Package 'lifecontingencies' reference manual

Title:	Financial and Actuarial Mathematics for Life Contingencies
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>.
Authors:	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]
Maintainer:	Giorgio Alfredo Spedicato <[email protected]>
License:	MIT + file LICENSE
Version:	1.3.12
Built:	2025-02-26 05:33:21 UTC
Source:	https://github.com/spedygiorgio/lifecontingencies

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: <[email protected]>

References

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

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) 

##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)

Function to evaluate the accumulated value.

Description

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

Usage

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

Arguments

`i`	Effective interest rate expressed in decimal form. E.g. 0.03 means 3%.
`n`	Number of terms of payment.
`m`	Deferring period, whose default value is zero.
`k`	Frequency of payment.
`type`	The Payment type, either `"advance"` for the annuity due (default) or `"arrears"` for the annuity immediate. Alternatively, one can use `"due"` or `"immediate"` respectively (can be abbreviated).

Details

The accumulated value is the future value of the terms of an annuity. Its mathematical expression is $s_{\left. {\overline {\, n \,}}\! \right| } = \left( {1 + i} \right)^n a_{\left. {\overline {\, n \,}}\! \right| }$

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 value 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.

Examples

#A man wants to save 100,000 to pay for his sons
#education in 10 years time. An education fund requires the investors to
#deposit equal installments annually at the end of each year. If interest of
#0.075 is paid, how much does the man need to save each year in order to
#meet his target?
R=100000/accumulatedValue(i=0.075,n=10)

#A man wants to save 100,000 to pay for his sons
#education in 10 years time. An education fund requires the investors to
#deposit equal installments annually at the end of each year. If interest of
#0.075 is paid, how much does the man need to save each year in order to
#meet his target?
R=100000/accumulatedValue(i=0.075,n=10)

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.

Examples

showClass("actuarialtable")
showClass("actuarialtable")

Function to evaluate the n-year endowment insurance

Description

This function evaluates the n-year endowment insurance.

Usage

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

Arguments

`actuarialtable`	An actuarial table object.
`x`	Insured age.
`n`	Length of the insurance.
`i`	Rate of interest. When missing the one included in the actuarialtable object is used.
`k`	Frequency of benefit payment.
`type`	A string, either `"EV"` for expected value of the actuarial present value (default) or `"ST"` for one stochastic realization of the underlying present value of benefits. Alternatively, one can use `"expected"` or `"stochastic"` respectively (can be abbreviated).
`power`	The power of the APV. Default is 1 (mean)

Details

The n-year endowment insurance provides a payment either in the year of death or at the end of the insured period.

Value

A numeric value.

Note

When type="EV" the function calls both Axn and Exn.

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.

Examples

#Actuarial Mathematics book example
#check the actuarial equality on the expected values Exn+Axn=AExn
data(soa08Act)
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)
#Actuarial Mathematics book example
#check the actuarial equality on the expected values Exn+Axn=AExn
data(soa08Act)
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)

Annuity function

Description

Function to calculate present value of annuities-certain.

Usage

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

Arguments

`i`	Effective interest rate expressed in decimal form. E.g. 0.03 means 3%. It can be a vector of interest rates of the same length of periods.
`n`	Periods for payments. If n = `infinity` then `annuity` returns the value of a perpetuity (either immediate or due).
`m`	Deferring period, whose default value is zero.
`k`	Yearly payments frequency. A payment of $k^-1$ is supposed to be performed at the end of each year.
`type`	The Payment type, either `"advance"` for the annuity due (default) or `"arrears"` for the annuity immediate. Alternatively, one can use `"due"` or `"immediate"` respectively (can be abbreviated).

Details

This function calculates the present value of a stream of fixed payments separated by equal interval of time. Annuity immediate has the fist payment at time t = 0, while an annuity due has the first payment at time t = 1.

Value

A string, either "immediate" or "due".

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.

