# Actuarial Risk Modeling and Life Contingencies

## Documentation for package ‘mqriskR’ version 0.1.0

- DESCRIPTION file.
- User guides, package vignettes and other documentation.

## Help Pages

A B C D E F G H I L M N P Q R S T U V Z

## -- A --

A2barx | Second moment of continuous whole life insurance PV
A2barxn | Second moment of continuous endowment insurance PV
A2barxn1 | Second moment of continuous term insurance PV
A2nAbarx | Second moment of continuous deferred insurance PV
A2nAx | Second moment of deferred insurance PV
A2nAx_m | Second moment of m-thly deferred insurance PV
A2nEx | Second moment of pure endowment PV
A2x | Second moment of whole life insurance PV
A2xn | Second moment of endowment insurance PV
A2xn1 | Second moment of term insurance PV
A2xn1_m | Second moment of m-thly term insurance PV
A2xn_m | Second moment of m-thly endowment insurance PV
A2x_m | Second moment of m-thly whole life insurance PV
AAL_PUC_db | Projected Unit Credit accrued liability for a DB plan
AAL_TUC_db | Traditional Unit Credit accrued liability for a DB plan
Abarx | Continuous whole life insurance APV
abarx | Annual annuity functions (Chapter 8)
Abarxj_md | Continuous multiple-decrement insurance APV \overline{A}_{x}^{(j)}
Abarxn | Continuous endowment insurance APV
abarxn | Annual annuity functions (Chapter 8)
Abarxn1 | Continuous term insurance APV
Abarxn1_udd | UDD approximation of continuous term insurance
Abarxn_udd | UDD approximation of continuous endowment insurance
abarxn_udd | UDD annuity approximations
Abarxy | Continuous joint-life whole life insurance
abarxy | Continuous joint-life whole life annuity
Abarxy1 | Continuous contingent insurance: benefit on death of (x) if before (y)
Abarxy2 | Continuous contingent insurance: benefit on death of (x) if after (y)
Abarxybar | Continuous last-survivor whole life insurance
abarxybar | Continuous last-survivor whole life annuity
abarxybar_ch12 | Continuous last-survivor whole life annuity
abarxy_ch12 | Continuous joint-life whole life annuity
Abarx_udd | UDD approximation of continuous whole life insurance
abarx_udd | UDD annuity approximations
abarx_woolhouse2 | Woolhouse 2-term annuity approximations
abarx_woolhouse3 | Woolhouse 3-term annuity approximations
abarx_y | Continuous reversionary annuity to (y) after death of (x)
Abaryx1 | Continuous contingent insurance: benefit on death of (y) if before (x)
Abaryx2 | Continuous contingent insurance: benefit on death of (y) if after (x)
abary_x | Continuous reversionary annuity to (x) after death of (y)
AB_cae | Accrued benefit for a career average earnings plan
AB_fas | Accrued benefit for a final average salary plan
adotx | Annual annuity functions (Chapter 8)
adotxn | Annual annuity functions (Chapter 8)
adotxn_m | Temporary m-thly annuity-due
adotxn_m_udd | UDD annuity approximations
adotxn_m_woolhouse2 | Woolhouse 2-term annuity approximations
adotxn_m_woolhouse3 | Woolhouse 3-term annuity approximations
adotxy | Joint-life whole life annuity-due
adotxybar | Last-survivor whole life annuity-due
adotxybarn | Last-survivor temporary annuity-due
adotxyn | Joint-life temporary annuity-due
adotx_m | Whole life m-thly annuity-due
adotx_m_udd | UDD annuity approximations
adotx_m_woolhouse2 | Woolhouse 2-term annuity approximations
adotx_m_woolhouse3 | Woolhouse 3-term annuity approximations
ag38_prefunding_ratio | AG 38 prefunding ratio
ag38_reserve_ul | AG 38 reserve calculation
alphaF | Full preliminary term first-year modified premium
annuity_annual | Annual annuity functions (Chapter 8)
annuity_approximations | Annuity approximations (Chapter 8)
annuity_approximations_udd | UDD annuity approximations
annuity_approximations_woolhouse2 | Woolhouse 2-term annuity approximations
annuity_approximations_woolhouse3 | Woolhouse 3-term annuity approximations
annuity_certain | Present value of a level annuity-certain
annuity_identity_abarx | Annuity-insurance relationships (Chapter 8)
