Package {lifecontingencies}

Type:	Package
Title:	Financial and Actuarial Mathematics for Life Contingencies
Version:	1.6.3
Maintainer:	Giorgio Alfredo Spedicato <spedicato_giorgio@yahoo.it>
Description:	Classes and methods that allow the user to manage life table, actuarial tables (also multiple decrements tables). Moreover, functions to easily perform demographic, financial and actuarial mathematics on life contingencies insurances calculations are contained therein. See Spedicato (2013) <doi:10.18637/jss.v055.i10>.
Depends:	R (≥ 4.4.0), methods
Imports:	ggplot2, parallel, utils, Rcpp (≥ 1.0.0), stats
Suggests:	demography, forecast, markovchain, testthat, knitr, formatR, StMoMo, survival, rmarkdown, quarto
License:	MIT + file LICENSE
Encoding:	UTF-8
LazyLoad:	yes
LazyData:	true
BugReports:	https://github.com/spedygiorgio/lifecontingencies/issues
BuildVignettes:	yes
VignetteBuilder:	utils, knitr, quarto
URL:	https://github.com/spedygiorgio/lifecontingencies
LinkingTo:	Rcpp
Config/roxygen2/version:	8.1.0
NeedsCompilation:	yes
Packaged:	2026-10-10 05:59:41 UTC; Utente
Author:	Giorgio Alfredo Spedicato [aut, cre], Christophe Dutang [ctb], Reinhold Kainhofer [ctb], Kevin J Owens [ctb], Ernesto Schirmacher [ctb], Gian Paolo Clemente [ctb], Ivan Williams [ctb]
Repository:	CRAN
Date/Publication:	2026-10-10 06:50:10 UTC

Package to perform actuarial mathematics on life contingencies and classical financial mathematics calculations.

Description

The lifecontingencies package performs standard financial, demographic and actuarial mathematics calculation. The main purpose of the package is to provide a comprehensive set of tools to perform risk assessment of life contingent insurances.

Details

Some functions have been powered by Rcpp code.

Warning

This package and functions herein are provided as is, without any guarantee regarding the accuracy of calculations. The author disclaims any liability arising by any losses due to direct or indirect use of this package.

Note

Work in progress.

Author(s)

Giorgio Alfredo Spedicato with contributions from Reinhold Kainhofer and Kevin J. Owens Maintainer: <spedicato_giorgio@yahoo.it>

References

The lifecontingencies Package: Performing Financial and Actuarial Mathematics Calculations in R, Giorgio Alfredo Spedicato, Journal of Statistical Software, 2013,55 , 10, 1-36

See Also

accumulatedValue, annuity

Examples

##financial mathematics example

#calculates monthly installment of a loan of 100,000, 
#interest rate 0.05

i=0.05
monthlyInt=(1+i)^(1/12)-1
Capital=100000
#Montly installment

R=1/12*Capital/annuity(i=i, n=10,k=12, type = "immediate")
R
balance=numeric(10*12+1)
capitals=numeric(10*12+1)
interests=numeric(10*12+1)
balance[1]=Capital
interests[1]=0
capitals[1]=0

for(i in (2:121))	{
			balance[i]=balance[i-1]*(1+monthlyInt)-R
			interests[i]=balance[i-1]*monthlyInt
			capitals[i]=R-interests[i]
			}
loanSummary=data.frame(rate=c(0, rep(R,10*12)), 
	balance, interests, capitals)

head(loanSummary)

tail(loanSummary)

##actuarial mathematics example

#APV of an annuity

		data(soaLt)
		soa08Act=with(soaLt, new("actuarialtable",interest=0.06,
		x=x,lx=Ix,name="SOA2008"))
		#evaluate and life-long annuity for an aged 65
		axn(soa08Act, x=65) 

Multiple decrement insurances and annuities

Description

Axn.mdt gives the actuarial present value (APV) of a term insurance on a multiple decrement table, paying at the end of the year of decrement a benefit that may depend on the cause of decrement. axn.mdt gives the APV of an annuity payable while the insured is still in the active (no decrement yet) state.

Usage

Axn.mdt(object, x, n, i, decrement, benefits = 1, m = 0)

axn.mdt(object, x, n, i, m = 0, k = 1, payment = "advance")

Arguments

Details

With v = (1+i)^{-1} the insurance APV is

\sum_{j \in J} b_j \sum_{h=0}^{n-1} v^{m+h+1}\,{}_{m+h}p^{(\tau)}_x\, q^{(j)}_{x+m+h},

which for a single cause reduces to the historical Axn.mdt. The annuity APV is \frac1k \sum_{h} v^{t_h}\, {}_{t_h}p^{(\tau)}_x, with payment times t_h = m, m+1/k, \ldots, m+n-1/k (in advance) or m+1/k, \ldots, m+n (in arrears). Benefit premiums and reserves follow from the equivalence principle as ratios/differences of the two.

Value

A numeric vector of APVs (one per element of x).

References

Finan, M. B. (2014). A Reading of the Theory of Life Contingency Models: A Preparation for Exam MLC/3L, Sections 68-69.

Examples

# Finan (2014), Example 69.1: 3-year term on (16), i = 10%
myTable <- data.frame(x = 16:18, lx = c(20000, 17600, 14520),
                      da = c(1300, 1870, 2380), doc = c(1100, 1210, 1331))
myMdt <- new("mdt", table = myTable, name = "Finan 69.1")
A <- Axn.mdt(myMdt, x = 16, n = 3, i = 0.10, decrement = "doc")
a <- axn.mdt(myMdt, x = 16, n = 3, i = 0.10)
20000 * A / a   # level annual premium: 1250

# Finan (2014), Example 68.1: benefit 1 for cause 1, 2 for cause 2
t681 <- data.frame(x = 50:51, lx = c(1200, 800),
                   d1 = c(100, 200), d2 = c(300, 300))
m681 <- new("mdt", table = t681)
Axn.mdt(m681, x = 50, n = 2, i = 0.5, decrement = c("d1", "d2"),
        benefits = c(1, 2))  # 0.6852

Multiple lifes insurances and annuities

Description

Function to evalate the multiple lives insurances and annuities

Usage

Axyzn(tablesList, x, n, i, m, k = 1, status = "joint", type = "EV", 
power=1)
axyzn(tablesList, x, n, i, m, k = 1, status = "joint", type = "EV", 
power=1, payment="advance")

Arguments

Details

In theory, these functions apply the same concept of life insurances on one head on multiple heads.

Value

The insurance value is returned.

Note

These functions are the more general version of axyn and Axyn.

Author(s)

Giorgio Alfredo Spedicato, Kevin J. Owens.

References

Broverman, S.A., Mathematics of Investment and Credit (Fourth Edition), 2008, ACTEX Publications.

See Also

axyn,Axyn.

Examples

	data(soaLt)
	soa08Act=with(soaLt, new("actuarialtable",interest=0.06,
	x=x,lx=Ix,name="SOA2008"))
	#evaluate and life-long annuity for an aged 65
	listOfTables=list(soa08Act, soa08Act) 
	#Check actuarial equality
	axyzn(listOfTables,x=c(60,70),status="last")
	axn(listOfTables[[1]],60)+axn(listOfTables[[2]],70)-
	axyzn(listOfTables,x=c(60,70),status="joint")	

Increasing annuity life contingencies

Description

This function evaluates increasing annuities

Usage

Iaxn(actuarialtable, x, n, i, m = 0, type = "EV", power=1)

Arguments

Details

This actuarial mathematics is generally exoteric. I have seen no valid example of it.

Value

The APV of the insurance

Warning

The function is provided as is, without any guarantee regarding the accuracy of calculation. We disclaim any liability for eventual losses arising from direct or indirect use of this software.

Note

The function is provided as is, without any guarantee regarding the accuracy of calculation. We disclaim any liability for eventual losses arising from direct or indirect use of this software.

Author(s)

Giorgio A. Spedicato

References

Actuarial Mathematics (Second Edition), 1997, by Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J.

See Also

axn,IAxn

Examples

		#using SOA illustrative life tables
		data(soaLt)
		soa08Act=with(soaLt, new("actuarialtable",interest=0.06,
		x=x,lx=Ix,name="SOA2008"))
		#evaluate the value of a lifetime increasing annuity for a subject aged 80
		Iaxn(actuarialtable=soa08Act, x=80, n=10)

Function to calculated accumulated increasing annuity future value.

Description

This function evaluates non - stochastic increasing annuities future values.

Usage

Isn(i, n, type = "immediate")

Arguments

Details

It calls increasingAnnuity after having capitalized by \left( 1 + i \right)^n

Value

A numeric value

Warning

The function is provided as is, without any guarantee regarding the accuracy of calculation. We disclaim any liability for eventual losses arising from direct or indirect use of this software.

Note

This function calls internally increasingAnnuity function.

Author(s)

Giorgio A. Spedicato

References

Broverman, S.A., Mathematics of Investment and Credit (Fourth Edition), 2008, ACTEX Publications.

