In spedygiorgio/lifecontingencies: Financial and Actuarial Mathematics for Life Contingencies

library(knitr)
rm(list=ls())

Intro

• The lifecontingencies package [@spedLifecon] will be introduced.
• As first half 2017 it is the first R [@rSoftware] package merging demographic and financial mathematics function in order to perform actuarial evaluation of life contingent insurances (annuities, life insurances, endowments, etc).
• The applied examples will shown: how to load the R package, how to perform basic financial mathematics and demographic calculations, how to price and reserve financial products.

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• The final example will show how to mix lifecontingencies and demography [@demographyR] function to assess the mortality development impact on annuities.
• The interested readers are suggested to look to the package's vignettes (also appeared in the Journal of Statistical Sofware) for a broader overview. [@dickson2009actuarial; and @mazzoleni2000appunti] provide and introduction of Actuarial Mathematics theory.
• Also [@rmetrics1] and [@charpentierCAS] discuss the software.

Loading the package

• The package is loaded using

library(lifecontingencies) #load the package

• It requires a recent version of R (>=3.0) and the markovchain package [@markovchainR]. The development version of the package requires also Rcpp package [@RcppR].

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Package's Financial Mathematics Functions

Interest functions

• interest2Discount, discount2Interest: from interest to discount and reverse;
• interest2Intensity, intensity2Interest: from intensity to interest and reverse;
• convertible2Effective, effective2Convertible: from convertible interest rate to effective one.

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• $(1+i)=\left(1+{\frac{i^{{(m)}}}{m}}\right)^{{m}}=e^{\delta}$
• $e^{{\delta }}=\left(1-{\frac{d^{{(m)}}}{m}}\right)^{{-m}}=(1-d)^{{-1}}$

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#interest and discount rates
interest2Discount(i=0.03)
discount2Interest(interest2Discount(i=0.03))
#convertible and effective interest rates
convertible2Effective(i=0.10,k=4)

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Annuities and future values

• annuity: present value (PV) of an annuity;
• accumulatedValue: future value of constant cash flows;
• decreasingAnnuity, increasingAnnuity: increasing and decreasing annuities.

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• $a_{\lcroof{n}}={\frac {1-\left(1+i\right)^{{-n}}}{i}}$
• $s_{\lcroof{n}}=a_{\lcroof{n}}*(1+i)^n=\frac{(1+i)^n-1}{i}$
• $\ddot{a}{\lcroof{n}}=a{\lcroof{n}}*(1+i)$

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annuity(i=0.05,n=5) #due
annuity(i=0.05,n=5,m=1) #immediate
annuity(i=0.05,n=5,k=12) #due, with 
# fractional payemnts

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• $\frac{\ddot{a}_{\lcroof{n}} - n*v^n}{i}=\left( Ia \right)$
• $\frac{n-a_{\lcroof{n}}}{i}=\left( Da \right)$
• $\left( Ia_{n} \right) +\left( Da_{n} \right) = \left( n+1 \right)*a_{\lcroof{n}}$

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irate=0.04; term=10
increasingAnnuity(i=irate,n=term)+decreasingAnnuity(i=irate,
n=term)-(term+1)*annuity(i=irate,n=term)

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Other functions

• presentValue: PV of possible varying CFs.
• duration, convexity: calculate duration and convexity of any stream of CFs.

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• $PV=\sum_{t \in T} CF_t*\left(1+i_t \right )^{-t}$
• $D={\frac{1}{P(0)}}\sum _{t=\tau }^{T}t{\frac{c_t}{(1+r)^{t}}}$
• $C=\frac{1}{P(1+r)^{2}} \sum {i=1}^{n}{\frac {t{i}(t_{i}+1)F_{i}}{\left(1+r\right)^{t_{i}}}}$

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#bond analysis
irate=0.03
cfs=c(10,10,10,100)
times=1:4
#compute PV, Duration and Convexity
presentValue(cashFlows = cfs, 
timeIds = times, 
interestRates = irate)
duration(cashFlows = cfs, 
timeIds = times, i = irate)
convexity(cashFlows = cfs,
timeIds = times, i = irate)

Intro

• Lifecontingencies offers a wide set of functions for demographic analysis;
• Survival and death probabilities, expected residual lifetimes and other function can be easily modeled with the R package;
• Creation and manipulation of life table is easy as well.