annuity_identity_abarxn | Annuity-insurance relationships (Chapter 8)
annuity_identity_adotx | Annuity-insurance relationships (Chapter 8)
annuity_identity_adotxn | Annuity-insurance relationships (Chapter 8)
annuity_identity_ax | Annuity-insurance relationships (Chapter 8)
annuity_identity_axn | Annuity-insurance relationships (Chapter 8)
annuity_identity_nabarx | Annuity-insurance relationships (Chapter 8)
annuity_identity_nadotx | Annuity-insurance relationships (Chapter 8)
annuity_identity_nax | Annuity-insurance relationships (Chapter 8)
annuity_mthly | m-thly contingent annuity functions (Chapter 8)
annuity_mthly_accum_due | Temporary m-thly annuity-due actuarial accumulated value
annuity_mthly_accum_immediate | Temporary m-thly annuity-immediate actuarial accumulated value
annuity_mthly_deferred_due | Deferred whole life m-thly annuity-due
annuity_mthly_deferred_immediate | Deferred whole life m-thly annuity-immediate
annuity_mthly_temp_due | Temporary m-thly annuity-due
annuity_mthly_temp_immediate | Temporary m-thly annuity-immediate
annuity_mthly_whole_due | Whole life m-thly annuity-due
annuity_mthly_whole_immediate | Whole life m-thly annuity-immediate
annuity_relationships | Annuity-insurance relationships (Chapter 8)
annuity_varying_payments | Varying-payment annuity functions (Chapter 8)
APV_gross_premiums | APV of gross premiums under a risk discount rate
APV_NR_db | APV of normal retirement benefit for a DB plan
AS_path | Projected asset share path {}_{k}AS
AS_path_md | General projected asset share path (multiple decrements)
AVz_dc | Accumulated value of defined contribution plan contributions
AV_path_ul_typeA | Account-value path for Type A universal life
AV_path_ul_typeB | Account-value path for Type B universal life
Ax | Whole life insurance APV
ax | Annual annuity functions (Chapter 8)
Axj_md | Discrete multiple-decrement insurance APV A_{x}^{(j)}
Axn | Endowment insurance APV
axn | Annual annuity functions (Chapter 8)
Axn1 | Term insurance APV
Axn1_m | m-thly term insurance APV
Axn1_m_udd | UDD approximation of m-thly term insurance
Axn1_spot | Spot-rate actuarial present value functions
Axn1_var | Variable-interest actuarial present value functions
axn_improved | Temporary annuity-immediate under mortality improvement
Axn_m | m-thly endowment insurance APV
axn_m | Temporary m-thly annuity-immediate
Axn_m_udd | UDD approximation of m-thly endowment insurance
axn_m_udd | UDD annuity approximations
axn_m_woolhouse2 | Woolhouse 2-term annuity approximations
axn_m_woolhouse3 | Woolhouse 3-term annuity approximations
Axn_spot | Spot-rate actuarial present value functions
axn_spot | Spot-rate actuarial present value functions
Axn_var | Variable-interest actuarial present value functions
axn_var | Variable-interest actuarial present value functions
Axy | Joint-life whole life insurance
axy | Joint-life whole life annuity-immediate
Axybar | Last-survivor whole life insurance
axybar | Last-survivor whole life annuity-immediate
Axybarn | Last-survivor endowment insurance
axybarn | Last-survivor temporary annuity-immediate
Axybarn1 | Last-survivor term insurance
axybarn_ch12 | Last-survivor temporary annuity-immediate
axybar_ch12 | Last-survivor whole life annuity-immediate
Axyn | Joint-life endowment insurance
axyn | Joint-life temporary annuity-immediate
Axyn1 | Joint-life term insurance
axyn_ch12 | Joint-life temporary annuity-immediate
axy_ch12 | Joint-life whole life annuity-immediate
ax_improved | Whole life annuity-immediate under mortality improvement
Ax_m | m-thly whole life insurance APV
ax_m | Whole life m-thly annuity-immediate
Ax_m_udd | UDD approximation of m-thly whole life insurance
ax_m_udd | UDD annuity approximations
ax_m_woolhouse2 | Woolhouse 2-term annuity approximations
ax_m_woolhouse3 | Woolhouse 3-term annuity approximations
ax_y | Reversionary annuity to (y) after death of (x)
ay_x | Reversionary annuity to (x) after death of (y)