See Also

accumulatedValue

Examples

Isn(n=10,i=0.03)

Various demographic functions

Description

Various demographic functions

Usage

Lxt(object, x, t = 1, fxt = 0.5)

Tx(object, x, fxt = 0.5, axOmega = 1 - fxt)

Arguments

Details

Tx il the sum of years lived since age x by the population of the life table, it is the sum of Lx. The function is provided as is, without any warranty regarding the accuracy of calculations. Use at own risk.

Value

A numeric value

Author(s)

Giorgio Alfredo Spedicato.

References

Actuarial Mathematics (Second Edition), 1997, by Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J.

Examples

data(soaLt)
soa08Act=with(soaLt, new("actuarialtable",interest=0.06,
x=x,lx=Ix,name="SOA2008"))
Lxt(soa08Act, 67,10)
#assumes SOA example life table to be load
data(soaLt)
soa08Act=with(soaLt, new("actuarialtable",interest=0.06,x=x,lx=Ix,name="SOA2008"))
Tx(soa08Act, 67)

SoA illustrative service table

Description

Bowers' book Illustrative Service Table

Usage

data(SoAISTdata)

Format

A data frame with 41 observations on the following 6 variables.

x

Attained age

lx

Surviving subjects ate the beginning of each age

death

Drop outs for death cause

withdrawal

Drop outs for withdrawal cause

inability

Drop outs for inability cause

retirement

Drop outs for retirement cause

Details

It is a data frame that can be used to create a multiple decrement table

Source

Optical recognized characters from below source with some few adjustments

References

Actuarial Mathematics (Second Edition), 1997, by Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J.

Examples

data(SoAISTdata)
head(SoAISTdata)

Uk AM AF 92 life tables

Description

Uk AM AF life tables

Usage

data(AF92Lt)

Format

The format is: Formal class 'lifetable' [package ".GlobalEnv"] with 3 slots ..@ x : int [1:111] 0 1 2 3 4 5 6 7 8 9 ... ..@ lx : num [1:111] 100000 99924 99847 99770 99692 ... ..@ name: chr "AF92"

Details

Probabilities for earliest (under 16) and lastest ages (over 92) have been derived using a Brass - Logit model fit on Society of Actuaries life table.

Source

See Uk life table.

References

https://www.actuaries.org.uk/learn-and-develop/continuous-mortality-investigation/cmi-mortality-and-morbidity-tables/92-series-tables

Examples

data(AF92Lt)
exn(AF92Lt)
data(AM92Lt)
exn(AM92Lt)

Function to evaluate the accumulated value.

Description

This function returns the value at time n of a series of equally spaced payments of 1.

Usage

accumulatedValue(i, n, m = 0, k = 1, type = "immediate")

Arguments

Details

The accumulated value is the future value of the terms of an annuity. Its mathematical expression is s_{\left. {\overline {\, n \,}}\! \right| } = {\left( {1 + i} \right)^n} a_{\left. {\overline {\, n \,}}\! \right| }. The payment timing follows the type argument: an annuity-immediate has payments at the end of each period, while an annuity-due has payments at the beginning of each period.

Value

A numeric value representing the calculated accumulated value.

Warning

The function is provided as is, without any guarantee regarding the accuracy of calculation. We disclaim any liability for eventual losses arising from direct or indirect use of this software.

Note

Accumulated values are derived from annuities by the following basic equation {s_{\left. {\overline {\, n \,}}\! \right| }} = {\left( {1 + i} \right)^n} = a_{\left. {\overline {\, n \,}}\! \right| }.

Author(s)

Giorgio A. Spedicato

References

Broverman, S.A., Mathematics of Investment and Credit (Fourth Edition), 2008, ACTEX Publications.

See Also

annuity

Class "actuarialtable"

Description

Objects of class "actuarialtable" inherit the structure of class "lifetable" adding just the slot for interest rate, interest.

Objects from the Class

Objects can be created by calls of the form new("actuarialtable", ...). Creation is the same as lifetable objects creation, the slot for interest must be added too.

Slots

interest:

Object of class "numeric" slot for interest rate, e.g. 0.03

x:

Object of class "numeric" age slot

lx:

Object of class "numeric" subjects at risk at age x

name:

Object of class "character" name of the actuarial table

Extends

Class "lifetable", directly.

Methods

coerce

signature(from = "actuarialtable", to = "data.frame"): moves from actuarialtable to data.frame

coerce

signature(from = "actuarialtable", to = "numeric"): coerce from actuarialtable to a numeric

getOmega

signature(object = "actuarialtable"): as for lifetable

print

signature(x = "actuarialtable"): tabulates the actuarial commutation functions

show

signature(object = "actuarialtable"): show method

summary

signature(object = "actuarialtable"): prints brief summary

Warning

The function is provided as is, without any warranty regarding the accuracy of calculations. The author disclaims any liability for eventual losses arising from direct or indirect use of this software.

Note

The interest slot will handle time-varying interest rates in the future.

Author(s)

Giorgio A. Spedicato

References

Actuarial Mathematics (Second Edition), 1997, by Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J.

See Also

axn,lifetable

Examples

showClass("actuarialtable")

Annuity

Description

This function calculates the present value of a stream of fixed payments separated by equal intervals of time.

Usage

annuity(i, n, m = 0, k = 1, type = "immediate")

Arguments

Details

For an annuity-immediate the first payment occurs at time 1/k; for an annuity-due the first payment occurs at time 0. Thus, for annual payments, an annuity-immediate has payment times 1,2,\ldots,n, whereas an annuity-due has payment times 0,1,\ldots,n-1.

Value

A numeric value representing the present value of the annuity.

Note

The value returned by annuity function derives from direct calculation of the discounted cash flow and not from formulas, like {a^{\left( m \right)}}_{\left. {\overline {\, n \,}}\! \right| } = \frac{{1 - {v^n}}}{{{i^{\left( m \right)}}}}. When m is greater than 1, the payment per period is assumed to be \frac{1}{m}.

Author(s)

Giorgio A. Spedicato

References

Broverman, S.A., Mathematics of Investment and Credit (Fourth Edition), 2008, ACTEX Publications.

Life insurance with arithmetic-variation benefit (increasing/decreasing, fractional claims)

Description

This help page groups two term life insurances with **arithmetic variation** of the benefit, both with optional deferment m and k fractional claim times (claims at end of subperiods):

• IAxn: increasing arithmetic term insurance (benefit grows linearly with time);

• DAxn: decreasing arithmetic term insurance (benefit declines linearly with time).

Usage

IAxn(
  actuarialtable,
  x,
  n,
  i = actuarialtable@interest,
  m = 0,
  k = 1,
  type = "EV",
  power = 1
)

DAxn(
  actuarialtable,
  x,
  n,
  i = actuarialtable@interest,
  m = 0,
  k = 1,
  type = "EV",
  power = 1
)

Arguments

Details

Let t_j = m + (j-1)/k, j=1,\dots,nk. With **fractional claims at end of subperiods**, the EV implementations follow the pattern already used in Axn:

IAxn (increasing):

\mathrm{IA}_{\overline{n}|}^{(k)} = \sum_{j=1}^{nk} \Big(\frac{j}{k}\Big)\, v^{t_j + 1/k}\; {}_{t_j}p_x\; q_{x+t_j}^{(1/k)},

where v=(1+i)^{-1}, computed via pxt(...) and qxt(..., t=1/k).

DAxn (decreasing) is analogous with benefit (n - (j-1)/k); see its subsection below.

DAxn (decreasing):

\mathrm{DA}_{\overline{n}|}^{(k)} = \sum_{j=1}^{nk} \Big(n - \frac{j-1}{k}\Big)\, v^{t_j + 1/k}\; {}_{t_j}p_x\; q_{x+t_j}^{(1/k)}, \qquad t_j = m + \frac{j-1}{k}.

See “Fractional timing conventions” above for claim timing assumptions.

Value

A numeric value: the APV (or one simulated realization if type="ST").

IAxn — Increasing arithmetic term

Computes the APV of an n-year **increasing** term insurance on a life aged x, with k fractional claim times and optional deferment m. The benefit at the j-th subperiod equals j/k.

Fractional timing conventions

For **insurance** benefits in this package, fractional claims are assumed to occur at the **end** of each subperiod (i.e., at t_j + 1/k). This matches the implementation that multiplies survival to t_j and a fractional death probability over the next subperiod:

v^{t_j + 1/k}\; {}_{t_j}p_x\; q_{x+t_j}^{(1/k)}.

By contrast, **annuities** use a payment-timing flag ("immediate" vs "due") which changes the evaluation times; insurance here has a fixed claim timing at end-subperiod.

DAxn — Decreasing arithmetic term

Computes the APV of an n-year **decreasing** term insurance on a life aged x, with k fractional claim times and optional deferment m. The benefit at the j-th subperiod equals n - (j-1)/k.

