Package 'AnnuityRIR' reference manual

Title:	Annuity Random Interest Rates
Description:	Annuity Random Interest Rates proposes different techniques for the approximation of the present and final value of a unitary annuity-due or annuity-immediate considering interest rate as a random variable. Cruz Rambaud et al. (2017) <doi:10.1007/978-3-319-54819-7_16>. Cruz Rambaud et al. (2015) <doi:10.23755/rm.v28i1.25>.
Authors:	Salvador Cruz Rambaud [aut], Fabrizio Maturo [aut, cre], Ana Maria Sanchez Perez [aut]
Maintainer:	Fabrizio Maturo <[email protected]>
License:	GPL (>= 2)
Version:	1.0-0
Built:	2025-02-28 04:32:48 UTC
Source:	https://github.com/fabriziomaturo/annuityrir

Compute the parameters of the beta distribution and plot normalized data.

Description

Compute the parameters of the beta distribution and plot normalized data.

Usage

beta_parameters(data)
beta_parameters(data)

Arguments

data

A vector of interest rates.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Source

Cruz Rambaud, S.; Maturo, F. and Sánchez Pérez A. M. (2015): "Approach of the value of an annuity when non-central moments of the capitalization factor are known: an R application with interest rates following normal and beta distributions". Ratio Mathematica, 28(1), pp. 15-30. doi: 10.23755/rm.v28i1.25.

Examples


# example 1
data=c(0.00,-0.05,-0.05,-0.06,-0.06,0.02,-0.06,-0.05,-0.04,-0.05,
-0.03,-0.06,0.04,-0.05,-0.08,-0.05,-0.12,-0.03,-0.05,-0.04,-0.06)
beta_parameters(data)

# example 2
data<-rnorm(n=200,m=0.075,sd=0.2)
beta_parameters(data)


# example 1
data=c(0.00,-0.05,-0.05,-0.06,-0.06,0.02,-0.06,-0.05,-0.04,-0.05,
-0.03,-0.06,0.04,-0.05,-0.08,-0.05,-0.12,-0.03,-0.05,-0.04,-0.06)
beta_parameters(data)

# example 2
data<-rnorm(n=200,m=0.075,sd=0.2)
beta_parameters(data)

Compute the final expected value of an $n$ -payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate $X$ , using the tetraparametric function approach.

Description

Compute the final expected value of an $n$ -payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate $X$ , using the tetraparametric function approach.

Usage

FV_post_artan(data,years)
FV_post_artan(data,years)

Arguments

`data`	A vector of interest rates.
`years`	The number of years of the income. Default is 10 years.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Source

Cruz Rambaud, S.; Maturo, F. and Sánchez Pérez A. M. (2017): Expected present and final value of an annuity when some non-central moments of the capitalization factor are unknown: Theory and an application using R. In Š. Hošková-Mayerová, et al. (Eds.), Mathematical-Statistical Models and Qualitative Theories for Economic and Social Sciences (pp. 233-248). Springer, Cham. doi:10.1007/978-3-319-54819-7_16.

Examples


#example 1
data=c(0.298,0.255,0.212,0.180,0.165,0.163,0.167,0.161,0.154,
0.128,0.079,0.059,0.042,-0.008,-0.012,-0.002)
FV_post_artan(data,6)

# example 2
data<-rnorm(n=30,m=0.03,sd=0.01)
FV_post_artan(data,10)
#example 1
data=c(0.298,0.255,0.212,0.180,0.165,0.163,0.167,0.161,0.154,
0.128,0.079,0.059,0.042,-0.008,-0.012,-0.002)
FV_post_artan(data,6)

# example 2
data<-rnorm(n=30,m=0.03,sd=0.01)
FV_post_artan(data,10)

Compute the final expected value of an $n$ -payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate $X$ , using the estimated moments of the beta distribution.

Description

Compute the final expected value of an $n$ -payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate $X$ , using the estimated moments of the beta distribution.

Usage

FV_post_beta_kmom(data,years)
FV_post_beta_kmom(data,years)

Arguments

`data`	A vector of interest rates.
`years`	The number of years of the income. Default is 10 years.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Source

Cruz Rambaud, S.; Maturo, F. and Sánchez Pérez A. M. (2015): “Approach of the value of an annuity when non-central moments of the capitalization factor are known: an R application with interest rates following normal and beta distributions”. Ratio Mathematica, 28(1), pp. 15-30. doi: 10.23755/rm.v28i1.25.

Examples


# example 1
data=c(0.00,-0.05,-0.05,-0.06,-0.06,0.02,-0.06,-0.05,-0.04,-0.05,
-0.03,-0.06,0.04,-0.05,-0.08,-0.05,-0.12,-0.03,-0.05,-0.04,-0.06)
FV_post_beta_kmom(data,8)

# example 2
data<-rnorm(n=200,m=0.075,sd=0.2)
FV_post_beta_kmom(data,8)


# example 1
data=c(0.00,-0.05,-0.05,-0.06,-0.06,0.02,-0.06,-0.05,-0.04,-0.05,
-0.03,-0.06,0.04,-0.05,-0.08,-0.05,-0.12,-0.03,-0.05,-0.04,-0.06)
FV_post_beta_kmom(data,8)

# example 2
data<-rnorm(n=200,m=0.075,sd=0.2)
FV_post_beta_kmom(data,8)

Compute the final expected value of an $n$ -payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate $X$ , using the method of Mood et al.

Description

Compute the final expected value of an $n$ -payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate $X$ , using the method of Mood et al.

Usage

FV_post_mood(data,years)
FV_post_mood(data,years)

Arguments

`data`	A vector of interest rates.
`years`	The number of years of the income. Default is 10 years.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Source

Cruz Rambaud, S.; Maturo, F. and Sánchez Pérez A. M. (2017): “Expected present and final value of an annuity when some non-central moments of the capitalization factor are unknown: Theory and an application using R”. In Š. Hošková-Mayerová, et al. (Eds.), Mathematical-Statistical Models and Qualitative Theories for Economic and Social Sciences (pp. 233-248). Springer, Cham. doi:10.1007/978-3-319-54819-7_16.

Examples

#example 1
data=c(0.298,0.255,0.212,0.180,0.165,0.163,0.167,0.161,0.154,
0.128,0.079,0.059,0.042,-0.008,-0.012,-0.002)
FV_post_mood(data,6)

# example 2
data<-rnorm(n=30,m=0.03,sd=0.01)
FV_post_mood(data,10)
#example 1
data=c(0.298,0.255,0.212,0.180,0.165,0.163,0.167,0.161,0.154,
0.128,0.079,0.059,0.042,-0.008,-0.012,-0.002)
FV_post_mood(data,6)

# example 2
data<-rnorm(n=30,m=0.03,sd=0.01)
FV_post_mood(data,10)

Compute the final expected value of an $n$ -payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate $X$ , using the estimated moments of the normal distribution.