## -- B --

betaF | Full preliminary term renewal modified premium

## -- C --

ch12_multilife_cont_params | Shared parameters for Chapter 12 continuous multi-life functions
chapter15_spot_interest_apv | Spot-rate actuarial present value functions
chapter15_variable_interest_apv | Variable-interest actuarial present value functions
coi_ul_typeB | Cost of insurance for Type B universal life
contribution_rate_target | Target contribution rate for a defined contribution plan
cov_term_deferred | Covariance of term and deferred insurance PVs
cov_term_endow | Covariance of term insurance and pure endowment PVs
cumhaz0 | Cumulative hazard for age-at-failure T0

## -- D --

Dabarxn | Varying-payment annuity functions (Chapter 8)
DAbarxn1 | Piecewise-continuous decreasing n-year term insurance
Dadotxn | Varying-payment annuity functions (Chapter 8)
Daxn | Varying-payment annuity functions (Chapter 8)
DAxn1 | Decreasing n-year term insurance
DbarAbarxn1 | Fully continuous decreasing n-year term insurance
decompGg_disc | Ordered gross gain decomposition
discount | Discount factor for compound interest
discounted_payback_period | Discounted payback period
dist0 | Distribution functions for age-at-failure T0
double_force_delta | Doubled force of interest
double_force_i | Effective annual interest at doubled force
dx | Compute deaths between ages x and x+1
dxj | Cause-specific decrements d_x^{(j)}
dxtau | Total decrements d_x^{(tau)}

## -- E --

EL0barAbarx | Chapter 9 premium, loss, and expense functions
EL0x | Chapter 9 premium, loss, and expense functions
EL0xn | Chapter 9 premium, loss, and expense functions
EL0xn1 | Chapter 9 premium, loss, and expense functions
ELtx | Mean present value of loss at duration t for whole life insurance
ex_complete | Complete expectation of life at age x
ex_complete_tab | Complete expectation of life from a life table
ex_curtate | Curtate expectation of life at age x
ex_curtate_tab | Curtate expectation of life from a life table
ex_temp_complete_tab | Temporary complete expectation of life from a life table
ex_temp_curtate_tab | Temporary curtate expectation of life from a life table

## -- F --

F0 | Distribution functions for age-at-failure T0
f0 | Distribution functions for age-at-failure T0
fnk_from_z | Forward rate f_{n,k} from spot rates
forward_matrix_from_z | Matrix of all determinable forward rates from spot rates
fx | Conditional density for Tx
fx_tab | Fractional conditional density from a life table

## -- G --

gain_loss_md | Gain or loss in a multiple-decrement model
GI_cont | Interest gain helper for continuous-style recursion
GI_disc | Interest gain for a discrete insurance contract
GMF_rollforward_ul | Guaranteed maturity fund roll-forward
GM_cont | Mortality gain helper for continuous-style recursion
GM_disc | Mortality gain for a discrete insurance contract
GTg_disc | Total gross gain for a discrete insurance contract
GT_cont | Total gain for a continuous-style one-step recursion
GT_disc | Total gain for a discrete insurance contract
Gx | Chapter 9 premium, loss, and expense functions

## -- H --

hazard0 | Hazard / force for age-at-failure T0
htVnAx | Deferred insurance reserve functions
htVx | h-pay whole life net level premium reserve