References

Bowers, N. L., Gerber, H. U., Hickman, J. C., Jones, D. A., Nesbitt, C. J. (1997). Actuarial Mathematics, 2nd ed., SOA.

See Also

Axn (level benefit), AExn, Exn, axn

Other life-contingency APVs: endowment_trio

Examples

## Setup (legacy examples)
data(soaLt)
soa08Act <- with(soaLt, new("actuarialtable", interest=0.06, x=x, lx=Ix, name="SOA2008"))

## IAxn: increasing arithmetic term, 10 years, age 25 (legacy)
IAxn(actuarialtable = soa08Act, x = 25, n = 10)

## More examples (k>1 and deferment)
IAxn(actuarialtable = soa08Act, x = 40, n = 20, k = 12)     # monthly claims
IAxn(actuarialtable = soa08Act, x = 40, n = 15, m = 5, k = 4) # deferred 5y, quarterly

## DAxn: decreasing arithmetic term, 10 years, age 25 (legacy)
DAxn(actuarialtable = soa08Act, x = 25, n = 10)
## More examples (k>1 and deferment)
DAxn(actuarialtable = soa08Act, x = 45, n = 10, k = 2)       # semiannual
DAxn(actuarialtable = soa08Act, x = 45, n = 12, m = 3, k = 12) # deferred 3y, monthly

Autoplot a life table

Description

Produces a ggplot2 representation of the number of survivors by age. The method is defined for 'lifetable'; S3 dispatch also makes it available for S4 subclasses such as 'actuarialtable'.

Usage

## S3 method for class 'lifetable'
autoplot(object, ...)

Arguments

Value

A 'ggplot2' plot object.

Examples

lt <- new("lifetable", x = 0:3, lx = c(100, 90, 50, 10))
ggplot2::autoplot(lt)

Functions to evaluate life insurance and annuities on two heads.

Description

These functions evaluates life insurances and annuities on two heads.

Usage

axyn(tablex, tabley, x, y, n, i, m, k = 1, status = "joint", type = "EV", 
payment="advance")
Axyn(tablex, x, tabley, y, n, i, m, k = 1, status = "joint", type = "EV")

Arguments

Details

Actuarial mathematics book formulas has been implemented.

Value

A numeric value returning APV of chosen insurance form.

Warning

The function is provided as is, without any warranty regarding the accuracy of calculations. The author disclaims any liability for eventual losses arising from direct or indirect use of this software.

Note

Deprecated functions. Use Axyzn and axyzn instead.

Author(s)

Giorgio A. Spedicato

References

Actuarial Mathematics (Second Edition), 1997, by Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J.

See Also

pxyt

Examples

## Not run: 
	data(soa08Act)
	#last survival status annuity
	axyn(tablex=soa08Act, tabley=soa08Act, x=65, y=70, 
		n=5,  status = "last",type = "EV")
    #first survival status annuity
	Axyn(tablex=soa08Act, tabley=soa08Act, x=65, y=70,
	status = "last",type = "EV")
	
## End(Not run)

Build an mdt object from a matrix of independent (ASDT) rates

Description

buildMdtFromIndependentRates constructs a multiple-decrement table (mdt) from a matrix of independent single-decrement rates q'^{(j)}_x, the inverse of independentRatesFromMdt.

Usage

buildMdtFromIndependentRates(
  x,
  qx.primes,
  radix = 1e+05,
  name = "ASDT-built mdt"
)

Arguments

Details

For each age the combined (absolute) rate of decrement j is obtained by the UDD-based integration formula

q^{(j)}_x = q'^{(j)}_x \int_0^1 \prod_{i \ne j} \bigl(1 - s\,q'^{(i)}_x\bigr)\,ds,

which is the same formula used by qxt.fromQxprime for a single age. The resulting absolute rates are multiplied by l^{(\tau)}_x to obtain the decrement counts, and the survivorship column is computed recursively from p^{(\tau)}_x = \prod_j (1 - q'^{(j)}_x).

Value

An mdt object.

See Also

independentRatesFromMdt for the reverse extraction, qxt.fromQxprime for the single-age formula.

Examples

# Finan (2014) Example 67.4:
# Three decrements (death, disability, retirement) at ages 60-61.
qp <- matrix(c(0.010, 0.030, 0.100,
                0.013, 0.050, 0.200), nrow = 2, byrow = TRUE,
              dimnames = list(NULL, c("death", "disability", "retirement")))
mdt674 <- buildMdtFromIndependentRates(x = 60:61, qx.primes = qp,
                                        radix = 1000, name = "Finan 67.4")
print(mdt674)

Italian Health Insurance Data

Description

A list of data.frames containing transition probabilities by age (row) and year of projections Transitions are split by males and females, and show probabilities of survival, death and transitions from Healty to Disabled

Usage

de_angelis_di_falco

Format

a list containing elevent items (data.frames), and an mdt data object (HealthyMaleTable2013)

Source

Paolo De Angelis, Luigi di Falco (a cura di). Assicurazioni sulla salute: caratteristiche, modelli attuariali e basi tecniche

Function to evaluate decreasing annuities.

Description

This function returns present values for decreasing annuities-certain.

Usage

decreasingAnnuity(i, n, type = "immediate")

Arguments

Details

A decreasing annuity has the following flows of payments: n, n-1, n-2, ..., 1, 0. For an annuity-immediate these payments occur at times 1,2,\ldots,n; for an annuity-due they occur at times 0,1,\ldots,n-1.

Value

A numeric value reporting the present value of the decreasing cash flows.

Warning

The function is provided as is, without any guarantee regarding the accuracy of calculation. The author disclaims any liability for eventual losses arising from direct or indirect use of this software.

Note

This function calls presentValue function internally.

Author(s)

Giorgio A. Spedicato

References

Broverman, S.A., Mathematics of Investment and Credit (Fourth Edition), 2008, ACTEX Publications.

See Also

annuity, increasingAnnuity, DAxn

Examples

# The present value of 10, 9, 8, ..., 0 payable at the end of the period for 10 years is
decreasingAnnuity(i = 0.03, n = 10)
# Assuming a 3% interest rate
sum((10:1)/(1 + .03)^(1:10))

Canada Mortality Rates for UP94 Series

Description

UP94 life tables underlying mortality rates

Usage

data(demoCanada)

Format

A data frame with 120 observations on the following 7 variables.

x

age

up94M

UP 94, males

up94F

UP 94, females

up942015M

UP 94 projected to 2015, males

up942015f

UP 94 projected to 2015, females

up942020M

UP 94 projected to 2020, males

up942020F

UP 94 projected to 2020, females

Details

Mortality rates are provided.

Source

Courtesy of Andrew Botros

References

Courtesy of Andrew Botros

Examples

data(demoCanada)
head(demoCanada)
#create the up94M life table
up94MLt<-probs2lifetable(probs=demoCanada$up94M,radix=100000,"qx",name="UP94")
#create the up94M actuarial table table
up94MAct<-new("actuarialtable", lx=up94MLt@lx, x=up94MLt@x,interest=0.02)

China Mortality Rates for life table construction

Description

Seven yearly mortality rates for each age

Usage

data(demoChina)

Format

A data frame with 106 observations on the following 8 variables.

age

Attained age

CL1

CL1 rates

CL2

CL2 rates

CL3

CL3 rates

CL4

CL4 rates

CL5

CL5 rates

CL6

CL6 rates

CL90-93

CL 90-93 rates

Details

See the source link for details.

Source

Society of Actuaries

References

https://mort.soa.org/

Examples

data(demoChina)
tableChinaCL1<-probs2lifetable(probs=demoChina$CL1,radix=1000,type="qx",name="CHINA CL1")

French population life tables

Description

Illustrative life tables from French population.

Usage

data(demoFrance)

Format

A data frame with 113 observations on the following 5 variables.

age

Attained age

TH00_02

Male 2000 life table

TF00_02

Female 2000 life table

TD88_90

1988 1990 life table

TV88_90

1988 1990 life table

Details

These tables are real French population life tables. They regard 88 - 90 and 00 - 02 experience.

Source

Actuaris - Winter Associes

Examples

data(demoFrance)
head(demoFrance)

German population life tables

Description

Dataset containing mortality rates for German population, male and females.

Usage

data(demoGermany)

Format

A data frame with 113 observations on the following 5 variables.

x

Attained age

qxMale

Male mortality rate

qxFemale

Female mortality rate

Details

Sterbetafel DAV 1994

Source

Private communicatiom

Examples

data(demoGermany)
head(demoGermany)

Italian population life tables for males and females

Description

This dataset reports five pairs of Italian population life tables. These table can be used to create life table objects and actuarial tables object.

Usage

data(demoIta)

Format

A data frame with 121 observations on the following 9 variables.

X

a numeric vector, representing ages from 0 to \omega.