Examples

# The present value of 5 payments of 1000 at one year interval that begins
# now when the interest rate is 2.5% is
1000 * annuity(i = 0.025, n = 5, type = "due")
# A man borrows a loan of 20,000 to purchase a car at
# a nominal annual rate of interest of 0.06. He will pay back the loan through monthly
# installments over 5 years, with the first installment to be made one month
# after the release of the loan. What is the monthly installment he needs to pay?
20000 / annuity(i = 0.06 / 12, n = 5 * 12)
# The present value of 5 payments of 1000 at one year interval that begins
# now when the interest rate is 2.5% is
1000 * annuity(i = 0.025, n = 5, type = "due")
# A man borrows a loan of 20,000 to purchase a car at
# a nominal annual rate of interest of 0.06. He will pay back the loan through monthly
# installments over 5 years, with the first installment to be made one month
# after the release of the loan. What is the monthly installment he needs to pay?
20000 / annuity(i = 0.06 / 12, n = 5 * 12)

Annuity immediate and due function.

Description

This function calculates actuarial value of annuities, given an actuarial table. Fractional and deferred annuities can be evaluated. Moreover it can be used to simulate the stochastic distribution of the annuity value.

Usage

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,payment = "advance", ...)

Arguments

`actuarialtable`	An actuarial table object.
`x`	Age of the annuitant. (can be a vector).
`n`	Number of terms of the annuity, if missing annuity is intended to be paid until death. (can be a vector).
`i`	Interest rate (default value the interest of the life table). (should be a scalar).
`m`	Deferring period. Assumed to be 1 whether missing. (can be a vector).
`k`	Number of fractional payments per period. Assumed to be 1 whether missing. (should be a scalar).
`type`	A string, either `"EV"` for expected value of the actuarial present value (default) or `"ST"` for one stochastic realization of the underlying present value of benefits. Alternatively, one can use `"expected"` or `"stochastic"` respectively (can be abbreviated).
`power`	The power of the APV. Default is 1 (mean)
`payment`	The Payment type, either `"advance"` for the annuity due (default) or `"arrears"` for the annuity immediate. Alternatively, one can use `"due"` or `"immediate"` respectively (can be abbreviated).
`...`	Arguments to be passed to `pxt()`.

Details

When "ST" has been selected a stochastic value representing a number drawn from the domain of

$a_{x}^{n}$

is drawn. "EV" calculates the classical APV.

Value

A numeric value.

Warning

Note

When either $x=\omega$ or $n=0$ zero is returned.

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.

Examples

		#assume SOA example life table to be load
		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) 
#assume SOA example life table to be load
		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)

Function to evaluate life insurance.

Description

This function evaluates n - years term and whole life insurance.

Usage

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

Arguments

`actuarialtable`	An actuarial table object.
`x`	Age of the insured. (can be a vector).
`n`	Coverage period, if missing the insurance is considered whole life $n=\omega-x-m$ . (can be a vector).
`i`	Interest rate (overrides the interest rate slot in `actuarialtable`). (should be a scalar).
`m`	Deferring period, even fractional, if missing assumed to be 0. (can be a vector).
`k`	Number of periods per year at the end of which the capital is payable in case of insured event, default=1 (capital payable at the end of death year). (should be a scalar).
`type`	A string, either `"EV"` for expected value of the actuarial present value (default) or `"ST"` for one stochastic realization of the underlying present value of benefits. Alternatively, one can use `"expected"` or `"stochastic"` respectively (can be abbreviated).
`power`	The power of the APV. Default is 1 (mean)
`...`	Arguments to be passed to `pxt()`.

Details

The variance calculation has not been implemented yet.

Value

A numeric value representing either the actuarial value of the coverage (when type="EV") or a number drawn from the underlying distribution of Axn.

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

It is possible that value returned by stochastic simulation are biased. Successive releases of this software will analyze the issue with detail.