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Package's demographic functions

Table creation and manipulation

• new lifetable or actuarialtable methods.
• print, plot show methods.
• probs2lifetable function to create tables from probabilities

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• ${ l_0, l_1, l_2, \ldots, l_{\omega} }$
• $L_{x}=\frac{l_{x}+l_{x+1}}{2}$
• $q_{x,t} = \frac{d_{x+t}}{l_{x}}$

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data("demoIta")
sim92<-new("lifetable",x=demoIta$X, 
           lx=demoIta$SIM92, name='SIM92')
getOmega(sim92)
tail(sim92)

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Life tables' functions

• dxt, deaths between age x and x + t, tdx
• pxt, survival probability between age x and x + t, tpx
• pxyzt, survival probability for two (or more) lives, tpxy
• qxt, death probability between age x and x + t, tqx
• qxyzt, death probability for two (or more) lives, tqxy
• Txt, number of person-years lived after exact age x, tTx
• mxt, central death rate, tmx
• exn, expected lifetime between age x and age x + n, nex
• exyz, n-year curtate lifetime of the joint-life status

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• $p_{x,t} = 1 - q_{x,t} = \frac{l_{x+t}}{l_{x}}$
• $e_{x,n}=\sum_{t=1}^{n}p_{x,t}$

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#two years death rate
qxt(sim92, x=65,2)
#expected residual lifetime between x and x+n
exn(sim92, x=25,n = 40)

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Simulation

• rLife, sample from the time until death distribution underlying a life table
• rLifexyz, sample from the time until death distribution underlying two or more lives

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#simulate 1000 samples of residual life time
res_lt<-rLife(n=1000,object = sim92,x=65)
hist(res_lt,xlab="Residual Life Time")

Assessing longevity impact on annuities using lifecontingencies and demography

• This part of the presentation will make use of the demography package to calibrate Lee Carter [@Lee1992] model, $log\left(\mu_{x,t} \right) =a_{x}+b_{x}*k_{t}\rightarrow p_{x,t}=exp^{-\mu_{x,t}}$ ,projecting mortality and implicit life tables.

#library(demography)
#italy.demo<-hmd.mx("ITA", username="[e-mail]", password="mortality")

#load the package and the italian tables
library(demography) 
#italyDemo<-hmd.mx("ITA", username="yourUN", 
#password="yourPW")
load(file="mortalityDatasets.RData") #load the dataset

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• Lee Carter model is calibrated using lca function of demography package.
• Then an arima model is used to project (extrapolate) the underlying $k_t$ over the historical period.

#calibrate lee carter
italy.leecarter<-lca(data=italyDemo,series="total",
                     max.age=103,adjust = "none")
#perform modeling of kt series
kt.model<-auto.arima(italy.leecarter$kt)
#projecting the kt
kt.forecast<-forecast(kt.model,h=100) 

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-The code below generates the matrix of prospective life tables

#indexing the kt
kt.full<-ts(union(italy.leecarter$kt, kt.forecast$mean),
            start=1872)  
#getting and defining the life tables matrix
mortalityTable<-exp(italy.leecarter$ax
+italy.leecarter$bx%*%t(kt.full)) 
rownames(mortalityTable)<-seq(from=0, to=103)
colnames(mortalityTable)<-seq(from=1872, 
to=1872+dim(mortalityTable)[2]-1)

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plot.ts(kt.full, main="historical and projected KT",xlab="year",
        ylab="kt",col="steelblue")
abline(v=2009,col="darkred",lwd=2.5)

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• now we need a function that returns the one-year death probabilities given a year of birth (cohort.

getCohortQx<-function(yearOfBirth)
{
  colIndex<-which(colnames(mortalityTable)
                  ==yearOfBirth) #identify 
  #the column corresponding to the cohort 
  #definex the probabilities from which 
  #the projection is to be taken
  maxLength<-min(nrow(mortalityTable)-1,
                 ncol(mortalityTable)-colIndex)
  qxOut<-numeric(maxLength+1)
  for(i in 0:maxLength)
    qxOut[i+1]<-mortalityTable[i+1,colIndex+i]
  #fix: we add a fictional omega age where 
  #death probability = 1
  qxOut<-c(qxOut,1)
  return(qxOut)
}

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• Now we use such function to obtain prospective life tables and to perform actuarial calculations. For example, we can compute the APV of an annuity on a workers' retiring at 65 assuming he were born in 1920, in 1950 and in 1980. We will use the interest rate of 1.5\% (the one used to compute Italian Social Security annuity factors).
• The first step is to generate the life and actuarial tables