SIM02

a numeric vector, 2002 cross section general population males life table

SIF02

a numeric vector, 2002 cross section general population females life table

SIM00

a numeric vector, 2000 cross section general population males life table

SIF00

a numeric vector, 2000 cross section general population females life table

SIM92

a numeric vector, 1992 cross section general population males life table

SIF92

a numeric vector, 1992 cross section general population females life table

SIM81

a numeric vector, 1981 cross sectional general population males life table

SIF81

a numeric vector, 1981 cross sectional general population females life table

SIM61

a numeric vector, 1961 cross sectional general population males life table

SIF61

a numeric vector, 1961 cross sectional general population females life table

RG48M

a numeric vector, RG48 projected males life table

RG48F

a numeric vector, RG48 projected females life table

IPS55M

a numeric vector, IPS55 projected males life table

IPS55F

a numeric vector, IPS55 projected females life table

SIM71

a numeric vector, 1971 cross sectional general population males life table

SIM51

a numeric vector, 1951 cross sectional general population males life table

SIM31

a numeric vector, 1931 cross sectional general population males life table

Details

These table contains the vectors of survival at the beginning of life years and are the building block of both lifetable and actuarialtable classes.

Source

These tables comes from Italian national statistical bureau (ISTAT) for SI series, government Ministry of Economics (Ragioneria Generale dello Stato) for RG48 or from Insurers' industrial association IPS55. RG48 represents the projected survival table for the 1948 born cohort, while IPS55 represents the projected survival table for the 1955 born cohort.

References

ISTAT, IVASS, Ordine Nazionale Attuari

Examples

	#load and show
	data(demoIta)
	head(demoIta)
	#create sim92 life and actuarial table
	lxsim92<-demoIta$SIM92

	lxsim92<-lxsim92[!is.na(lxsim92) & lxsim92!=0]
	xsim92<-seq(0,length(lxsim92)-1,1)
	#create the table
	sim92lt=new("lifetable",x=xsim92,lx=lxsim92,name="SIM92")
	plot(sim92lt)

Japan Mortality Rates for life table construction

Description

Two yearly mortality rates for each age

Usage

data(demoJapan)

Format

A data frame with 110 observations on the following 3 variables.

JP8587M

Male life table

JP8587F

Female life table

age

Attained age

Details

Dowloaded in 2012 from Society of Actuaries (SOA) mortality table web site

Source

SOA mortality web site

Examples

data(demoJapan)
head(demoJapan)

UK life tables

Description

AM and AF one year mortality rate. Series of 1992

Usage

data(demoUk)

Format

A data frame with 74 observations on the following 3 variables:

Age

Annuitant age

AM92

One year mortality rate (males)

AF92

One year mortality rate (males)

Details

This data set shows the one year survival rates for males and females of the 1992 series. It has been taken from the Institute of Actuaries. The series cannot be directly used to create a life table since neither rates are not provided for ages below 16 nor for ages over 90. Various approach can be used to complete the series.

Source

Institute of Actuaries

References

https://www.actuaries.org.uk/learn-and-develop/continuous-mortality-investigation/cmi-mortality-and-morbidity-tables/92-series-tables

Examples

data(demoUk)
head(demoUk)

United States Social Security life tables

Description

This data set contains period life tables for years 1990, 2000 and 2007. Both males and females life tables are reported.

Usage

demoUsa

Format

A data.frame containing people surviving at the beginning of "age" at 2007, 2000, and 1990 split by gender

Details

Reported age is truncated at the last age with lx>0.

Source

See https://www.ssa.gov/oact/NOTES/as120/LifeTables_Body.html

Examples

data(demoUsa)
head(demoUsa)

Compute the duration or the convexity of a series of CF

Description

Compute the duration or the convexity of a series of CF

Usage

duration(cashFlows, timeIds, i, k = 1, macaulay = TRUE)

convexity(cashFlows, timeIds, i, k = 1)

Arguments

Details

The Macaulay duration is defined as \sum\limits_t^{T} \frac{t*CF_{t}\left( 1 + \frac{i}{k} \right)^{ - t*k}}{P}, while \sum\limits_{t}^{T} t*\left( t + \frac{1}{k} \right) * CF_t \left(1 + \frac{y}{k} \right)^{ - k*t - 2}

Value

A numeric value representing either the duration or the convexity of the cash flow series

References

Broverman, S.A., Mathematics of Investment and Credit (Fourth Edition), 2008, ACTEX Publications.

Examples

#evaluate the duration/convexity of a coupon payment
cf=c(10,10,10,10,10,110)
t=c(1,2,3,4,5,6)
duration(cf, t, i=0.03)
convexity(cf, t, i=0.03)

Endowment, insurance, pure endowment, and survival annuity APVs (shared topic)

Description

This help page groups four classical life-contingency present values:

• Exn: pure endowment, pays 1 at time n if alive.

• Axn: term/whole life insurance, pays 1 at death within n years (or up to limiting age if n is missing), with fractional claim timing.

• AExn: n-year endowment insurance, i.e. Axn + Exn.

• axn: survival annuity (immediate/due), with deferment m and k payments per year.

Usage

Exn(actuarialtable, x, n, i = actuarialtable@interest, type = "EV", power = 1)

axn(
  actuarialtable,
  x,
  n,
  i = actuarialtable@interest,
  m,
  k = 1,
  type = "EV",
  power = 1,
  payment = "advance",
  ...
)

Axn(
  actuarialtable,
  x,
  n,
  i = actuarialtable@interest,
  m,
  k = 1,
  type = "EV",
  power = 1,
  ...
)

AExn(
  actuarialtable,
  x,
  n,
  i = actuarialtable@interest,
  k = 1,
  type = "EV",
  power = 1
)

Arguments

Details

Exn: E_x^n = v^n \, {}_np_x with v=(1+i)^{-1}.

Axn: With fractional claims,

A_{\overline{n}|}^{(k)} = \sum_{j=1}^{nk} v^{t_j + 1/k}\; {}_{t_j}p_x\; q_{x+t_j}^{(1/k)},

where t_j = m + (j-1)/k, computed via pxt(...) and qxt(..., t=1/k).

AExn: returns Axn(...) + Exn(...) with aligned arguments.

Vectorization: Exn and AExn accept vectors for x, n and i; the arguments are recycled to a common length and one value per element is returned.

axn: Survival annuity with payment timing "immediate" (arrears) or "due" (advance), deferment m and k payments per year (see function-specific parameters).

axn — Survival annuity (immediate/due), with deferment m and k fractional payments. For type="EV" the annuity is computed as

a_{\overline{n}|}^{(k)} = \sum_{j=1}^{nk} \frac{1}{k}\, v^{t_j}\, {}_{t_j}p_x,

where t_j are the payment times depending on payment and m.

Axn — Life insurance (term / whole life), fractional claim times. Vectorized in x, n, m. k must be scalar.

AExn — n-year endowment insurance, computed as Axn + Exn.

Value

A numeric value (or vector for vectorized inputs): the APV in expected value, or one simulated realization when type="ST".

Exn — Pure endowment

Computes the actuarial present value (APV) of a pure endowment that pays 1 at time n provided survival to x+n.

References

Bowers, N. L., Gerber, H. U., Hickman, J. C., Jones, D. A., Nesbitt, C. J. (1997). Actuarial Mathematics, 2nd ed., SOA.

See Also

Axn, AExn, axn, Exn

Other life-contingency APVs: arithmetic_variation_insurances

Examples

## Common setup used in legacy docs
data(soaLt)
soa08Act <- with(soaLt, new("actuarialtable", interest=0.06, x=x, lx=Ix, name="SOA2008"))

## Exn (pure endowment)
Exn(soa08Act, x=30, n=35)

## Axn (term / whole life insurance)
# 10-year term, semiannual claims:
Axn(soa08Act, x=50, n=10, k=2)
# Whole life (n inferred), monthly:
Axn(soa08Act, x=30, k=12)

# Several ages and terms at once:
Exn(soa08Act, x=c(30,40,50), n=c(35,25,15))

## AExn = Axn + Exn  (legacy book-check)
AExn(soa08Act, x=35, n=30, i=0.06)
Exn(soa08Act, x=35, n=30, i=0.06) + Axn(soa08Act, x=35, n=30, i=0.06)
AExn(soa08Act, x=c(35,45), n=c(30,20))

## axn (survival annuity, legacy example)
# Life-long annuity for age 65:
axn(soa08Act, x=65)

## axn specific legacy examples
# Immediate (arrears) vs due (advance), quarterly, 15-year term deferred 5 years:
axn(soa08Act, x=60, n=15, m=5, k=4, payment="immediate")
axn(soa08Act, x=60, n=15, m=5, k=4, payment="due")
# Vectorization over x/n:
axn(soa08Act, x=c(60,65), n=c(10,20), k=12, payment="due")

Expected residual life.

Description

Expected future lifetime of a life aged x, either over the whole remaining lifespan or over a temporary period of n years.

Usage

exn(object, x, n, type = "curtate", fxt = 0.5, axOmega = 1 - fxt)

Arguments

Details

For type = "curtate" the function returns the (temporary) curtate expectation of life

e_{x:\overline{n}|} = \sum_{k=1}^{n} {}_k p_x ,

that is the expected number of complete future years lived by (x) within the next n years. With n missing it is the curtate expectation of life e_x = E[K_x].