Description

Usage

FV_post_norm_kmom(data,years)
FV_post_norm_kmom(data,years)

Arguments

`data`	A vector of interest rates.
`years`	The number of years of the income. Default is 10 years.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Source

Examples


# example 1
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
data=data/100
FV_post_norm_kmom(data,8)


# example 1
data<-rnorm(n=200,m=0.075,sd=0.2)
norm_test_jb(data) #test data
FV_post_norm_kmom(data,8)


# example 1
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
data=data/100
FV_post_norm_kmom(data,8)


# example 1
data<-rnorm(n=200,m=0.075,sd=0.2)
norm_test_jb(data) #test data
FV_post_norm_kmom(data,8)

Compute the final expected value of an $n$ -payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate $X$ , using the quadratic discount method.

Description

Compute the final expected value of an $n$ -payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate $X$ , using the quadratic discount method.

Usage

FV_post_quad(data,years)
FV_post_quad(data,years)

Arguments

`data`	A vector of interest rates.
`years`	The number of years of the income. Default is 10 years.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Source

Examples

#example 1
data=c(0.298,0.255,0.212,0.180,0.165,0.163,0.167,0.161,0.154,
0.128,0.079,0.059,0.042,-0.008,-0.012,-0.002)
FV_post_quad(data,8)

# example 2
data<-rnorm(n=30,m=0.03,sd=0.01)
FV_post_quad(data,10)
#example 1
data=c(0.298,0.255,0.212,0.180,0.165,0.163,0.167,0.161,0.154,
0.128,0.079,0.059,0.042,-0.008,-0.012,-0.002)
FV_post_quad(data,8)

# example 2
data<-rnorm(n=30,m=0.03,sd=0.01)
FV_post_quad(data,10)

Compute the final expected value of an n-payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate $X$ , using the tetraparametric function approach.

Description

Compute the final expected value of an n-payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate $X$ , using the tetraparametric function approach.

Usage

FV_pre_artan(data,years)
FV_pre_artan(data,years)

Arguments

`data`	A vector of interest rates.
`years`	The number of years of the income. Default is 10 years.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Source

Examples

#example 1
data=c(0.298,0.255,0.212,0.180,0.165,0.163,0.167,0.161,0.154,
0.128,0.079,0.059,0.042,-0.008,-0.012,-0.002)
FV_pre_artan(data,6)

# example 2
data<-rnorm(n=30,m=0.03,sd=0.01)
FV_pre_artan(data,10)
#example 1
data=c(0.298,0.255,0.212,0.180,0.165,0.163,0.167,0.161,0.154,
0.128,0.079,0.059,0.042,-0.008,-0.012,-0.002)
FV_pre_artan(data,6)

# example 2
data<-rnorm(n=30,m=0.03,sd=0.01)
FV_pre_artan(data,10)

Compute the final expected value of an $n$ -payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate $X$ , using the estimated moments of the beta distribution.

Description

Compute the final expected value of an $n$ -payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate $X$ , using the estimated moments of the beta distribution.

Usage

FV_pre_beta_kmom(data,years)
FV_pre_beta_kmom(data,years)

Arguments

`data`	A vector of interest rates.
`years`	The number of years of the income. Default is 10 years.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Source

Examples


# example 1
data=c(0.00,-0.05,-0.05,-0.06,-0.06,0.02,-0.06,-0.05,-0.04,-0.05,
-0.03,-0.06,0.04,-0.05,-0.08,-0.05,-0.12, -0.03,-0.05,-0.04,-0.06)
FV_pre_beta_kmom(data,8)

# example 2
data<-rnorm(n=200,m=0.075,sd=0.2)
FV_pre_beta_kmom(data,8)

# example 1
data=c(0.00,-0.05,-0.05,-0.06,-0.06,0.02,-0.06,-0.05,-0.04,-0.05,
-0.03,-0.06,0.04,-0.05,-0.08,-0.05,-0.12, -0.03,-0.05,-0.04,-0.06)
FV_pre_beta_kmom(data,8)

# example 2
data<-rnorm(n=200,m=0.075,sd=0.2)
FV_pre_beta_kmom(data,8)

Compute the final expected value of an $n$ -payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate $X$ , using the method of Mood et al.

Description

Compute the final expected value of an $n$ -payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate $X$ , using the method of Mood et al.

Usage

FV_pre_mood(data,years)
FV_pre_mood(data,years)

Arguments

`data`	A vector of interest rates.
`years`	The number of years of the income. Default is 10 years.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Source

Examples

#example 1
data=c(0.298,0.255,0.212,0.180,0.165,0.163,0.167,0.161,0.154,
0.128,0.079,0.059,0.042,-0.008,-0.012,-0.002)
FV_pre_mood(data,6)

# example 2
data<-rnorm(n=30,m=0.03,sd=0.01)
FV_pre_mood(data,10)
#example 1
data=c(0.298,0.255,0.212,0.180,0.165,0.163,0.167,0.161,0.154,
0.128,0.079,0.059,0.042,-0.008,-0.012,-0.002)
FV_pre_mood(data,6)

# example 2
data<-rnorm(n=30,m=0.03,sd=0.01)
FV_pre_mood(data,10)

Compute the final expected value of an $n$ -payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate $X$ , using the estimated moments of the normal distribution.

Description

Usage

FV_pre_norm_kmom(data,years)
FV_pre_norm_kmom(data,years)

Arguments

`data`	A vector of interest rates.
`years`	The number of years of the income. Default is 10 years.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Source

Examples


# example 1
data<-rnorm(n=30,m=0.03,sd=0.01)
norm_test_jb(data) #test data
FV_pre_norm_kmom(data,8)

# example 1
data<-rnorm(n=200,m=0.075,sd=0.2)
norm_test_jb(data) #test data
FV_pre_norm_kmom(data,8)


# example 1
data<-rnorm(n=30,m=0.03,sd=0.01)
norm_test_jb(data) #test data
FV_pre_norm_kmom(data,8)

# example 1
data<-rnorm(n=200,m=0.075,sd=0.2)
norm_test_jb(data) #test data
FV_pre_norm_kmom(data,8)

Compute the final expected value of an $n$ -payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate $X$ , using the quadratic discount method.

Description

Compute the final expected value of an $n$ -payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate $X$ , using the quadratic discount method.

Usage

FV_pre_quad(data,years)
FV_pre_quad(data,years)

Arguments

`data`	A vector of interest rates.
`years`	The number of years of the income. Default is 10 years.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Source

Examples

#example 1
data=c(0.298,0.255,0.212,0.180,0.165,0.163,0.167,0.161,0.154,
0.128,0.079,0.059,0.042,-0.008,-0.012,-0.002)
FV_pre_quad(data,6)

# example 2
data<-rnorm(n=30,m=0.03,sd=0.01)
FV_pre_quad(data,10)
#example 1
data=c(0.298,0.255,0.212,0.180,0.165,0.163,0.167,0.161,0.154,
0.128,0.079,0.059,0.042,-0.008,-0.012,-0.002)
FV_pre_quad(data,6)

# example 2
data<-rnorm(n=30,m=0.03,sd=0.01)
FV_pre_quad(data,10)

Compute the exact moments of a distribution.

