bar{A})_{x:\overline{n}|}^{1}

• (\bar{D}\bar{A})_{x:\overline{n}|}^{1}

• (D\bar{A})_{x:\overline{n}|}^{1}

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

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))

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

Value

A data.frame with class "life_table".

Extract life-table survivor values