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.

Examples

	
		#assume SOA example life table to be load
		data(soaLt)
		soa08Act=with(soaLt, new("actuarialtable",interest=0.06,
		x=x,lx=Ix,name="SOA2008"))
		#evaluate the value of a 40 years term life insurance for an aged 25
		Axn(actuarialtable=soa08Act, x=25, n=40) 
		#check an relevant life contingencies relationship
		k=12
		i=0.06
		j=real2Nominal(i,k)
		Axn(soa08Act, 30,k=12)
		i/j*Axn(soa08Act, 30,k=1)
	#assume SOA example life table to be load
		data(soaLt)
		soa08Act=with(soaLt, new("actuarialtable",interest=0.06,
		x=x,lx=Ix,name="SOA2008"))
		#evaluate the value of a 40 years term life insurance for an aged 25
		Axn(actuarialtable=soa08Act, x=25, n=40) 
		#check an relevant life contingencies relationship
		k=12
		i=0.06
		j=real2Nominal(i,k)
		Axn(soa08Act, 30,k=12)
		i/j*Axn(soa08Act, 30,k=1)

Multiple decrement life insurance

Description

Function to evaluate multiple decrement insurances

Usage

Axn.mdt(object, x, n, i, decrement)
Axn.mdt(object, x, n, i, decrement)

Arguments

`object`	an `mdt` or `actuarialtable` object
`x`	policyholder's age
`n`	contract duration
`i`	interest rate
`decrement`	decrement category

Value

The scalar representing APV of the insurance

Warning

The function is experimental and very basic. Testing is still needed. Use at own risk!

Examples

#creates a temporary mdt
myTable<-data.frame(x=41:43,lx=c(800,776,752),d1=rep(8,3),d2=rep(16,3))
myMdt<-new("mdt",table=myTable,name="ciao")
Axn.mdt(myMdt, x=41,n=2,i=.05,decrement="d2")
#creates a temporary mdt
myTable<-data.frame(x=41:43,lx=c(800,776,752),d1=rep(8,3),d2=rep(16,3))
myMdt<-new("mdt",table=myTable,name="ciao")
Axn.mdt(myMdt, x=41,n=2,i=.05,decrement="d2")

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")
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

`tablex`	Life X `lifetable` object.
`tabley`	Life Y `lifetable` object.
`x`	Age of life X.
`y`	Age of life Y.
`n`	Insured duration. Infinity if missing.
`i`	Interest rate. Default value is those implied in `actuarialtable`.
`m`	Deferring period. Default value is zero.
`k`	Fractional payments or periods where insurance is payable.
`status`	Either `"joint"` for the joint-life status model or `"last"` for the last-survivor status model (can be abbreviated).
`type`	A string, either `"EV"` for expected value of the actuarial present value (default) or `"ST"` for one stochastic realization of the underlying present value of benefits. Alternatively, one can use `"expected"` or `"stochastic"` respectively (can be abbreviated).
`payment`	The Payment type, either `"advance"` for the annuity due (default) or `"arrears"` for the annuity immediate. Alternatively, one can use `"due"` or `"immediate"` respectively (can be abbreviated).

Details

Actuarial mathematics book formulas has been implemented.

Value

A numeric value returning APV of chosen insurance form.

Warning

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.

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)
## 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)

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")
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

`tablesList`	A list whose elements are either lifetable or actuarialtable class objects.
`x`	A vector of the same size of tableList that contains the initial ages.
`n`	Lenght of the insurance.
`i`	Interest rate
`m`	Deferring period.
`k`	Fractional payment frequency.
`status`	Either `"joint"` for the joint-life status model or `"last"` for the last-survivor status model (can be abbreviated).
`type`	A string, either `"EV"` for expected value of the actuarial present value (default) or `"ST"` for one stochastic realization of the underlying present value of benefits. Alternatively, one can use `"expected"` or `"stochastic"` respectively (can be abbreviated).
`power`	The power of the APV. Default is 1 (mean).
`payment`	The Payment type, either `"advance"` for the annuity due (default) or `"arrears"` for the annuity immediate. Alternatively, one can use `"due"` or `"immediate"` respectively (can be abbreviated).