For type = "complete" the function returns the (temporary) complete expectation of life

\mathring{e}_{x:\overline{n}|} = \int_0^n {}_t p_x \, dt = \frac{{}_nL_x}{l_x} ,

evaluated under the uniform distribution of deaths (UDD) assumption within each year, i.e. L_x = l_x - 0.5 d_x (see Lxt). With n missing it is \mathring{e}_x = T_x / l_x = E[T_x].

Under UDD the two quantities are related by \mathring{e}_{x:\overline{n}|} = e_{x:\overline{n}|} + 0.5\,(1 - {}_n p_x), which reduces to \mathring{e}_x = e_x + 0.5 when n covers the whole remaining lifespan.

The last tabulated age \omega is treated as a closed interval: those alive at \omega are assumed to die on average half a year later. Published tables that close the table with an open interval (e.g. L_{\omega} = l_{\omega}/m_{\omega}, as in the NCHS tables) can therefore show a slightly larger complete life expectancy.

Value

A numeric value representing the expected life span.

Author(s)

Giorgio Alfredo Spedicato

References

Actuarial Mathematics (Second Edition), 1997, by Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J.

See Also

lifetable, Tx, Lxt

Examples

#loads and show
data(soa08Act)
#curtate expectation of life at birth
exn(object=soa08Act, x=0)
#complete expectation of life at birth (curtate + 0.5 under UDD)
exn(object=soa08Act, x=0,type="complete")
#temporary 20-year expectations at age 50
exn(object=soa08Act, x=50, n=20)
exn(object=soa08Act, x=50, n=20, type="complete")

Function to return the decrements defined in the mdt class

Description

This function list the character decrements of the mdf class

Usage

getDecrements(object)

Arguments

Details

A character vector is returned

Value

A character vector listing the decrements defined in the class

Note

To be updated

Author(s)

Giorgio Alfredo Spedicato

References

Marcel Finan A Reading of the Theory of Life Contingency Models: A Preparation for Exam MLC/3L

See Also

getOmega

Examples

#create a new table
tableDecr=data.frame(d1=c(150,160,160),d2=c(50,75,85))
newMdt<-new("mdt",name="testMDT",table=tableDecr)
getDecrements(newMdt)

Functions to obtain the present value of a life contingency given the time to death

Description

It returns the present value of a life contingency, specified by its APV symbol, known the time to death ob the sibjects

Usage

getLifecontingencyPv(deathsTimeX, lifecontingency, object, x, t, i = object@interest, 
m = 0, k = 1, payment = "advance")
getLifecontingencyPvXyz(deathsTimeXyz, lifecontingency, tablesList, x, t, i, m = 0, 
k = 1, status = "joint", payment = "advance")

Arguments

Details

This function is a wrapper to the many internal functions that give the PV known the age of death.

Value

A vector or matrix of size number of rows of deathTimeXyz / deathTimeXy

Warning

The function is provided as is, without any warranty regarding the accuracy of calculations. The author disclaims any liability for eventual losses arising from direct or indirect use of this software.

Note

Multiple life function needs to be tested

Author(s)

Spedicato Giorgio

References

Actuarial Mathematics (Second Edition), 1997, by Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J.

See Also

rLifeContingenciesXyz, rLifeContingencies

Examples

#simulate the PV values for some life contingencies given some death times
data(soa08Act)
testgetLifecontingencyPvXyzAxyz<-getLifecontingencyPvXyz(deathsTimeXyz=
matrix(c(50,50,51,43,44,22,12,56,20,24,53,12),
ncol=2),
lifecontingency = "Axyz",tablesList = list(soa08Act, soa08Act), i = 0.03, t=30,x=c(40,50),
m=0, k=1,status="last")
testgetLifecontingencyPvAxn<-getLifecontingencyPv(deathsTimeX = seq(0, 110, by=1), 
lifecontingency = "Axn", object=soa08Act, 
		x=40,t=20, m=0, k=1)

Function to return the terminal age of a life table.

Description

This function returns the \omega value of a life table object, that is, the last attainable age within a life table.

Usage

getOmega(object)

Arguments

Value

A numeric value representing the \omega value of a life table object

Warning

The function is provided as is, without any guarantee regarding the accuracy of calculation. We disclaim any liability for eventual losses arising from direct or indirect use of this software.

Author(s)

Giorgio A. Spedicato

References

Actuarial Mathematics (Second Edition), 1997, by Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J.

See Also

actuarialtable

Examples

		#assumes SOA example life table to be load
		data(soaLt)
		soa08=with(soaLt, new("lifetable",
		x=x,lx=Ix,name="SOA2008"))
		#the last attainable age under SOA life table is
		getOmega(soa08) 

Increasing annuity.

Description

This function evaluates non-stochastic increasing annuities.

Usage

increasingAnnuity(i, n, type = "immediate")

Arguments

Details

An increasing annuity shows the following flow of payments: 1,2,\ldots,n-1,n. For an annuity-immediate these payments occur at times 1,2,\ldots,n; for an annuity-due they occur at times 0,1,\ldots,n-1.

Value

The value of the annuity.

Warning

The function is provided as is, without any guarantee regarding the accuracy of calculation. We disclaim any liability for eventual losses arising from direct or indirect use of this software.

Note

This function calls internally presentValue function.

Author(s)

Giorgio A. Spedicato

References

Broverman, S.A., Mathematics of Investment and Credit (Fourth Edition), 2008, ACTEX Publications.

See Also

decreasingAnnuity, IAxn

Examples

# The present value of an increasing sequence of payments payable at the end
# of each period for 10 periods is
increasingAnnuity(i = 0.03, n = 10)
# Assuming a 3% interest rate

Extract the full Associated Single Decrement Table (ASDT) from an mdt object

Description

independentRatesFromMdt returns a matrix of ASDT independent rates q'^{(j)}_x for every combination of age and decrement in the supplied multiple-decrement table.

Usage

independentRatesFromMdt(object, x, t = 1)

Arguments

Details

The independent rate for decrement j at age x is obtained under the Uniform Distribution of Deaths (UDD) assumption as

q'^{(j)}_x = 1 - \bigl(1 - q^{(\tau)}_x\bigr)^{q^{(j)}_x / q^{(\tau)}_x},

which is the same formula used by qxt.prime.fromMdt for a single age/decrement pair. independentRatesFromMdt is a convenience wrapper that applies this extraction to all ages and all decrements at once, returning the result as a tidy matrix.

Value

A numeric matrix with one row per age and one column per decrement. Row names are the ages, column names the decrement identifiers.

See Also

qxt.prime.fromMdt for a single age/decrement pair, buildMdtFromIndependentRates for the inverse operation.

Examples

valdezDf <- data.frame(
  x = 50:54,
  lx = c(4832555, 4821937, 4810206, 4797185, 4782737),
  heart = c(5168, 5363, 5618, 5929, 6277),
  accidents = c(1157, 1206, 1443, 1679, 2152),
  other = c(4293, 5162, 5960, 6840, 7631))
valdezMdt <- new("mdt", name = "ValdezExample", table = valdezDf)

# ASDT matrix on the ages actually supplied
independentRatesFromMdt(valdezMdt, x = 50:54)

# Subset of ages
independentRatesFromMdt(valdezMdt, x = 50:52)

Functions to switch from interest to intensity and vice versa.

Description

There functions switch from interest to intensity and vice - versa.

Usage

intensity2Interest(intensity)

interest2Intensity(i)

Arguments

Details

Simple financial mathematics formulas are applied.

Value

A numeric value.

Author(s)

Giorgio A. Spedicato

References

Broverman, S.A., Mathematics of Investment and Credit (Fourth Edition), 2008, ACTEX Publications.

See Also

real2Nominal, nominal2Real

Examples

# a force of interest of 0.02 corresponds to an APR of 
intensity2Interest(intensity=0.02)
#an interest rate equal to 0.02 corresponds to a force of interest of of 
interest2Intensity(i=0.02)

Functions to switch from interest to discount rates

Description

These functions switch from interest to discount rates and vice - versa

Usage

interest2Discount(i)

discount2Interest(d)

Arguments

Details

The following formula (and its inverse) rules the relationships:

\frac{i}{{1 + i}} = d

Value

A numeric value

Author(s)

Giorgio Alfredo Spedicato

References

Broverman, S.A., Mathematics of Investment and Credit (Fourth Edition), 2008, ACTEX Publications.

See Also

intensity2Interest,nominal2Real

Examples

discount2Interest(d=0.04)

Class "lifetable"

Description

lifetable objects allow to define and use life tables with the aim to evaluate survival probabilities and mortality rates easily. Such values represent the building blocks used to estimate life insurances actuarial mathematics.

Objects from the Class

Objects can be created by calls of the form new("lifetable", ...). Two vectors are needed. The age vector and the population at risk vector.