Description

Compute the exact moments of a distribution.

Usage

moment(x,order,central, absolute, na.rm)
moment(x,order,central, absolute, na.rm)

Arguments

`x`	A vector X of interest rates.
`order`	The order of moment that should be computed. Default is 1.
`central`	If central moments are to be computed. Default is "FALSE".
`absolute`	If absolute moments are to be computed. Default is "FALSE".
`na.rm`	If missing values should be removed. Default is "FALSE".

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Source

Examples


#example 1
data=c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
moment(data,3)





#example 1
data=c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
moment(data,3)

Fit the data to a normal curve and compute the moments of the normal distribution according to the definition (as integral).

Description

Fit the data to a normal curve and compute the moments of the normal distribution according to the definition (as integral).

Usage

norm_mom(data,order)
norm_mom(data,order)

Arguments

`data`	A vector X of interest rates.
`order`	The order of moment that should be computed.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Source

Examples


#example 1
data=c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
data=data/100
norm_mom(data,5)





#example 1
data=c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
data=data/100
norm_mom(data,5)

Compute the Jarque-Bera test for checking the assumption of normality of the interest rates distribution and returns the parameters of the fitted normal distribution.

Description

Compute the Jarque-Bera test for checking the assumption of normality of the interest rates distribution and returns the parameters of the fitted normal distribution.

Usage

norm_test_jb(data)
norm_test_jb(data)

Arguments

data

A vector of interest rates.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Source

Examples


#example 1
data=c(0.298,0.255,0.212,0.180,0.165,0.163,0.167,0.161,
0.154,0.128,0.079,0.059,0.042,-0.008,-0.012,-0.002)
norm_test_jb(data)

# example 2
data<-rnorm(n=30,m=0.03,sd=0.01)
norm_test_jb(data)

# example 3
data=runif(999, min = 0, max = 1)
norm_test_jb(data)

# example 4
data=c(0.00,-0.05,-0.05,-0.06,-0.06,0.02,-0.06,-0.05,-0.04,-0.05,
-0.03,-0.06,0.04,-0.05,-0.08,-0.05,-0.12,-0.03,-0.05,-0.04,-0.06)
norm_test_jb(data)
#example 1
data=c(0.298,0.255,0.212,0.180,0.165,0.163,0.167,0.161,
0.154,0.128,0.079,0.059,0.042,-0.008,-0.012,-0.002)
norm_test_jb(data)

# example 2
data<-rnorm(n=30,m=0.03,sd=0.01)
norm_test_jb(data)

# example 3
data=runif(999, min = 0, max = 1)
norm_test_jb(data)

# example 4
data=c(0.00,-0.05,-0.05,-0.06,-0.06,0.02,-0.06,-0.05,-0.04,-0.05,
-0.03,-0.06,0.04,-0.05,-0.08,-0.05,-0.12,-0.03,-0.05,-0.04,-0.06)
norm_test_jb(data)

Plot the final expected value of an $n$ -payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate $X$ , using the estimated moments of the beta distribution.

Description

Plot the final expected value of an $n$ -payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate $X$ , using the estimated moments of the beta distribution.

Usage

plot_FV_post_beta_kmom(data,years,lwd,lty)
plot_FV_post_beta_kmom(data,years,lwd,lty)

Arguments

`data`	A vector of interest rates.
`years`	The number of years of the income. Default is 10 years.
`lwd`	The width of the curve. Default is 1.5.
`lty`	The style of the curve. Default is 1.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Examples


# example 1
data<-runif(34, 0,1)
plot_FV_post_beta_kmom(data,8)


# example 1
data<-runif(34, 0,1)
plot_FV_post_beta_kmom(data,8)

Plot the final expected value of an $n$ -payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate $X$ , using the estimated moments of the normal distribution.

Description

Usage

plot_FV_post_norm_kmom(data,years,lwd,lty)
plot_FV_post_norm_kmom(data,years,lwd,lty)

Arguments

`data`	A vector of interest rates.
`years`	The number of years of the income. Default is 10 years.
`lwd`	The width of the curve. Default is 1.5.
`lty`	The style of the curve. Default is 1.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Examples


# example 1
data<-rnorm(n=30,m=0.03,sd=0.01)
plot_FV_post_norm_kmom(data,8)

# example 2
data<-rnorm(n=200,m=0.075,sd=0.2)
plot_FV_post_norm_kmom(data,8)
# example 1
data<-rnorm(n=30,m=0.03,sd=0.01)
plot_FV_post_norm_kmom(data,8)

# example 2
data<-rnorm(n=200,m=0.075,sd=0.2)
plot_FV_post_norm_kmom(data,8)

Plot the final expected value of an $n$ -payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate $X$ , using the estimated moments of the beta distribution.

Description

Plot the final expected value of an $n$ -payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate $X$ , using the estimated moments of the beta distribution.

Usage

plot_FV_pre_beta_kmom(data,years,lwd,lty)
plot_FV_pre_beta_kmom(data,years,lwd,lty)

Arguments

`data`	A vector of interest rates.
`years`	The number of years of the income. Default is 10 years.
`lwd`	The width of the curve. Default is 1.5.
`lty`	The style of the curve. Default is 1.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Examples


# example 1
data<-runif(34, 0,1)
plot_FV_pre_beta_kmom(data,8)



# example 1
data<-runif(34, 0,1)
plot_FV_pre_beta_kmom(data,8)

Plot the final expected value of an $n$ -payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate $X$ , using the estimated moments of the normal distribution.

Description

Usage

plot_FV_pre_norm_kmom(data,years,lwd,lty)
plot_FV_pre_norm_kmom(data,years,lwd,lty)

Arguments

`data`	A vector of interest rates.
`years`	The number of years of the income. Default is 10 years.
`lwd`	The width of the curve. Default is 1.5.
`lty`	The style of the curve. Default is 1.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Examples


# example 1
data<-rnorm(n=30,m=0.03,sd=0.01)
plot_FV_pre_norm_kmom(data,8)


# example 2
data<-rnorm(n=200,m=0.075,sd=0.2)
plot_FV_pre_norm_kmom(data,8)




# example 1
data<-rnorm(n=30,m=0.03,sd=0.01)
plot_FV_pre_norm_kmom(data,8)


# example 2
data<-rnorm(n=200,m=0.075,sd=0.2)
plot_FV_pre_norm_kmom(data,8)

Plot the final expected values of an $n$ -payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate $X$ , using different approaches.