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.

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")	
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")

Decreasing life insurance

Description

This function evaluates the n-year term decreasing life insurance. Both actuarial value and stochastic random sample can be returned.

Usage

	DAxn(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

`actuarialtable`	An actuarial table object.
`x`	Age of the insured.
`n`	Length of the insurance period.
`i`	Interest rate, when present it overrides the interest rate of the actuarial table object.
`m`	Deferring period, even fractional, assumed 1 whether missing.
`k`	Number of fractional payments per period. Assumed to be 1 whether missing.
`type`	A string, either `"EV"` for expected value of the actuarial present value (default) or `"ST"` for one stochastic realization of the underlying present value of benefits. Alternatively, one can use `"expected"` or `"stochastic"` respectively (can be abbreviated).
`power`	The power of the APV. Default is 1 (mean)

Details

Formulas of Bowes book have been implemented.

Value

A numeric value representing the expected value or the simulated 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

Neither fractional payments nor stochastic calculations have been implemented yet.

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.

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 10 years decreasing term life insurance for an aged 25
		DAxn(actuarialtable=soa08Act, x=25, n=10) 
#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 10 years decreasing term life insurance for an aged 25
		DAxn(actuarialtable=soa08Act, x=25, n=10)

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
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 return present values for decreasing annuities - certain.

Usage

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

Arguments

`i`	A numeric value representing the interest rate.
`n`	The number of periods.
`type`	The Payment type, either `"advance"` for the annuity due (default) or `"arrears"` for the annuity immediate. Alternatively, one can use `"due"` or `"immediate"` respectively (can be abbreviated).

Details

A decreasing annuity has the following flows of payments: n, n-1, n-2, ..., 1, 0.

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.

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
	#should be
	sum((10:1)/(1+.03)^(1:10))
#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
	#should be
	sum((10:1)/(1+.03)^(1:10))

Canada Mortality Rates for UP94 Series

Description

UP94 life tables underlying mortality rates

Usage

data(demoCanada)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$up94M,radix=100000,"qx",name="UP94")
#create the up94M actuarial table table
up94MAct<-new("actuarialtable", lx=up94MLt@lx, x=up94MLt@x,interest=0.02)
data(demoCanada)
head(demoCanada)
#create the up94M life table
up94MLt<-probs2lifetable(probs=demoCanada$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)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$CL1,radix=1000,type="qx",name="CHINA CL1")
data(demoChina)
tableChinaCL1<-probs2lifetable(probs=demoChina$CL1,radix=1000,type="qx",name="CHINA CL1")

French population life tables

Description

Illustrative life tables from French population.

Usage

data(demoFrance)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)
data(demoFrance)
head(demoFrance)

German population life tables

Description

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

Usage

data(demoGermany)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)
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)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)
#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)
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)
data(demoJapan)
head(demoJapan)

UK life tables

Description

AM and AF one year mortality rate. Series of 1992

Usage

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

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

Arguments

`cashFlows`	A vector representing the cash flows amounts.
`timeIds`	Cash flows times
`i`	APR interest, i.e. nominal interest rate compounded m-thly.
`k`	Compounding frequency for the nominal interest rate.
`macaulay`	Use the Macaulay formula

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)
#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)

Expected residual life.

Description

Expected residual life.

Usage

exn(object, x, n, type = "curtate")
exn(object, x, n, type = "curtate")

Arguments

`object`	A lifetable/actuarialtable object.
`x`	Attained age
`n`	Time until which the expected life should be calculated. Assumed omega - x whether missing.
`type`	Either `"Tx"`, `"complete"` or `"continuous"` for continuous future lifetime, `"Kx"` or `"curtate"` for curtate furture lifetime (can be abbreviated).