## -- I --

IAbarx | Piecewise-continuous increasing whole life insurance
Iabarx | Varying-payment annuity functions (Chapter 8)
Iabarxn | Varying-payment annuity functions (Chapter 8)
Iadotx | Varying-payment annuity functions (Chapter 8)
Iadotxn | Varying-payment annuity functions (Chapter 8)
IAx | Increasing whole life insurance
Iax | Varying-payment annuity functions (Chapter 8)
Iaxn | Varying-payment annuity functions (Chapter 8)
IAxn1 | Increasing n-year term insurance
IbarAbarx | Fully continuous increasing whole life insurance
IbarAbarxn1 | Fully continuous increasing n-year term insurance
iMA_eiul | Monthly-average index growth rate
Income_dc | Retirement income from a defined contribution accumulation
interest_convert | Convert between compound-interest quantities
iP_eiul | Point-to-point index growth rates
IRR_profit | Internal rate of return of a profit signature
i_credit_eiul | Credited rates from raw index growth rates

## -- L --

life_table | Construct a life table
lx | Extract life-table survivor values
lx_select | Extract select-table survivor value
lx_to_S0 | Convert life-table values to survival probabilities

## -- M --

markov_nstep_prob | n-step transition probability for a discrete-time Markov chain
md_table | Build a multiple-decrement table
meanVx | Mean reserve for whole life insurance
mortality_improvement | Mortality improvement projection functions (Chapter 8)
multilife_ch12 | Multi-life functions for Chapter 12
multilife_ch12_ext | Extended multi-life functions for Chapter 12
mux_tab | Fractional force of mortality from a life table

## -- N --

nAbarx | Continuous deferred insurance APV
nabarx | Annual annuity functions (Chapter 8)
nAbarx_udd | UDD approximation of continuous deferred insurance
nabarx_udd | UDD annuity approximations
nadotx | Annual annuity functions (Chapter 8)
nadotx_m | Deferred whole life m-thly annuity-due
nadotx_m_udd | UDD annuity approximations
nadotx_m_woolhouse2 | Woolhouse 2-term annuity approximations
nadotx_m_woolhouse3 | Woolhouse 3-term annuity approximations
nAx | Deferred insurance APV
nax | Annual annuity functions (Chapter 8)
naxn_improved | Deferred temporary annuity-immediate under mortality improvement
nAx_m | m-thly deferred insurance APV
nax_m | Deferred whole life m-thly annuity-immediate
nAx_m_udd | UDD approximation of m-thly deferred insurance
nax_m_udd | UDD annuity approximations
nax_m_woolhouse2 | Woolhouse 2-term annuity approximations
nax_m_woolhouse3 | Woolhouse 3-term annuity approximations
NC_EAN_db | Entry Age Normal normal cost for a DB plan
NC_PUC_db | Projected Unit Credit normal cost for a DB plan
NC_TUC_db | Traditional Unit Credit normal cost for a DB plan
ndx | Compute deaths over an n-year interval from a life table
nEx | Pure endowment APV
nExy | Joint-life pure endowment
nExybar | Last-survivor pure endowment
nEx_spot | Spot-rate actuarial present value functions
nEx_var | Variable-interest actuarial present value functions
nkqx | Curtate death probability from a life table
nmxq | Deferred death probability from a life table
nmxq_select | Deferred select-life death probability
NPV_partial | Partial net present values
NPV_profit | Net present value of a profit signature
npx | Compute n-year survival probability from a life table
npxtau_md | {}_n p_x^{(tau)} from a multiple-decrement table
npx_select | Select-life survival probability
nqx | Compute n-year death probability from a life table
nqxj_md | {}_n q_x^{(j)} from a multiple-decrement table
nqxtau_md | {}_n q_x^{(tau)} from a multiple-decrement table
nqx_select | Select-life death probability