Description

Plot the final expected values of an $n$ -payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate $X$ , using different approaches.

Usage

plot_FVs_post(data,years,lwd,lty1,lty2,lty3)
plot_FVs_post(data,years,lwd,lty1,lty2,lty3)

Arguments

`data`	A vector of interest rates.
`years`	The number of years of the income. Default is 10 years.
`lwd`	The width of the curve. Default is 1.5.
`lty1`	The style of the curve for the "arctan" approximation. Default is 1.
`lty2`	The style of the curve for the "cubic" approximation. Default is 2.
`lty3`	The style of the curve for the "mood" approximation. Default is 3.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Examples

#example 1
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
data=data/100
plot_FVs_post(data)

# example 2
data<-rnorm(n=30,m=0.03,sd=0.003)
plot_FVs_post(data)
#example 1
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
data=data/100
plot_FVs_post(data)

# example 2
data<-rnorm(n=30,m=0.03,sd=0.003)
plot_FVs_post(data)

Plot the final expected values of an $n$ -payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate $X$ , using different approaches.

Description

Plot the final expected values of an $n$ -payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate $X$ , using different approaches.

Usage

plot_FVs_pre(data,years,lwd,lty1,lty2,lty3)
plot_FVs_pre(data,years,lwd,lty1,lty2,lty3)

Arguments

`data`	A vector of interest rates.
`years`	The number of years of the income. Default is 10 years.
`lwd`	The width of the curve. Default is 1.5.
`lty1`	The style of the curve for the "arctan" approximation. Default is 1.
`lty2`	The style of the curve for the "cubic" approximation. Default is 2.
`lty3`	The style of the curve for the "mood" approximation. Default is 3.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Examples

#example 1
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
data=data/100
plot_FVs_pre(data)

# example 2
data<-rnorm(n=30,m=0.03,sd=0.003)
plot_FVs_pre(data)
#example 1
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
data=data/100
plot_FVs_pre(data)

# example 2
data<-rnorm(n=30,m=0.03,sd=0.003)
plot_FVs_pre(data)

Plot the present expected values of an $n$ -payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate $X$ , using different approaches.

Description

Plot the present expected values of an $n$ -payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate $X$ , using different approaches.

Usage

plot_PVs_post(data,years,lwd,lty1,lty2,lty3,lty4,lty5,lty6)
plot_PVs_post(data,years,lwd,lty1,lty2,lty3,lty4,lty5,lty6)

Arguments

`data`	A vector of interest rates.
`years`	The number of years of the income. Default is 10 years.
`lwd`	The width of the curve. Default is 1.5.
`lty1`	The style of the curve for the "arctan" approximation. Default is 1.
`lty2`	The style of the curve for the "cubic" approximation. Default is 2.
`lty3`	The style of the curve for the "mood with positive moments" approximation. Default is 3.
`lty4`	The style of the curve for the "mood with negative moments" approximation. Default is 4.
`lty5`	The style of the curve for the exact value. Default is 5.
`lty6`	The style of the curve for "triangular distribution" approximation. Default is 6.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Examples


# example 1
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
data=data/100
plot_PVs_post(data)

# example 2
data<-rnorm(n=30,m=0.03,sd=0.003)
plot_PVs_post(data)

# example 1
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
data=data/100
plot_PVs_post(data)

# example 2
data<-rnorm(n=30,m=0.03,sd=0.003)
plot_PVs_post(data)

Plot the present expected values of an $n$ -payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate $X$ , using different approaches.

Description

Plot the present expected values of an $n$ -payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate $X$ , using different approaches.

Usage

plot_PVs_pre(data,years,lwd,lty1,lty2,lty3,lty4,lty5,lty6)
plot_PVs_pre(data,years,lwd,lty1,lty2,lty3,lty4,lty5,lty6)

Arguments

`data`	A vector of interest rates.
`years`	The number of years of the income. Default is 10 years.
`lwd`	The width of the curve. Default is 1.5.
`lty1`	The style of the curve for the "arctan" approximation. Default is 1.
`lty2`	The style of the curve for the "cubic" approximation. Default is 2.
`lty3`	The style of the curve for the "mood with positive moments" approximation. Default is 3.
`lty4`	The style of the curve for the "mood with negative moments" approximation. Default is 4.
`lty5`	The style of the curve for the exact value. Default is 5.
`lty6`	The style of the curve for "triangular distribution" approximation. Default is 6.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Examples


# example 1
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
data=data/100
plot_PVs_pre(data)

# example 2
data<-rnorm(n=30,m=0.03,sd=0.003)
plot_PVs_pre(data)

# example 1
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
data=data/100
plot_PVs_pre(data)

# example 2
data<-rnorm(n=30,m=0.03,sd=0.003)
plot_PVs_pre(data)

Compute present expected value of an $n$ -payment annuity, with payments of 1 unit each, made at the end of every year (annuity-immediate), valued at the rate $X$ , using the tetraparametric function approach.

Description

Compute present expected value of an $n$ -payment annuity, with payments of 1 unit each, made at the end of every year (annuity-immediate), valued at the rate $X$ , using the tetraparametric function approach.

Usage

PV_post_artan(data,years)
PV_post_artan(data,years)

Arguments

`data`	A vector of interest rates.
`years`	The number of years of the income. Default is 10 years.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Source

Examples


#example 1
data=c(0.298,0.255,0.212,0.180,0.165,0.163,0.167,0.161,0.154,
0.128,0.079,0.059,0.042,-0.008,-0.012,-0.002)
PV_post_artan(data)

# example 2
data<-rnorm(n=30,m=0.03,sd=0.01)
PV_post_artan(data)

# example 3
data<-rnorm(n=30,m=0.03,sd=0.2)
PV_post_artan(data)

#example 1
data=c(0.298,0.255,0.212,0.180,0.165,0.163,0.167,0.161,0.154,
0.128,0.079,0.059,0.042,-0.008,-0.012,-0.002)
PV_post_artan(data)

# example 2
data<-rnorm(n=30,m=0.03,sd=0.01)
PV_post_artan(data)

# example 3
data<-rnorm(n=30,m=0.03,sd=0.2)
PV_post_artan(data)

Compute the present expected value of an $n$ -payment annuity, with payments of 1 unit each made at the end of every year (annuity-due), valued at the rate $X$ , using the cubic discount method.

Description

Compute the present expected value of an $n$ -payment annuity, with payments of 1 unit each made at the end of every year (annuity-due), valued at the rate $X$ , using the cubic discount method.