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.

Examples

#loads and show
data(soa08Act)
exn(object=soa08Act, x=0)
exn(object=soa08Act, x=0,type="complete")
#loads and show
data(soa08Act)
exn(object=soa08Act, x=0)
exn(object=soa08Act, x=0,type="complete")

Function to evaluate the pure endowment

Description

Function to evaluate the pure endowment

Usage

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

Arguments

`actuarialtable`	An actuarial table object.
`x`	Age of the insured.
`n`	Length of the contract.
`i`	Interest rate (it overwrites the `actuarialtable` one)
`type`	A string, eithed "EV" (default value), "ST" (stocastic realization) or "VR" if the value of the variance is needed.
`power`	The power of the APV. Default is 1 (mean)

Value

The APV of the contract

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.

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"))
#evaluate the pure endowment for a man aged 30 for a time span of 35
Exn(soa08Act, x=30, n=35) 
#assumes SOA example life table to be load
data(soaLt)
soa08Act=with(soaLt, new("actuarialtable",interest=0.06, x=x,lx=Ix,name="SOA2008"))
#evaluate the pure endowment for a man aged 30 for a time span of 35
Exn(soa08Act, x=30, n=35)

Function to return the decrements defined in the mdt class

Description

This function list the character decrements of the mdf class

Usage

getDecrements(object)
getDecrements(object)

Arguments

object

A mdt class object

Details

A character vector is returned

Value

A character vector listing the decrements defined in the class

Note

To be updated

Author(s)

Giorgio Spedicato

References

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

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)
#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")
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

`deathsTimeX`	Time to death
`lifecontingency`	lifecontingency symbol
`object`	life table(s)
`x`	age(s) of the policyholder(s)
`t`	term of the contract
`i`	interest rate
`m`	deferrement
`k`	fractional payments
`payment`	The Payment type, either `"advance"` for the annuity due (default) or `"arrears"` for the annuity immediate. Alternatively, one can use `"due"` or `"immediate"` respectively (can be abbreviated).
`deathsTimeXyz`	matrix of death times from birth
`tablesList`	list of table of the same size of num column of deathTimeXyz.
`status`	Either `"joint"` for the joint-life status model or `"last"` for the last-survivor status model (can be abbreviated).

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

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.

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)
#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)
getOmega(object)

Arguments

object

A life table object.

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.

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) 

#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 life contingencies

Description

This function evaluates increasing annuities

Usage

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

Arguments

`actuarialtable`	An `actuarialtable` object.
`x`	The age of the insured head.
`n`	The duration of the insurance
`i`	The interest rate that overrides the one in the `actuarialtable` object.
`m`	The deferring period.
`type`	Yet only "EV" is implemented.
`power`	The power of the APV. Default is 1 (mean)

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.

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)
#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)

Increasing life insurance

Description

This function evaluates the APV of an increasing life insurance. The amount payable at the end of year of death are: $1, 2, \ldots, n-1, n$ . N can be set as $\omega-x-1$ .

Usage

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

Arguments

`actuarialtable`	The actuarial table used to perform life - contingencies calculations.
`x`	The age of the insured.
`n`	The term of life insurance. If missing n is set as $n=\omega - x - m -1$ .
`i`	Interest rate (overrides the interest rate of the actuarialtable object).
`m`	The deferring period. If missing, m is set as 0.
`k`	Number of fractional payments per period. Assumed to be 1 whether missing.
`type`	A string, either `"EV"` for expected value of the actuarial present value (default) or `"ST"` for one stochastic realization of the underlying present value of benefits. Alternatively, one can use `"expected"` or `"stochastic"` respectively (can be abbreviated).
`power`	The power of the APV. Default is 1 (mean).

Details

The stochastic value feature has not been implemented yet.