## -- P --

PAB_cae | Projected annual benefit under a career average earnings DB plan
PAB_fas | Projected annual benefit under a final average salary DB plan
PbarAbarx | Chapter 9 premium, loss, and expense functions
PbarAbarxn | Chapter 9 premium, loss, and expense functions
PbarAbarxn1 | Chapter 9 premium, loss, and expense functions
Pbarx | Chapter 9 premium, loss, and expense functions
Pbarxn | Chapter 9 premium, loss, and expense functions
Pbarxn1 | Chapter 9 premium, loss, and expense functions
Pbar_trapz_ms | Continuous premium approximation \overline{P} by trapezoidal rule
Pi_signature | Profit signature from a profit vector
PnAdotx | Deferred annuity-due premium
PnAx | Chapter 9 premium, loss, and expense functions
Pnax | Deferred annuity-immediate premium
PnAx_m | Chapter 9 premium, loss, and expense functions
PnEx | Chapter 9 premium, loss, and expense functions
premium_ch9 | Chapter 9 premium, loss, and expense functions
profit_margin | Profit margin
Pr_vector_disc | Profit vector for a discrete profit-analysis model
pv_cashflows | Present value of cash flows at time 0
pv_spot_cashflows | Present value of cash flows using spot rates
Px | Chapter 9 premium, loss, and expense functions
Pxn | Chapter 9 premium, loss, and expense functions
Pxn1 | Chapter 9 premium, loss, and expense functions
Pxn1_m | Chapter 9 premium, loss, and expense functions
Pxn_m | Chapter 9 premium, loss, and expense functions
pxtau | Survival probability p_x^{(tau)}
pxtau_ul | One-year persistency rates for universal life
Px_m | Chapter 9 premium, loss, and expense functions
px_proj | Project one-year survival probability under mortality improvement
px_to_lx | Construct life-table values from p_x values

## -- Q --

qxprime_mudd | Independent probabilities q_x^{\prime(j)} from dependent probabilities q_x^{(j)} under MUDD
qxprime_sudd | Independent probabilities q_x^{\prime(j)} from dependent probabilities q_x^{(j)} under SUDD
qxtau | Total probability of decrement q_x^{(tau)}
qx_dep_cf | Dependent probabilities q_x^{(j)} from independent probabilities q_x^{\prime(j)} under constant force
qx_dep_sudd | Dependent probabilities q_x^{(j)} from independent probabilities q_x^{\prime(j)} under SUDD
qx_proj | Project one-year death probability under mortality improvement
qx_tab | Compute one-year death probability from a life table
qx_to_lx | Construct life-table values from q_x values

## -- R --

replacement_ratio_db | Replacement ratio for a defined benefit plan
replacement_ratio_dc | Replacement ratio for a defined contribution plan
reserve_ch10 | Reserve functions for Chapter 10
reserve_ch10_ext | Extended reserve functions for Chapter 10
reserve_ch11 | Reserve functions for Chapter 11
reserve_ch11_ext | Extended reserve functions for Chapter 11
reserve_deferred_insurance_ch10 | Deferred insurance reserve functions
rt_ul | Ratio r_t = AV_t / GMF_t capped at 1

## -- S --

S0 | Survival function for age-at-failure T0
S0_to_lx | Convert survival probabilities to life-table values
salary_scale | Salary scale under constant annual growth
sbarxn | Annual annuity functions (Chapter 8)
sdotxn | Annual annuity functions (Chapter 8)
sdotxn_m | Temporary m-thly annuity-due actuarial accumulated value
sdotxn_m_udd | UDD annuity approximations
sdotxn_m_woolhouse2 | Woolhouse 2-term annuity approximations
select_life_table | Construct a select life table
solve_yield | Solve the yield rate by the equation of value
sxn | Annual annuity functions (Chapter 8)
sxn_m | Temporary m-thly annuity-immediate actuarial accumulated value
sxn_m_udd | UDD annuity approximations
sxn_m_woolhouse2 | Woolhouse 2-term annuity approximations