Usage

PV_post_cubic(data,years)
PV_post_cubic(data,years)

Arguments

`data`	A vector of interest rates.
`years`	The number of years of the income. Default is 10 years.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Examples

#example 1
data=c(0.298,0.255,0.212,0.180,0.165,0.163,0.167,0.161,0.154,
0.128,0.079,0.059,0.042,-0.008,-0.012,-0.002)
PV_post_cubic(data)

#example 2
data<-rnorm(n=30,m=0.03,sd=0.01)
PV_post_cubic(data)

# example 3
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
data=data/100
PV_post_cubic(data)
#example 1
data=c(0.298,0.255,0.212,0.180,0.165,0.163,0.167,0.161,0.154,
0.128,0.079,0.059,0.042,-0.008,-0.012,-0.002)
PV_post_cubic(data)

#example 2
data<-rnorm(n=30,m=0.03,sd=0.01)
PV_post_cubic(data)

# example 3
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
data=data/100
PV_post_cubic(data)

Computes the present value of an annuity-immediate considering only non-central moments of negative orders.

Description

Computes the present value of an annuity-immediate considering only non-central moments of negative orders.

Usage

PV_post_exact(data,years)
PV_post_exact(data,years)

Arguments

`data`	A vector of interest rates.
`years`	The number of years of the income. Default is 10 years.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Examples


# example 1
data=c(0.0177, 0.0185, 0.0185, 0.0184, 0.0184, 0.0183, 0.0185, 0.0185, 0.0188, 0.0185, 
0.0180, 0.0184, 0.0191, 0.0185, 0.0184, 0.0185, 0.0186, 0.0185, 0.0188, 0.0186)
PV_post_exact(data,10)



# example 1
data=c(0.0177, 0.0185, 0.0185, 0.0184, 0.0184, 0.0183, 0.0185, 0.0185, 0.0188, 0.0185, 
0.0180, 0.0184, 0.0191, 0.0185, 0.0184, 0.0185, 0.0186, 0.0185, 0.0188, 0.0186)
PV_post_exact(data,10)

Compute the present expected value of an $n$ -payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate $X$ , with the method of Mood et al. using some negative moments of the distribution.

Description

Compute the present expected value of an $n$ -payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate $X$ , with the method of Mood et al. using some negative moments of the distribution.

Usage

PV_post_mood_nm(data,years)
PV_post_mood_nm(data,years)

Arguments

`data`	A vector of interest rates.
`years`	The number of years of the income. Default is 10 years.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Source

Mood, A. M.; Graybill, F. A. and Boes, D. C. (1974). Introduction to the Theory of Statistics (3rd Ed.). New York: McGraw Hill.

Rice, J. A. (1995). Mathematical Statistics and Data Analysis (2nd Ed.). California: Ed. Duxbury Press.

Examples

#example 1
data=c(0.298,0.255,0.212,0.180,0.165,0.163,0.167,0.161,0.154,
0.128,0.079,0.059,0.042,-0.008,-0.012,-0.002)
PV_post_mood_nm(data)

# example 2
data<-rnorm(n=30,m=0.03,sd=0.01)
PV_post_mood_nm(data)

# example 3
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
data=data/100
PV_post_mood_nm(data)
#example 1
data=c(0.298,0.255,0.212,0.180,0.165,0.163,0.167,0.161,0.154,
0.128,0.079,0.059,0.042,-0.008,-0.012,-0.002)
PV_post_mood_nm(data)

# example 2
data<-rnorm(n=30,m=0.03,sd=0.01)
PV_post_mood_nm(data)

# example 3
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
data=data/100
PV_post_mood_nm(data)

Compute the present expected value of an $n$ -payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate $X$ , with the method of Mood et al. using some positive moments of the distribution.

Description

Usage

PV_post_mood_pm(data,years)
PV_post_mood_pm(data,years)

Arguments

`data`	A vector of interest rates.
`years`	The number of years of the income. Default is 10 years.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Source

Mood, A. M.; Graybill, F. A. and Boes, D. C. (1974). Introduction to the Theory of Statistics (3rd Ed.). New York: McGraw Hill.

Rice, J. A. (1995). Mathematical Statistics and Data Analysis (2nd Ed.). California: Ed. Duxbury Press.

Examples

#example 1
data=c(0.298,0.255,0.212,0.180,0.165,0.163,0.167,0.161,0.154,
0.128,0.079,0.059,0.042,-0.008,-0.012,-0.002)
PV_post_mood_pm(data)

# example 2
data<-rnorm(n=30,m=0.03,sd=0.01)
PV_post_mood_pm(data)

# example 3
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
data=data/100
PV_post_mood_pm(data)
#example 1
data=c(0.298,0.255,0.212,0.180,0.165,0.163,0.167,0.161,0.154,
0.128,0.079,0.059,0.042,-0.008,-0.012,-0.002)
PV_post_mood_pm(data)

# example 2
data<-rnorm(n=30,m=0.03,sd=0.01)
PV_post_mood_pm(data)

# example 3
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
data=data/100
PV_post_mood_pm(data)

Compute the present value of an annuity-immediate considering only non-central moments of negative orders. The calculation is performed by using the function triangular\_moments\_3 for the moments greater than $-2$ (in absolute value).

Description

Compute the present value of an annuity-immediate considering only non-central moments of negative orders. The calculation is performed by using the function triangular\_moments\_3 for the moments greater than $-2$ (in absolute value).

Usage

PV_post_triang_3(data,years)
PV_post_triang_3(data,years)

Arguments

`data`	A vector of interest rates expressed as percentages.
`years`	The number of years of the income. Default is 10 years.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Examples


data=c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
PV_pre_triang_3(data,10)

data=c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
PV_pre_triang_3(data,10)

Compute the present value of an annuity-immediate considering only non-central moments of negative orders. The calculation is performed by using the moments of the fitted triangular distribution of the random variable "capitalization factor" $U$ (which are obtained from the definition of negative moment of a continuous random variable).

Description

Compute the present value of an annuity-immediate considering only non-central moments of negative orders. The calculation is performed by using the moments of the fitted triangular distribution of the random variable "capitalization factor" $U$ (which are obtained from the definition of negative moment of a continuous random variable).

Usage

PV_post_triang_dis(data,years)
PV_post_triang_dis(data,years)

Arguments

`data`	A vector of interest rates expressed as percentages.
`years`	The number of years of the income. Default is 10 years.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Examples


data=c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
PV_post_triang_dis(data,10)

data=c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
PV_post_triang_dis(data,10)

Compute the present expected value of an $n$ -payment annuity, with payments of 1 unit each, made at the beginning of every year (annuity-due), valued at the rate $X$ , using the tetraparametric function approach.

Description

Compute the present expected value of an $n$ -payment annuity, with payments of 1 unit each, made at the beginning of every year (annuity-due), valued at the rate $X$ , using the tetraparametric function approach.