Value

A numeric value.

Warning

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.

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"))
		#evaluate the value of a 10 years increasing term life insurance for an aged 25
		IAxn(actuarialtable=soa08Act, x=25, n=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"))
		#evaluate the value of a 10 years increasing term life insurance for an aged 25
		IAxn(actuarialtable=soa08Act, x=25, n=10)

Increasing annuity.

Description

This function evaluates non - stochastic increasing annuities.

Usage

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

Arguments

`i`	A numeric value representing the interest rate.
`n`	The number of periods.
`type`	The Payment type, either `"advance"` for the annuity due (default) or `"arrears"` for the annuity immediate. Alternatively, one can use `"due"` or `"immediate"` respectively (can be abbreviated).

Details

An increasing annuity shows the following flow of payments: $1,2,\ldots,n-1,n$

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.

Examples

	#the present value of 1,2,...,n-1, n sequence of payments, 
	#payable at the end of the period
	#for 10 periods is
	increasingAnnuity(i=0.03, n=10)
	#assuming a 3% interest rate
#the present value of 1,2,...,n-1, n sequence of payments, 
	#payable at the end of the period
	#for 10 periods is
	increasingAnnuity(i=0.03, n=10)
	#assuming a 3% interest rate

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)
intensity2Interest(intensity)

interest2Intensity(i)

Arguments

`intensity`	Intensity rate
`i`	interest rate

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.

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)
# 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)
interest2Discount(i)

discount2Interest(d)

Arguments

`i`	Interest rate
`d`	Discount rate

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.

Examples

discount2Interest(d=0.04)
discount2Interest(d=0.04)

Function to calculated accumulated increasing annuity future value.

Description

This function evaluates non - stochastic increasing annuities future values.

Usage

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

Arguments

`i`	Interest rate.
`n`	Terms.
`type`	Either "due" for annuity due or "immediate" for annuity immediate.

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.

Examples

Isn(n=10,i=0.03)
Isn(n=10,i=0.03)

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
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", y = "ANY"): 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

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.

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)
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)

Various demographic functions

Description

Various demographic functions

Usage

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

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

Tx(object, x)

Arguments

`object`	a `lifetable` or `actuarialtable` object
`x`	age of the subject
`t`	duration of the calculation
`fxt`	correction constant, default 0.5

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)
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)

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, ...). They store absolute decrements

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
coerce: signature(from = "mdt", to = "data.frame"): coercing to markovchainList objects
summary: signature(object = "mdt"): it returns summary information about the object

Note

Currently only decrements storage of the class is defined.

Author(s)

Giorgio Spedicato

References

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

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)
#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)

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)), ...)

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

`tablesList`	A list whose elements are either `lifetable` or `actuarialtable` class objects.
`x`	A vector of the same size of tableList that contains the initial ages.
`t`	The duration.
`status`	Either `"joint"` for the joint-life status model or `"last"` for the last-survivor status model (can be abbreviated).
`type`	Either `"Tx"` for continuous future lifetime, `"Kx"` for curtate furture lifetime (can be abbreviated).
`fractional`	Assumptions for fractional age. One of `"linear"`, `"hyperbolic"`, `"constant force"` (can be abbreviated).
`...`	Options to be passed to `pxt`.

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.

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"))
#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)
mx2qx(mx, ax = 0.5)

Arguments

`mx`	mortality rates vector
`ax`	the average number of years lived between ages x and x +1 by individuals who die in that interval

Details

Function to convert mortality rates to probabilities of death

Value

A vector of death probabilities

Examples


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

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

Central mortality rate

Description

This function returns the central mortality rate demographic function.

Usage

mxt(object, x, t)
mxt(object, x, t)

Arguments

`object`	a `lifetable` or `actuarialtable` object
`x`	subject's age
`t`	period on which the rate is evaluated

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)
#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")
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

`i`	The rate to be converted.
`k`	The original / target compounting frequency.
`type`	Either "interest" (default) or "nominal".