Slots

x:

Object of class "numeric", representing the sequence 0,1,\ldots, \omega

lx:

Object of class "numeric", representing the number of lives at the beginning of age x. It is a non increasing sequence. The last element of vector x is supposed to be > 0.

name:

Object of class "character", reporting the name of the table

Methods

coerce

signature(from = "lifetable", to = "data.frame"): method to create a data - frame from a lifetable object

coerce

signature(from = "lifetable", to = "markovchainList"): coerce method from lifetable to markovchainList; available only when the optional markovchain package is installed

coerce

signature(from = "lifetable", to = "numeric"): brings to numeric

coerce

signature(from = "data.frame", to = "lifetable"): brings to life table

getOmega

signature(object = "lifetable"): returns the maximum attainable life age

plot

signature(x = "lifetable"): plot method

head

signature(x = "lifetable"): head method

print

signature(x = "lifetable"): method to print the survival probability implied in the table

show

signature(object = "lifetable"): identical to plot method

summary

signature(object = "lifetable"): it returns summary information about the object

Warning

The function is provided as is, without any warranty regarding the accuracy of calculations. The author disclaims any liability for eventual losses arising from direct or indirect use of this software.

Note

t may be missing in pxt, qxt, ext. It assumes value equal to 1 in such case.

Author(s)

Giorgio A. Spedicato

References

Actuarial Mathematics (Second Edition), 1997, by Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J.

See Also

actuarialtable

Examples

showClass("lifetable")
data(soa08)
summary(soa08)
#the last attainable age under SOA life table is
getOmega(soa08) 
#head and tail
data(soaLt)
tail(soaLt)
head(soaLt)

Class "mdt"

Description

A class to store multiple decrement tables

Objects from the Class

Objects can be created by calls of the form new("mdt", name, table, bottomCompletionSurvival, ...). They store absolute decrements. bottomCompletionSurvival (default 0.99) is the one-year survival ratio used to synthetically back-fill ages below the lowest age supplied in table; it does not affect rows for ages actually supplied.

Slots

name:

The name of the table

table:

A data frame containing at least the number of decrements

Methods

getDecrements

signature(object = "mdt"): return the name of decrements

getOmega

signature(object = "mdt"): maximum attainable age

initialize

signature(.Object = "mdt"): method to initialize the class

print

signature(x = "mdt"): tabulate absolute decrement rates

show

signature(object = "mdt"): show rates of decrement

coerce

signature(from = "mdt", to = "markovchainList"): coercing to markovchainList objects; available only when the optional markovchain package is installed

coerce

signature(from = "mdt", to = "data.frame"): coercing to data.frame

summary

signature(object = "mdt"): it returns summary information about the object

Note

Currently only decrements storage of the class is defined.

Author(s)

Giorgio Alfredo Spedicato

References

Marcel Finan A Reading of the Theory of Life Contingency Models: A Preparation for Exam MLC/3L

See Also

lifetable

Examples

#shows the class definition
showClass("mdt")
#create a new table
tableDecr=data.frame(d1=c(150,160,160),d2=c(50,75,85))
newMdt<-new("mdt",name="testMDT",table=tableDecr)

Convert an mdt to long (time, status) format for survival analysis

Description

mdtToLong reshapes a multiple decrement table into an aggregated long data set with one row per (exit time, cause) and a count column, ready for competing-risks tools such as survival::survfit(Surv(time, status) ~ 1, weights = count) (Aalen-Johansen cumulative incidence) or for any analysis based on survival::Surv.

Usage

mdtToLong(object, x, t, exitTime = c("end", "mid"), dropZero = TRUE)

Arguments

Details

With exitTime = "end" the Aalen-Johansen estimate of the cumulative incidence of cause j at time k computed on the weighted long data coincides with qxt(object, x, k, decrement = j); see the multiple decrement vignette.

Value

A data.frame with columns time (years since age x), age (age at exit or censoring), status (a factor whose first level, "censored", is followed by the decrement names, as expected by survival::Surv for multi-state data) and count (number of lives, possibly non-integer).

Examples

valdezDf <- data.frame(
  x = 50:54,
  lx = c(4832555, 4821937, 4810206, 4797185, 4782737),
  heart = c(5168, 5363, 5618, 5929, 6277),
  accidents = c(1157, 1206, 1443, 1679, 2152),
  other = c(4293, 5162, 5960, 6840, 7631))
valdezMdt <- new("mdt", name = "ValdezExample", table = valdezDf)
long <- mdtToLong(valdezMdt, x = 50, t = 5)
head(long)
if (requireNamespace("survival", quietly = TRUE)) {
  fit <- survival::survfit(survival::Surv(time, status) ~ 1,
                           data = long, weights = count)
  summary(fit, times = 1:5)$pstate
}

Median age at death of a life table

Description

S4 method for the median generic: the median age at death implied by a lifetable, i.e. the age at which the survivorship column l_x has fallen to half of its value at age (linear interpolation on l_x).

Usage

## S4 method for signature 'lifetable'
median(x, na.rm = FALSE, ...)

Arguments

Value

The median age at death (a numeric value).

See Also

quantile, modalAge, exn

Examples

data(soa08Act)
median(soa08Act)              # median age at death from birth
median(soa08Act, age = 65)    # median age at death given survival to 65

Modal age at death of a life table

Description

Returns the age at which the number of deaths d_x is largest, i.e. the Lexis modal age at death. (R's base mode returns the storage type of an object and cannot be used for this, so a dedicated function is provided.)

Usage

modalAge(object, startAge = NULL, interpolate = FALSE)

Arguments

Details

The death counts are d_x = l_x - l_{x+1}; the artificial mass at the last, open age interval (all remaining survivors) is excluded from the search.

Value

The modal age at death (a numeric value).

See Also

median, quantile

Examples

data(soa08Act)
modalAge(soa08Act)
modalAge(soa08Act, startAge = 10, interpolate = TRUE)

Functions to deals with multiple life models

Description

These functions evaluate multiple life survival probabilities, either for joint or last life status. Arbitrary life probabilities can be generated as well as random samples of lifes.

Usage

exyzt(tablesList, x, t = Inf, status = "joint",  type = "Kx", ...)

pxyzt(tablesList, x, t, status = "joint", 
fractional=rep("linear", length(tablesList)), ...)

qxyzt(tablesList, x, t, status = "joint",  
fractional=rep("linear",length(tablesList)), ...)

Arguments

Details

These functions extends pxyt family to an arbitrary number of life contingencies.

Value

An estimate of survival / death probability or expected lifetime, or a matrix of ages.

Note

The procedure is experimental.

Author(s)

Giorgio Alfredo, Spedicato

References

Broverman, S.A., Mathematics of Investment and Credit (Fourth Edition), 2008, ACTEX Publications.

See Also

pxt,exn

Examples

#assessment of curtate expectation of future lifetime of the joint-life status
#generate a sample of lifes
data(soaLt)
soa08Act=with(soaLt, new("actuarialtable",interest=0.06,x=x,lx=Ix,name="SOA2008"))
tables=list(males=soa08Act, females=soa08Act)
xVec=c(60,65)
test=rLifexyz(n=50000, tablesList = tables,x=xVec,type="Kx")
#check first survival status
t.test(x=apply(test,1,"min"),mu=exyzt(tablesList=tables, x=xVec,status="joint"))
#check last survival status
t.test(x=apply(test,1,"max"),mu=exyzt(tablesList=tables, x=xVec,status="last"))

Mortality rates to Death probabilities

Description

Function to convert mortality rates to probabilities of death

Usage

mx2qx(mx, ax = 0.5)

Arguments

Details

Function to convert mortality rates to probabilities of death

Value

A vector of death probabilities

See Also

mxt, qxt, qx2mx

Examples

#using some recursion
qx2mx(mx2qx(.2))

Central mortality rate

Description

This function returns the central mortality rate demographic function.

Usage

mxt(object, x, t)

Arguments

Value

A numeric value representing the central mortality rate between age x and x+t.

References

Actuarial Mathematics (Second Edition), 1997, by Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J.

Examples

#assumes SOA example life table to be load
data(soaLt)
soa08Act=with(soaLt, new("actuarialtable",interest=0.06,x=x,lx=Ix,name="SOA2008"))
#compare mx and qx 
mxt(soa08Act, 60,10)
qxt(soa08Act, 60,10)

Functions to switch from nominal / effective / convertible rates

Description

Functions to switch from nominal / effective / convertible rates

Usage

nominal2Real(i, k = 1, type = "interest")

convertible2Effective(i, k = 1, type = "interest")

real2Nominal(i, k = 1, type = "interest")

effective2Convertible(i, k = 1, type = "interest")

Arguments

Details

effective2Convertible and convertible2Effective wrap the other two functions.

Value

A numeric value.

Note

Convertible rates are synonims of nominal rates

References

Broverman, S.A., Mathematics of Investment and Credit (Fourth Edition), 2008, ACTEX Publications.