Usage

PV_pre_artan(data,years)
PV_pre_artan(data,years)

Arguments

`data`	A vector of interest rates.
`years`	The number of years of the income. Default is 10 years.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Source

Examples


#example 1
data=c(0.298,0.255,0.212,0.180,0.165,0.163,0.167,0.161,0.154,0.128,
0.079,0.059,0.042,-0.008,-0.012,-0.002)
PV_pre_artan(data)

# example 2
data<-rnorm(n=30,m=0.03,sd=0.01)
PV_pre_artan(data)

#example 1
data=c(0.298,0.255,0.212,0.180,0.165,0.163,0.167,0.161,0.154,0.128,
0.079,0.059,0.042,-0.008,-0.012,-0.002)
PV_pre_artan(data)

# example 2
data<-rnorm(n=30,m=0.03,sd=0.01)
PV_pre_artan(data)

Compute the present expected value of an $n$ -payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate $X$ , using the cubic discount method.

Description

Compute the present expected value of an $n$ -payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate $X$ , using the cubic discount method.

Usage

PV_pre_cubic(data,years)
PV_pre_cubic(data,years)

Arguments

`data`	A vector of interest rates.
`years`	The number of years of the income. Default is 10 years.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Examples

#example 1
data=c(0.298,0.255,0.212,0.180,0.165,0.163,0.167,0.161,0.154,
0.128,0.079,0.059,0.042,-0.008,-0.012,-0.002)
PV_pre_cubic(data)

#example 2
data<-rnorm(n=30,m=0.03,sd=0.01)
PV_pre_cubic(data)

# example 3
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
data=data/100
PV_pre_cubic(data)
#example 1
data=c(0.298,0.255,0.212,0.180,0.165,0.163,0.167,0.161,0.154,
0.128,0.079,0.059,0.042,-0.008,-0.012,-0.002)
PV_pre_cubic(data)

#example 2
data<-rnorm(n=30,m=0.03,sd=0.01)
PV_pre_cubic(data)

# example 3
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
data=data/100
PV_pre_cubic(data)

Compute the present value of an annuity-due considering only non-central moments of negative orders.

Description

Compute the present value of an annuity-due considering only non-central moments of negative orders.

Usage

PV_pre_exact(data,years)
PV_pre_exact(data,years)

Arguments

`data`	A vector of interest rates.
`years`	The number of years of the income. Default is 10 years.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Examples


# example 1
data=c(0.0177, 0.0185, 0.0185, 0.0184, 0.0184, 0.0183, 0.0185, 0.0185, 0.0188, 
0.0185, 0.0180, 0.0184, 0.0191, 0.0185, 0.0184, 0.0185, 0.0186, 0.0185, 0.0188, 0.0186)
PV_pre_exact(data,10)



# example 1
data=c(0.0177, 0.0185, 0.0185, 0.0184, 0.0184, 0.0183, 0.0185, 0.0185, 0.0188, 
0.0185, 0.0180, 0.0184, 0.0191, 0.0185, 0.0184, 0.0185, 0.0186, 0.0185, 0.0188, 0.0186)
PV_pre_exact(data,10)

Compute the present expected value of an $n$ -payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate $X$ , with the method of Mood et al. using some negative moments of the distribution.

Description

Compute the present expected value of an $n$ -payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate $X$ , with the method of Mood et al. using some negative moments of the distribution.

Usage

PV_pre_mood_nm(data,years)
PV_pre_mood_nm(data,years)

Arguments

`data`	A vector of interest rates.
`years`	The number of years of the income. Default is 10 years.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Examples

#example 1
data=c(0.298,0.255,0.212,0.180,0.165,0.163,0.167,0.161,0.154,
0.128,0.079,0.059,0.042,-0.008,-0.012,-0.002)
PV_pre_mood_nm(data)

# example 2
data<-rnorm(n=30,m=0.03,sd=0.01)
PV_pre_mood_nm(data)

# example 3
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,
1.84,1.85,1.86,1.85,1.88,1.86)
data=data/100
PV_pre_mood_nm(data)
#example 1
data=c(0.298,0.255,0.212,0.180,0.165,0.163,0.167,0.161,0.154,
0.128,0.079,0.059,0.042,-0.008,-0.012,-0.002)
PV_pre_mood_nm(data)

# example 2
data<-rnorm(n=30,m=0.03,sd=0.01)
PV_pre_mood_nm(data)

# example 3
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,
1.84,1.85,1.86,1.85,1.88,1.86)
data=data/100
PV_pre_mood_nm(data)

Compute the present expected value of an $n$ -payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate $X$ , with the method of Mood et al. using some positive moments of the distribution.

Description

Usage

PV_pre_mood_pm(data,years)
PV_pre_mood_pm(data,years)

Arguments

`data`	A vector of interest rates.
`years`	The number of years of the income. Default is 10 years.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Source

Examples

#example 1
data=c(0.298,0.255,0.212,0.180,0.165,0.163,0.167,0.161,0.154,
0.128,0.079,0.059,0.042,-0.008,-0.012,-0.002)
PV_pre_mood_pm(data)

# example 2
data<-rnorm(n=30,m=0.3,sd=0.01)
PV_pre_mood_pm(data)

# example 3
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
data=data/100
PV_pre_mood_pm(data)
#example 1
data=c(0.298,0.255,0.212,0.180,0.165,0.163,0.167,0.161,0.154,
0.128,0.079,0.059,0.042,-0.008,-0.012,-0.002)
PV_pre_mood_pm(data)

# example 2
data<-rnorm(n=30,m=0.3,sd=0.01)
PV_pre_mood_pm(data)

# example 3
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
data=data/100
PV_pre_mood_pm(data)

Compute the present value of an annuity-due considering only non-central moments of negative orders. The calculation is performed by using the function $triangular\_moments\_3$ for the moments greater than $-2$ (in absolute value).

Description

Compute the present value of an annuity-due considering only non-central moments of negative orders. The calculation is performed by using the function $triangular\_moments\_3$ for the moments greater than $-2$ (in absolute value).

Usage

PV_pre_triang_3(data,years)
PV_pre_triang_3(data,years)

Arguments

`data`	A vector of interest rates expressed as percentages.
`years`	The number of years of the income. Default is 10 years.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Examples


data=c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
PV_pre_triang_3(data,10)

data=c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
PV_pre_triang_3(data,10)

Compute the present value of an annuity-due considering only non-central moments of negative orders. The calculation is performed by using the moments of the fitted triangular distribution of the random variable "capitalization factor" $U$ (which are obtained from the definition of negative moment of a continuous random variable)

Description

Compute the present value of an annuity-due considering only non-central moments of negative orders. The calculation is performed by using the moments of the fitted triangular distribution of the random variable "capitalization factor" $U$ (which are obtained from the definition of negative moment of a continuous random variable)

Usage

PV_pre_triang_dis(data,years)
PV_pre_triang_dis(data,years)

Arguments

`data`	A vector of interest rates expressed as percentages.
`years`	The number of years of the income. Default is 10 years.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Examples


data=c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
PV_pre_triang_dis(data,10)

data=c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
PV_pre_triang_dis(data,10)

Compute the negatives moments (different from orders 1 and 2) of the fitted triangular distribution of the random variable X.