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.

Examples

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

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

Arguments

`cashFlows`	Vector of cashFlow, must be coherent with `timeIds`
`timeIds`	Vector of points of time where `cashFlows` are due.
`interestRates`	A numeric value or a time-size vector of interest rate used to discount cahs flow.
`probabilities`	Optional vector of probabilities.
`power`	Power to square discount and cash flows. Default is set to 1

Details

probabilities is optional, a sequence of 1 length of timeIds is assumed. Interest rate shall be a fixed number or a vector of the same size of timeIds. power parameters is generally useless beside life contingencies insurances evaluations.

Value

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

Warning

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 of payments one per year for three years
 t=c(1,2,3) #years
 p=c(1,1,1) #assume payments certainty
 #assume 3% of interest rate
presentValue(cashFlows=cf, timeIds=t, interestRates=0.03, probabilities=p)
#simple example
 cf=c(10,10,10)	#$10 of payments one per year for three years
 t=c(1,2,3) #years
 p=c(1,1,1) #assume payments certainty
 #assume 3% of interest rate
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")
probs2lifetable(probs, radix = 10000, type = "px", name = "ungiven")

Arguments

`probs`	A real valued vector representing either one year survival or death probabilities. The last value in the vector must be either 1 or 0, depending if it represents death or survival probabilities respectively.
`radix`	The radix of the life table.
`type`	Character value either "px" or "qx" indicating how probabilities must be interpreted.
`name`	The character value to be put in the corresponding slot of returned object.

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.

Examples

fakeSurvivalProbs=seq(0.9,0,by=-0.1)
newTable=probs2lifetable(fakeSurvivalProbs,type="px",name="fake")
head(newTable)
tail(newTable)
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)
dxt(object, x, t, decrement)
pxt(object, x, t, fractional = "linear", decrement)
qxt(object, x, t, fractional = "linear", decrement)

Arguments

`object`	A `lifetable` object.
`x`	Age of life `x`. (can be a vector for `pxt, qxt`).
`t`	Period until which the age shall be evaluated. Default value is 1. (can be a vector for `pxt, qxt`).
`fractional`	Assumptions for fractional age. One of `"linear"`, `"hyperbolic"`, `"constant force"` (can be abbreviated).
`decrement`	The reason of decrement (only for `mdt` class objects). Can be either an ordinal number or the name of decrement

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.

Value

A numeric value representing requested probability.

Warning

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.

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" )
	#add another example for MDT
#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" )
	#add another example for MDT

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")
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

`objectx`	`lifetable` for life X.
`objecty`	`lifetable` for life Y.
`x`	Age of life X.
`y`	Age of life Y.
`t`	Time until survival has to be evaluated.
`status`	Either `"joint"` for the joint-life status model or `"last"` for the last-survivor status model (can be abbreviated).

Value

A numeric value representing joint survival probability.

Warning

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.

Examples

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

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

## End(Not run)

Death Probabilities to Mortality Rates

Description

Function to convert death probabilities to mortality rates

Usage

qx2mx(qx, ax = 0.5)
qx2mx(qx, ax = 0.5)

Arguments

`qx`	death probabilities
`ax`	the average number of years lived between ages x and x +1 by individuals who die in that interval

Details

Function to convert death probabilities to mortality rates

Value

A vector of mortality rates

Examples

data(soa08Act)
soa08qx<-as(soa08Act,"numeric")
soa08mx<-qx2mx(qx=soa08qx)
soa08qx2<-mx2qx(soa08mx)
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)
qxt.prime.fromMdt(object, x, t = 1, decrement)

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

Arguments

`object`	a mdj object
`x`	age
`t`	period (default 1)
`decrement`	type (necessary)
`qx.prime`	single ASDT decrement of which corresponding decrement is desired
`other.qx.prime`	ASDT decrements other than `qx.prime`