See Also

real2Nominal

Examples

#a nominal rate of 0.12 equates an APR of
nominal2Real(i=0.12, k = 12, "interest")

Plot a multiple decrement table

Description

S4 method for the plot generic: visualises the decrement structure of an mdt object. Three views are available: a stacked-area chart of death counts d^{(j)}_x (default), a stacked bar chart, or a line chart of decrement probabilities q^{(j)}_x = d^{(j)}_x / l^{(\tau)}_x.

Usage

## S4 method for signature 'mdt,missing'
plot(x, y, type = c("area", "bar", "probability"), ...)

Arguments

Value

A ggplot2 object (returned invisibly).

Examples

valdezDf <- data.frame(
  x = 50:54,
  lx = c(4832555, 4821937, 4810206, 4797185, 4782737),
  heart = c(5168, 5363, 5618, 5929, 6277),
  accidents = c(1157, 1206, 1443, 1679, 2152),
  other = c(4293, 5162, 5960, 6840, 7631))
valdezMdt <- new("mdt", name = "ValdezExample", table = valdezDf)
plot(valdezMdt)
plot(valdezMdt, type = "bar")
plot(valdezMdt, type = "probability")

Present value of a series of cash flows

Description

This function evaluates the present values of a series of cash flows, given occurrence time. Probabilities of occurrence can also be taken into account.

Usage

presentValue(cashFlows, timeIds, interestRates, probabilities, power = 1)

Arguments

Details

Present value of a series of cash flows.

Evaluate the present value (or actuarial present value when probabilities are provided) of a vector of cash flows occurring at given time instants and discounted by provided interest rates.

probabilities is optional; when omitted a sequence of 1's with the same length as timeIds is assumed. Interest rate may be a fixed number or a vector of the same size as timeIds. The power parameter is normally unused except in special actuarial evaluations.

Value

A numeric scalar representing the present value of the cash flow vector, or the actuarial present value if probabilities are provided.

Note

This simple function is the kernel working core of the package. Actuarial and financial mathematics ground on it.

Author(s)

Giorgio A. Spedicato

References

Broverman, S.A., Mathematics of Investment and Credit (Fourth Edition), 2008, ACTEX Publications.

Examples

# simple example
cf <- c(10,10,10)    # $10 payments one per year for three years
t  <- c(1,2,3)       # years
p  <- c(1,1,1)       # payments certainty
presentValue(cashFlows = cf, timeIds = t, interestRates = 0.03, probabilities = p)

Life table from probabilities

Description

This function returns a newly created lifetable object given either survival or death (one year) probabilities)

Usage

probs2lifetable(probs, radix = 10000, type = "px", name = "ungiven")

Arguments

Details

The \omega value is the length of the probs vector.

Value

A lifetable object.

Warning

The function is provided as is, without any guarantee regarding the accuracy of calculation. We disclaim any liability for eventual losses arising from direct or indirect use of this software.

Note

This function allows to use mortality projection given by other softwares with the lifecontingencies package.

Author(s)

Giorgio A. Spedicato

References

Actuarial Mathematics (Second Edition), 1997, by Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J.

See Also

actuarialtable

Examples

fakeSurvivalProbs=seq(0.9,0,by=-0.1)
newTable=probs2lifetable(fakeSurvivalProbs,type="px",name="fake")
head(newTable)
tail(newTable)

Functions to evaluate survival, death probabilities and deaths.

Description

These functions evaluate raw survival and death probabilities between age x and x+t

Usage

dxt(object, x, t, decrement)
pxt(object, x, t, fractional = "linear", decrement)
qxt(object, x, t, fractional = "linear", decrement)

Arguments

Details

Fractional assumptions are:

• linear: linear interpolation between consecutive ages, i.e. assume uniform distribution.

• constant force of mortality : constant force of mortality, also known as exponential interpolation.

• hyperbolic: Balducci assumption, also known as harmonic interpolation.

Note that fractional="uniform", "exponential", "harmonic" or "Balducci" is also authorized. See references for details.

dxt, pxt and qxt are S4 generics with methods for lifetable (and hence actuarialtable) and mdt objects; other packages can add methods for new table classes.

For mdt objects with a decrement, {}_tq_x^{(j)} = {}_td_x^{(j)} / l_x^{(\tau)} and pxt returns its complement 1 - {}_tq_x^{(j)}; ages must be tabulated integers and fractional t is interpolated linearly (UDD) within the year.

Value

A numeric value representing requested probability.

Warning

The function is provided as is, without any warranty regarding the accuracy of calculations. The author disclaims any liability for eventual losses arising from direct or indirect use of this software.

Note

Function dxt accepts also fractional value of t. Linear interpolation is used in such case. These functions are called by many other functions.

Author(s)

Giorgio A. Spedicato

References

Actuarial Mathematics (Second Edition), 1997, by Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J.

See Also

exn, lifetable

Examples

	#dxt example
	data(soa08Act)
	dxt(object=soa08Act, x=90, t=2)
	#qxt example
	qxt(object=soa08Act, x=90, t=2)
	#pxt example
	pxt(object=soa08Act, x=90, t=2, "constant force" )
	#MDT example (Valdez)
	valdezMdt <- new("mdt", name = "ValdezExample", table = data.frame(
	  x = 50:54, lx = c(4832555, 4821937, 4810206, 4797185, 4782737),
	  heart = c(5168, 5363, 5618, 5929, 6277),
	  accidents = c(1157, 1206, 1443, 1679, 2152),
	  other = c(4293, 5162, 5960, 6840, 7631)))
	qxt(valdezMdt, x = 50, t = 3, decrement = "heart")
	qxt(valdezMdt, x = 50, t = 3, decrement = c("heart", "accidents"))
	pxt(valdezMdt, x = 50, t = 3)

Functions to evaluate joint survival probabilities.

Description

These functions evaluate survival and death probabilities for two heads.

Usage

exyt(objectx, objecty, x, y, t, status = "joint")

pxyt(objectx, objecty, x, y, t, status = "joint")

qxyt(objectx, objecty, x, y, t,  status = "joint")

Arguments

Value

A numeric value representing joint survival probability.

Warning

The function is provided as is, without any warranty regarding the accuracy of calculations. The author disclaims any liability for eventual losses arising from direct or indirect use of this software. Also it is being Deprecated and asap removed from the package.

Note

These functions are used to evaluate two or more life contingencies.

Author(s)

Giorgio A. Spedicato, Kevin J. Owens.

References

Actuarial Mathematics (Second Edition), 1997, by Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J.

See Also

exyt

Examples

## Not run: 
data(soa08Act)
pxyt(soa08Act, soa08Act, 65, 70,10)
pxyt(soa08Act, soa08Act, 65, 70,10, "last")

## End(Not run)

Quantiles of the age-at-death distribution of a life table

Description

S4 method for the quantile generic: quantiles of the age at death implied by a lifetable, obtained by linear interpolation on the survivorship column l_x.

Usage

## S4 method for signature 'lifetable'
quantile(x, probs = seq(0, 1, 0.25), age = min(x@x), names = TRUE, ...)

Arguments

Value

A numeric vector of age-at-death quantiles.

See Also

median, modalAge

Examples

data(soa08Act)
quantile(soa08Act)
quantile(soa08Act, probs = c(0.1, 0.9), age = 65)

Death Probabilities to Mortality Rates

Description

Function to convert death probabilities to mortality rates

Usage

qx2mx(qx, ax = 0.5)

Arguments

Details

Function to convert death probabilities to mortality rates

Value

A vector of mortality rates

See Also

mxt, qxt, mx2qx

Examples

data(soa08Act)
soa08qx<-as(soa08Act,"numeric")
soa08mx<-qx2mx(qx=soa08qx)
soa08qx2<-mx2qx(soa08mx)

Return Associated single decrement from absolute rate of decrement

Description

Return Associated single decrement from absolute rate of decrement

Usage

qxt.prime.fromMdt(object, x, t = 1, decrement)

qxt.fromQxprime(qx.prime, other.qx.prime, t = 1)

Arguments

Value

a single value (AST)

Functions

• qxt.fromQxprime(): Obtain decrement from single decrements

Examples

#Creating the valdez mdf

valdezDf<-data.frame(
x=c(50:54),
lx=c(4832555,4821937,4810206,4797185,4782737),
hearth=c(5168, 5363, 5618, 5929, 6277),
accidents=c(1157, 1206, 1443, 1679,2152),
other=c(4293,5162,5960,6840,7631))
valdezMdt<-new("mdt",name="ValdezExample",table=valdezDf) 

qxt.prime.fromMdt(object=valdezMdt,x=53,decrement="other")

#Finan example 67.2

qxt.fromQxprime(qx.prime = 0.01,other.qx.prime = c(0.03,0.06))