Description

Compute the negatives moments (different from orders 1 and 2) of the fitted triangular distribution of the random variable X.

Usage

triangular_moments_3(data,order)
triangular_moments_3(data,order)

Arguments

`data`	A vector X of interest rates.
`order`	The order of moment that should be computed.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Examples


#example 1
data=c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
triangular_moments_3(data,3)
triangular_moments_3(data,4)

# example 2 - first 10 negative moments of fitted triangular distribution 
#(an example from normal distributed simulated data)
data<-rnorm(n=200,m=0.75,sd=0.2)
triangular_parameters(data)
first10negmoments=rep(NA,10)  #except first and second
for (i in 3:10) first10negmoments[i]=triangular_moments_3(data,i)
first10negmoments



#example 1
data=c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
triangular_moments_3(data,3)
triangular_moments_3(data,4)

# example 2 - first 10 negative moments of fitted triangular distribution 
#(an example from normal distributed simulated data)
data<-rnorm(n=200,m=0.75,sd=0.2)
triangular_parameters(data)
first10negmoments=rep(NA,10)  #except first and second
for (i in 3:10) first10negmoments[i]=triangular_moments_3(data,i)
first10negmoments

Compute the negatives moments (different from orders 1 and 2) of the fitted triangular distribution of the random variable "capitalization factor" $U$ .

Description

Compute the negatives moments (different from orders 1 and 2) of the fitted triangular distribution of the random variable "capitalization factor" $U$ .

Usage

triangular_moments_3_U(data,order)
triangular_moments_3_U(data,order)

Arguments

`data`	A vector X of interest rates.
`order`	The order of moment that should be computed.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Examples


#example 1
data=c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
triangular_moments_3_U(data,3)
triangular_moments_3_U(data,4)

# example 2 - first 10 negative moments of fitted triangular distribution 
#(an example from normal distributed simulated data)
data<-rnorm(n=200,m=0.75,sd=0.2)
triangular_parameters(data)
first10negmoments=rep(NA,10)  #except first and second
for (i in 3:10) first10negmoments[i]=triangular_moments_3_U(data,i)
first10negmoments



#example 1
data=c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
triangular_moments_3_U(data,3)
triangular_moments_3_U(data,4)

# example 2 - first 10 negative moments of fitted triangular distribution 
#(an example from normal distributed simulated data)
data<-rnorm(n=200,m=0.75,sd=0.2)
triangular_parameters(data)
first10negmoments=rep(NA,10)  #except first and second
for (i in 3:10) first10negmoments[i]=triangular_moments_3_U(data,i)
first10negmoments

Compute the negative moments of the fitted triangular distribution of the random variable $X$ according to the definition (as integral).

Description

Compute the negative moments of the fitted triangular distribution of the random variable $X$ according to the definition (as integral).

Usage

triangular_moments_dis(data,order)
triangular_moments_dis(data,order)

Arguments

`data`	A vector of interest rates as percentage.
`order`	The order of moment of the triangular distribution

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Examples


# example 1
data=c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
triangular_moments_dis(data,1)
triangular_moments_dis(data,2)
triangular_moments_dis(data,3)
triangular_moments_dis(data,4)

# example 2 - first 10 negative moments of fitted triangular distribution 
#(an example from normal distributed simulated data)
data<-rnorm(n=200,m=0.75,sd=0.2)
triangular_parameters(data)
first10negmoments=rep(NA,10)
for (i in 1:10) first10negmoments[i]=triangular_moments_dis(data,i)
first10negmoments



# example 1
data=c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
triangular_moments_dis(data,1)
triangular_moments_dis(data,2)
triangular_moments_dis(data,3)
triangular_moments_dis(data,4)

# example 2 - first 10 negative moments of fitted triangular distribution 
#(an example from normal distributed simulated data)
data<-rnorm(n=200,m=0.75,sd=0.2)
triangular_parameters(data)
first10negmoments=rep(NA,10)
for (i in 1:10) first10negmoments[i]=triangular_moments_dis(data,i)
first10negmoments

Compute the negative moments of the fitted triangular distribution of the random variable "capitalization factor" $U$ according to the definition (as integral).

Description

Compute the negative moments of the fitted triangular distribution of the random variable "capitalization factor" $U$ according to the definition (as integral).

Usage

triangular_moments_dis_U(data,order)
triangular_moments_dis_U(data,order)

Arguments

`data`	A vector of interest rates as percentage.
`order`	The order of moment of the triangular distribution

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Examples


# example 1
data=c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
triangular_moments_dis_U(data,1)
triangular_moments_dis_U(data,2)
triangular_moments_dis_U(data,3)
triangular_moments_dis_U(data,4)

# example 2 - first 10 negative moments of fitted triangular distribution 
#(an example from normal distributed simulated data)
data<-rnorm(n=200,m=0.75,sd=0.2)
triangular_parameters(data)
first10negmoments=rep(NA,10)
for (i in 1:10) first10negmoments[i]=triangular_moments_dis_U(data,i)
first10negmoments



# example 1
data=c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
triangular_moments_dis_U(data,1)
triangular_moments_dis_U(data,2)
triangular_moments_dis_U(data,3)
triangular_moments_dis_U(data,4)

# example 2 - first 10 negative moments of fitted triangular distribution 
#(an example from normal distributed simulated data)
data<-rnorm(n=200,m=0.75,sd=0.2)
triangular_parameters(data)
first10negmoments=rep(NA,10)
for (i in 1:10) first10negmoments[i]=triangular_moments_dis_U(data,i)
first10negmoments

Compute the parameters and plot the fitted triangular distribution of the random variable $X$ .

Description

Compute the parameters and plot the fitted triangular distribution of the random variable $X$ .

Usage

triangular_parameters(data)
triangular_parameters(data)

Arguments

data

A vector of interest rates.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Examples


# example 1
data=c(0.00,-0.05,-0.05,-0.06,-0.06,0.02,-0.06,-0.05,-0.04,-0.05,
-0.03,-0.06,0.04,-0.05,-0.08,-0.05,-0.12,-0.03,-0.05,-0.04,-0.06)
triangular_parameters(data)

# example 2
data<-rnorm(n=200,m=0.75,sd=0.2)
triangular_parameters(data)


# example 3
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
triangular_parameters(data)

# example 1
data=c(0.00,-0.05,-0.05,-0.06,-0.06,0.02,-0.06,-0.05,-0.04,-0.05,
-0.03,-0.06,0.04,-0.05,-0.08,-0.05,-0.12,-0.03,-0.05,-0.04,-0.06)
triangular_parameters(data)

# example 2
data<-rnorm(n=200,m=0.75,sd=0.2)
triangular_parameters(data)


# example 3
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
triangular_parameters(data)

Return the parameters of the fitted triangular distribution of the random variable "capitalization factor" $U$ .