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))

#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,
  payment = "advance"
)

rLifeContingenciesXyz(
  n,
  lifecontingency,
  tablesList,
  x,
  t,
  i,
  m = 0,
  k = 1,
  status = "joint",
  parallel = FALSE,
  payment = "advance"
)
rLifeContingencies(
  n,
  lifecontingency,
  object,
  x,
  t,
  i = object@interest,
  m = 0,
  k = 1,
  parallel = FALSE,
  payment = "advance"
)

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

Arguments

`n`	Size of sample
`lifecontingency`	A character string, either `"Exn"`, `"Axn"`, `"axn"`, `"IAxn"` or `"DAxn"`
`object`	An `actuarialtable` object.
`x`	Policyholder's age at issue time; for `rLifeContingenciesXyz` a numeric vector of the same length of `object`, containing the policyholders' ages
`t`	The lenght of the insurance. Must be specified according to the present value of benefits definition.
`i`	The interest rate, whose default value is the `actuarialtable` interest rate slot value.
`m`	Deferring period, default value is zero.
`k`	Fractional payment, default value is 1.
`parallel`	Uses the parallel computation facility.
`payment`	The Payment type, either `"advance"` for the annuity due (default) or `"arrears"` for the annuity immediate. Alternatively, one can use `"due"` or `"immediate"` respectively (can be abbreviated).
`tablesList`	A list of `actuarialtable` objects
`status`	Either `"joint"` for the joint-life status model or `"last"` for the last-survivor status model (can be abbreviated).

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

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

Arguments

`n`	Number of variates to generate
`object`	An object of class lifetable
`x`	The attained age of subject x, default value is 0
`k`	Number of periods within the year when it is possible death to happen, default value is 1
`type`	Either `"Tx"` for continuous future lifetime, `"Kx"` for curtate furture lifetime (can be abbreviated).
`tablesList`	An list of lifetables

Details

Following relation holds for the future life time: $T_x=K_x+0.5$

Value

A numeric vector of n elements.

Note

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

Arguments

`n`	Number of simulations.
`object`	The `mdt` object to simulate from.
`x`	the period to simulate from.
`t`	the period until to simulate.
`t0`	initial status (default is "alive").
`include.t0`	should initial status to be included (default is TRUE)?

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 functin uses rmarkovchain function from markovchain package to simulate the chain

Author(s)

Giorgio Spedicato

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)
ciao<-rmdt(n=5,object = myMdt,x = 0,t = 4,include.t0=FALSE,t0="alive")

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)
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)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) ...
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)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)
## Not run: 
	data(soa08Act)

## End(Not run)

SoA illustrative service table

Description

Bowers' book Illustrative Service Table

Usage

data(SoAISTdata)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)
data(SoAISTdata)
head(SoAISTdata)

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)
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)
data(soaLt)
head(soaLt)

Uk AM AF 92 life tables

Description

Uk AM AF life tables

Usage

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

Package 'lifecontingencies'

Help Index

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

Description

Details

Warning

Note

Author(s)

References

See Also

Examples

Function to evaluate the accumulated value.

Description

Usage

Arguments

Details

Value

Warning

Note

Author(s)

References

See Also

Examples

Class "actuarialtable"

Description

Objects from the Class

Slots

Extends

Methods

Warning

Note

Author(s)

References

See Also

Examples

Function to evaluate the n-year endowment insurance

Description

Usage

Arguments

Details

Value

Note

Author(s)

References

See Also

Examples

Annuity function

Description

Usage

Arguments

Details

Value

Note

Author(s)

References

See Also

Examples

Annuity immediate and due function.

Description

Usage

Arguments

Details

Value

Warning

Note

Author(s)

References

See Also

Examples

Function to evaluate life insurance.

Description

Usage

Arguments

Details

Value

Warning

Note

Author(s)

References

See Also

Class `"actuarialtable"`