#generate the life tables
qx1920<-getCohortQx(yearOfBirth = 1920)
lt1920<-probs2lifetable(probs=qx1920,type="qx",
name="Table 1920")
at1920<-new("actuarialtable",x=lt1920@x,
lx=lt1920@lx,interest=0.015)
qx1950<-getCohortQx(yearOfBirth = 1950)
lt1950<-probs2lifetable(probs=qx1950,
type="qx",name="Table 1950")
at1950<-new("actuarialtable",x=lt1950@x,
lx=lt1950@lx,interest=0.015)
qx1980<-getCohortQx(yearOfBirth = 1980)
lt1980<-probs2lifetable(probs=qx1980,
type="qx",name="Table 1980")
at1980<-new("actuarialtable",x=lt1980@x,
lx=lt1980@lx,interest=0.015)

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• Now we can evaluate $\ddot{a}{65}$ and $\mathring{e}{65}$ for workers born in 1920, 1950 and 1980 respectively.

cat("Results for 1920 cohort","\n")
c(exn(at1920,x=65),axn(at1920,x=65))
cat("Results for 1950 cohort","\n")
c(exn(at1950,x=65),axn(at1950,x=65))
cat("Results for 1980 cohort","\n")
c(exn(at1980,x=65),axn(at1980,x=65))

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Intro on Actuarial Mathematics Funcs

• The lifecontingencies package allows to compute all classical life contingent insurances.
• Stochastic calculation varying expected lifetimes are possible as well.
• This makes the lifecontingencies package a nice tool to perform actuarial computation at command line on life insurance tasks.

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Creating actuarial tables

• Actuarial tables are stored as S4 object.
• The $l_x$, $x$, interest rate and a name are required.
• The print method return a classical actuarial table (commutation functions)

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data("demoIta")
sim92Act<-new("actuarialtable",x=demoIta$X, 
lx=demoIta$SIM92, name='SIM92')
sif92Act<-new("actuarialtable",x=demoIta$X,
lx=demoIta$SIF92, name='SIF92')
head(sim92Act)

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Life insurances functions

Classical life contingent insurances

• Exn, pure endowment: $\Ax{\pureendow{x}{n}}$
• axn, annuity: $\ddot{a}{x} = \sum{k=0}^{\omega-x} v^{k} * p_{x,k}=\sum_{k=0}^{\omega-x} \ddot{a}{\lcroof{K+1}} p{x,k} q_{x+k,1}$
• Axn, life insurance: $\Ax{\termxn}=\sum_{k=0}^{n-1} v^{k-1} * p_{x,k} * q_{x+k,1}$
• AExn, endowment: $\Ax{\endowxn}=\Ax{\pureendow{x}{n}}+\Ax{\termxn}$

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100000*Exn(sim92Act,x=25,n=40)
100000*AExn(sim92Act,x=25,n=40)
1000*12*axn(sim92Act,x=65,k=12)
100000*Axn(sim92Act,x=25,n=40)

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Additional life contingent insurances

• Increasing and decreasing term insurances and annuities
• $\left(n-1\right) * \Ax{\termxn} = \IA_{\termxn} + \DA_{\termxn}$

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IAxn(sim92Act,x=40,n=10)+DAxn(sim92Act,x=40,n=10)
(10+1)*Axn(sim92Act,x=40,n=10)

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Insurances on multiple lifes

• First survival and last survival status, both for insurances and annuities
• $A_{xy}$+$A_{\overline{xy}}$=$A_x$+$A_y$
• $a_{xy}$+$a_{\overline{xy}}$=$a_x$+$a_y$

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fr_pay=12
1000*fr_pay*axyzn(tablesList = list(sim92Act,sif92Act),
x = c(64,67),status="last",k=fr_pay)
1000*fr_pay*(axn(sim92Act,x=64,k=fr_pay)+axn(sif92Act,
x=67,k=fr_pay)-axyzn(tablesList = list(sim92Act,sif92Act),
x = c(64,67),status="joint",k=fr_pay))

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Simulation

• It is possible to simulate the PV of insured benefit distributions.
• The rLifeContingencies function is used for single life benefit insurance.
• The rLifeContingenciesXyz function is used for multiple lifes benefits.

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hist(rLifeContingencies(n = 1000,lifecontingency = "Axn",
x = 40,object = sim92Act,getOmega(sim92Act)-40),
main="Life Insurance on 40 PV distribution")

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Bibliography {.allowframebreaks}

spedygiorgio/lifecontingencies documentation built on Feb. 29, 2024, 2:59 p.m.