Description

Return the parameters of the fitted triangular distribution of the random variable "capitalization factor" $U$ .

Usage

triangular_parameters_U(data)
triangular_parameters_U(data)

Arguments

data

A vector of interest rates expressed as percentage.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Examples


# example 1
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
triangular_parameters_U(data)

# example 1
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
triangular_parameters_U(data)

Compute the variance of the present value of an annuity using "discrete random variable" approach.

Description

Compute the variance of the present value of an annuity using "discrete random variable" approach.

Usage

variance_drv(data,years)
variance_drv(data,years)

Arguments

`data`	A vector X of interest rates.
`years`	The number of years of the income. Default is 10 years.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Examples


# example 1
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
data=data/100
variance_drv(data)

# example 1
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
data=data/100
variance_drv(data)

Compute the variance of the present value of an annuity-immediate using the Mood et al. approximation and some non-central moments of negative order.

Description

Compute the variance of the present value of an annuity-immediate using the Mood et al. approximation and some non-central moments of negative order.

Usage

variance_post_mood_nm(data,years)
variance_post_mood_nm(data,years)

Arguments

`data`	A vector X of interest rates.
`years`	The number of years of the income. Default is 10 years.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Examples


# example 1
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
data=data/100
variance_post_mood_nm(data)

# example 1
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
data=data/100
variance_post_mood_nm(data)

Compute the variance of the present value of an annuity-immediate using the Mood et al. approximation and some non-central moments of positive order.

Description

Compute the variance of the present value of an annuity-immediate using the Mood et al. approximation and some non-central moments of positive order.

Usage

variance_post_mood_pm(data,years)
variance_post_mood_pm(data,years)

Arguments

`data`	A vector X of interest rates.
`years`	The number of years of the income. Default is 10 years.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Examples


# example 1
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
data=data/100
variance_post_mood_pm(data)

# example 1
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
data=data/100
variance_post_mood_pm(data)

Compute the variance of the present value of an annuity-due using the Mood et al. approximation and some non-central moments of negative order.

Description

Compute the variance of the present value of an annuity-due using the Mood et al. approximation and some non-central moments of negative order.

Usage

variance_pre_mood_nm(data,years)

variance_pre_mood_nm(data,years)

Arguments

`data`	A vector X of interest rates.
`years`	The number of years of the income. Default is 10 years.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Examples


# example 1
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
data=data/100
variance_pre_mood_nm(data)

# example 1
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
data=data/100
variance_pre_mood_nm(data)

Compute the variance of the present value of an annuity-due using the Mood et al. approximation and some non-central moments of positive order.

Description

Compute the variance of the present value of an annuity-due using the Mood et al. approximation and some non-central moments of positive order.

Usage

variance_pre_mood_pm(data,years)
variance_pre_mood_pm(data,years)

Arguments

`data`	A vector X of interest rates.
`years`	The number of years of the income. Default is 10 years.

Author(s)

Salvador Cruz Rambaud, Fabrizio Maturo, Ana María Sánchez Pérez

Examples


# example 1
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
data=data/100
variance_pre_mood_pm(data)

# example 1
data = c(1.77,1.85,1.85,1.84,1.84,1.83,1.85,1.85,1.88,1.85,1.80,1.84,1.91,1.85,1.84,1.85,
1.86,1.85,1.88,1.86)
data=data/100
variance_pre_mood_pm(data)

Package 'AnnuityRIR'

Help Index

Compute the parameters of the beta distribution and plot normalized data.

Description

Usage

Arguments

Author(s)

Source

Examples

Compute the final expected value of an nnn-payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate XXX, using the tetraparametric function approach.

Description

Usage

Arguments

Author(s)

Source

Examples

Compute the final expected value of an nnn-payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate XXX, using the estimated moments of the beta distribution.

Description

Usage

Arguments

Author(s)

Source

Examples

Compute the final expected value of an nnn-payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate XXX, using the method of Mood et al.

Description

Usage

Arguments

Author(s)

Source

Examples

Compute the final expected value of an nnn-payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate XXX, using the estimated moments of the normal distribution.

Description

Usage

Arguments

Author(s)

Source

Examples

Compute the final expected value of an nnn-payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate XXX, using the quadratic discount method.

Description

Usage

Arguments

Author(s)

Source

Examples

Compute the final expected value of an n-payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate XXX, using the tetraparametric function approach.

Description

Usage

Arguments

Author(s)

Source

Examples

Compute the final expected value of an nnn-payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate XXX, using the estimated moments of the beta distribution.

Description

Usage

Arguments

Author(s)

Source

Examples

Compute the final expected value of an nnn-payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate XXX, using the method of Mood et al.

Description

Usage

Arguments

Author(s)

Source

Examples

Compute the final expected value of an nnn-payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate XXX, using the estimated moments of the normal distribution.

Description

Usage

Arguments

Author(s)

Source

Examples

Compute the final expected value of an nnn-payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate XXX, using the quadratic discount method.

Description

Usage

Arguments

Author(s)

Source

Examples

Compute the exact moments of a distribution.

Compute the final expected value of an $n$ -payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate $X$ , using the tetraparametric function approach.

Compute the final expected value of an $n$ -payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate $X$ , using the estimated moments of the beta distribution.

Compute the final expected value of an $n$ -payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate $X$ , using the method of Mood et al.

Compute the final expected value of an $n$ -payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate $X$ , using the estimated moments of the normal distribution.

Compute the final expected value of an $n$ -payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate $X$ , using the quadratic discount method.

Compute the final expected value of an n-payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate $X$ , using the tetraparametric function approach.

Compute the final expected value of an $n$ -payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate $X$ , using the estimated moments of the beta distribution.

Compute the final expected value of an $n$ -payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate $X$ , using the method of Mood et al.

Compute the final expected value of an $n$ -payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate $X$ , using the estimated moments of the normal distribution.

Compute the final expected value of an $n$ -payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate $X$ , using the quadratic discount method.

Plot the final expected value of an $n$ -payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate $X$ , using the estimated moments of the beta distribution.

Plot the final expected value of an $n$ -payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate $X$ , using the estimated moments of the normal distribution.

Plot the final expected value of an $n$ -payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate $X$ , using the estimated moments of the beta distribution.

Plot the final expected value of an $n$ -payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate $X$ , using the estimated moments of the normal distribution.

Plot the final expected values of an $n$ -payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate $X$ , using different approaches.

Plot the final expected values of an $n$ -payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate $X$ , using different approaches.

Plot the present expected values of an $n$ -payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate $X$ , using different approaches.

Plot the present expected values of an $n$ -payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate $X$ , using different approaches.

Compute present expected value of an $n$ -payment annuity, with payments of 1 unit each, made at the end of every year (annuity-immediate), valued at the rate $X$ , using the tetraparametric function approach.