Function to generate samples from the life contingencies stochastic variables

Description

Function to generate samples from the life contingencies stochastic variables

Usage

rLifeContingencies(
  n,
  lifecontingency,
  object,
  x,
  t,
  i = object@interest,
  m = 0,
  k = 1,
  parallel = FALSE,
  nthreads = NULL,
  payment = "advance"
)

rLifeContingenciesXyz(
  n,
  lifecontingency,
  tablesList,
  x,
  t,
  i,
  m = 0,
  k = 1,
  status = "joint",
  parallel = FALSE,
  nthreads = NULL,
  payment = "advance"
)

Arguments

Value

A numeric vector

Examples

## Not run: 
	#assumes SOA example life table to be load
	data(soaLt)
	soa08Act=with(soaLt, new("actuarialtable",interest=0.06, x=x,lx=Ix,name="SOA2008"))
	out<-rLifeContingencies(n=1000, lifecontingency="Axn",object=soa08Act, x=40,
	t=getOmega(soa08Act)-40, m=0)
	APV=Axn(soa08Act,x=40)
	#check if out distribution is unbiased
	t.test(x=out, mu=APV)$p.value>0.05

## End(Not run)
## Not run: 
data(soa08Act)
n=10000
lifecontingency="Axyz"
tablesList=list(soa08Act,soa08Act)
x=c(60,60); i=0.06; m=0; status="joint"; t=30; k=1
APV=Axyzn(tablesList=tablesList,x=x,n=t,m=m,k=k,status=status,type="EV")
samples<-rLifeContingenciesXyz(n=n,lifecontingency = lifecontingency,tablesList = tablesList,
x=x,t=t,m=m,k=k,status=status, parallel=FALSE)
APV
mean(samples)

## End(Not run) 

Function to generate random future lifetimes

Description

Function to generate random future lifetimes

Usage

rLife(n, object, x = 0, k = 1, type = "Tx")

rLifexyz(n, tablesList, x, k = 1, type = "Tx")

Arguments

Details

Under the uniform distribution of deaths assumption the complete future lifetime is the curtate one plus an independent uniform fraction of year, T_x=K_x+U with U\sim Unif(0,1), hence E[T_x]=E[K_x]+0.5. For type="Tx" the function returns K_x+0.5/k, i.e. a deterministic mid-period fraction, so only the expected value (not the full distribution) of T_x is reproduced.

Value

A numeric vector of n elements.

Note

The function is provided as is, without any warranty regarding the accuracy of calculations. The author disclaims any liability for eventual losses arising from direct or indirect use of this software.

References

Actuarial Mathematics (Second Edition), 1997, by Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J.

See Also

lifetable, exn

Examples

## Not run: 
##get 20000 random future lifetimes for the Soa life table at birth
data(soa08Act)
lifes=rLife(n=20000,object=soa08Act, x=0, type="Tx")
# check if the expected life at birth derived from the life table is statistically equal
# to the expected value of the sample
t.test(x=lifes, mu=exn(soa08Act, x=0, type="continuous"))

## End(Not run)
## Not run: 
#assessment of curtate expectation of future lifetime of the joint-life status
#generate a sample of lifes
data(soaLt)
soa08Act=with(soaLt, new("actuarialtable",interest=0.06,x=x,lx=Ix,name="SOA2008"))
tables=list(males=soa08Act, females=soa08Act)
xVec=c(60,65)
test=rLifexyz(n=50000, tablesList = tables,x=xVec,type="Kx")
#check first survival status
t.test(x=apply(test,1,"min"),mu=exyzt(tablesList=tables, x=xVec,status="joint"))
#check last survival status
t.test(x=apply(test,1,"max"),mu=exyzt(tablesList=tables, x=xVec,status="last"))

## End(Not run)

Simulate from a multiple decrement table

Description

Simulate from a multiple decrement table

Usage

rmdt(n = 1, object, x = 0, t = 1, t0 = "alive", include.t0 = TRUE)

Arguments

Value

A matrix with n columns (the length of simulation) and either t (if initial status is not included) or t+1 rows.

Details

The function uses rmarkovchain from the optional markovchain package to simulate the chain: it stops with an informative error if that package is not installed.

Author(s)

Giorgio Alfredo Spedicato

See Also

rLifeContingenciesXyz,rLifeContingencies

Examples

mdtDf<-data.frame(x=c(0,1,2,3),death=c(100,50,30,10),lapse=c(150,20,2,0))
myMdt<-new("mdt",name="example Mdt",table=mdtDf)
if (requireNamespace("markovchain", quietly = TRUE)) {
  ciao<-rmdt(n=5,object = myMdt,x = 0,t = 4,include.t0=FALSE,t0="alive")
}

Society of Actuaries Illustrative Life Table object.

Description

This is the table that appears in the classical book Actuarial Mathematics in Appendix 2A and used throughout the book to illustrate life contingent calculations. The Society of Actuaries has been using this table when administering US actuarial professional MLC preliminary examinations.

Usage

data(soa08)

Format

Formal class 'lifetable' [package "lifecontingencies"] with 3 slots ..@ x : int [1:141] 0 1 2 3 4 5 6 7 8 9 ... ..@ lx : num [1:141] 100000 97958 97826 97707 97597 ... ..@ name: chr "SOA Illustrative Life Table"

Details

This table is a blend of Makeham's mortality law for ages 13 and above and some ad hoc values for ages 0 to 12.

The parameters for Makeham's mortality law are

1000 * mu(x) = 0.7 + 0.05 * 10^(0.04 * x)

where mu(x) is the force of mortality.

The published Illustrative Life Table just shows ages 0 to 110 but in the computing exercises of chapter 3 the authors explain that the table's age range is from 0 to 140.

Note

This table is based on US 1990 general population mortality.

References

Actuarial Mathematics (Second Edition), 1997, by Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J.

Examples

data(soa08)
## maybe str(soa08) ; plot(soa08) ...

Society of Actuaries Illustrative Life Table with interest rate at 6

Description

An object of class actuarialtable built from the SOA illustrative life table. Interest rate is 6

Usage

data(soa08Act)

Format

Formal class 'actuarialtable' [package "lifecontingencies"] with 4 slots ..@ interest: num 0.06 ..@ x : int [1:141] 0 1 2 3 4 5 6 7 8 9 ... ..@ lx : num [1:141] 100000 97958 97826 97707 97597 ... ..@ name : chr "SOA Illustrative Life Table"

Details

This table is a blend of Makeham's mortality law for ages 13 and above and some ad hoc values for ages 0 to 12.

The parameters for Makeham's mortality law are

1000 * mu(x) = 0.7 + 0.05 * 10^(0.04 * x)

where mu(x) is the force of mortality.

The published Illustrative Life Table just shows ages 0 to 110 but in the computing exercises of chapter 3 the authors explain that the table's age range is from 0 to 140.

References

Actuarial Mathematics (Second Edition), 1997, by Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J.

Examples

## Not run: 
	data(soa08Act)

## End(Not run)

Society of Actuaries life table

Description

This table has been used by the classical book Actuarial Mathematics and by the Society of Actuaries for US professional examinations.

Usage

data(soaLt)

Format

A data.frame with 111 obs on the following 2 variables:

x

a numeric vector

Ix

a numeric vector

Details

Early ages have been found elsewere since miss in the original data sources; SOA did not provide population at risk data for certain spans of age (e.g. 1-5, 6-9, 11-14 and 16-19)

References

Actuarial Mathematics (Second Edition), 1997, by Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J.

Examples

data(soaLt)
head(soaLt)

Variance and standard deviation of the future lifetime

Description

varxn and sdxn return the variance and the standard deviation of the future lifetime of a life aged x, as the natural second-moment companions of exn (which returns the mean).

Usage

varxn(object, x, n, type = "Kx")

sdxn(object, x, n, type = "Kx")

Arguments

Details

For the curtate future lifetime the (temporary) variance is computed exactly from the life table as

\mathrm{Var}(\min(K_x, n)) = \sum_{k=1}^{n}(2k-1)\,{}_k p_x - \left(\sum_{k=1}^{n}{}_k p_x\right)^2 .

For the complete future lifetime the variance is returned only for the whole remaining lifespan (n missing), using the uniform-distribution-of-deaths relation \mathrm{Var}(T_x) = \mathrm{Var}(K_x) + 1/12 (because, under UDD, T_x = K_x + U with U\sim\mathrm{Unif}(0,1) independent of K_x). A temporary complete variance is not provided; use type = "Kx" for a temporary period.

Value

A numeric value: the variance (varxn) or standard deviation (sdxn) of the (temporary) future lifetime.

Author(s)

Giorgio Alfredo Spedicato

References

Dickson, D.C.M., Hardy, M.R., Waters, H.R. (2013), Actuarial Mathematics for Life Contingent Risks, 2nd ed., Cambridge University Press.

See Also

exn

Examples

data(soa08Act)
# variance and sd of the curtate future lifetime at birth
varxn(soa08Act, x = 0)
sdxn(soa08Act, x = 0)
# complete future lifetime: Var(T) = Var(K) + 1/12
varxn(soa08Act, x = 65, type = "complete")
# temporary (20-year) curtate future lifetime at age 40
varxn(soa08Act, x = 40, n = 20)