## -- T --

thiele_backward_path | Backward Euler reserve path from maturity
thiele_backward_step | One backward Euler-style Thiele step
thiele_dVdt | Reserve derivative from Thiele's equation
thiele_dVdt_01 | Reserve derivatives for the disability model with recovery
thiele_path_01 | Backward reserve path for the disability model with recovery
tp00_tp01_euler | Euler approximation for {}_{t}p_{x}^{00} and {}_{t}p_{x}^{01}
tPnAx | Chapter 9 premium, loss, and expense functions
tPnEx | Chapter 9 premium, loss, and expense functions
tPx | Chapter 9 premium, loss, and expense functions
tpx | Conditional survival probability for Tx
tPxn | Chapter 9 premium, loss, and expense functions
tPxn1 | Chapter 9 premium, loss, and expense functions
tpxprimej_cf | Single-decrement survival probability {}_t p_x^{\prime(j)} under constant force
tpxtau_ul | Cumulative persistency to the end of each policy year
tpxy | Joint-life survival probability
tpxybar | Last-survivor survival probability
tpx_improved | Multi-year survival probability under mortality improvement
tpx_tab | Fractional survival probability from a life table
tpx_tau_cf | Total survival probability {}_t p_x^{(tau)} under constant forces
tqx | Conditional failure probability for Tx
tqxj_cf | Cause-specific probability {}_t q_x^{(j)} under constant forces
tqxprimej_cf | Single-decrement failure probability {}_t q_x^{\prime(j)} under constant force
tqxprime_mudd | Independent probabilities {}_t q_x^{\prime(j)} under MUDD
tqxy | Joint-life failure probability
tqxy1 | Probability that (x) fails before (y) within n years
tqxy2 | Probability that (x) fails after (y) within n years
tqxybar | Last-survivor failure probability
tqx_tab | Fractional failure probability from a life table
tqyx1 | Probability that (y) fails before (x) within n years
tqyx2 | Probability that (y) fails after (x) within n years
tsVx | Fractional-duration whole life reserve
tsVxn | Fractional-duration endowment reserve
tsVxn1 | Fractional-duration term reserve
tVbarAbarx | Fully continuous whole life reserve
tVbarx | Whole life reserve with continuous premiums
tVEx | Whole life expense reserve
tVFx | Full preliminary term reserve for whole life insurance
tVGx | Whole life gross premium reserve
tVnAdotx | Deferred annuity-due reserve
tVnAx | Deferred insurance reserve functions
tVnax | Deferred annuity-immediate reserve
tVnEx | Pure endowment net level premium reserve
tVx | Whole life net level premium reserve
tVxn | Endowment insurance net level premium reserve
tVxn1 | Term insurance net level premium reserve
tVxn1_ret | Term insurance reserve by retrospective method
tVxn_ret | Endowment insurance reserve by retrospective method
tVx_m | Whole life reserve with m-thly premiums
tVx_ret | Whole life net level premium reserve by retrospective method

## -- U --

udd_continuous_multiplier | UDD multiplier for continuous insurance approximations
udd_mthly_multiplier | UDD multiplier for m-thly insurance approximations

## -- V --

varL0barAbarx | Chapter 9 premium, loss, and expense functions
varL0x | Chapter 9 premium, loss, and expense functions
varL0xn | Chapter 9 premium, loss, and expense functions
varL0xn1 | Chapter 9 premium, loss, and expense functions
varLtx | Variance of present value of loss at duration t for whole life insurance
var_Abarx | Variance of continuous whole life insurance PV
var_Abarxn | Variance of continuous endowment insurance PV
var_Abarxn1 | Variance of continuous term insurance PV
var_Ax | Variance of whole life insurance PV
var_Axn | Variance of endowment insurance PV
var_Axn1 | Variance of term insurance PV
var_Axn1_m | Variance of m-thly term insurance PV
var_Axn_m | Variance of m-thly endowment insurance PV
var_Ax_m | Variance of m-thly whole life insurance PV
var_nAbarx | Variance of continuous deferred insurance PV
var_nAx | Variance of deferred insurance PV
var_nAx_m | Variance of m-thly deferred insurance PV
var_nEx | Variance of pure endowment PV
Vprefloor_crvm_ul | Pre-floor CRVM reserve for universal life
vt_var | Discount factors under a variable annual interest scenario
V_zeroized | Zeroized reserves for a discrete death-only contract

## -- Z --

z_from_coupon_annual | Bootstrap annual spot rates from annual coupon-bond yields
z_from_coupon_semi | Bootstrap semiannual nominal spot rates from coupon-bond yields
z_from_fn1 | Spot rates from forward one-year rates