tugas aktuaria 3.1.1, 3.1.2, 3.1.4, 3.1.5

soal 3.1.1

contoh.3.1.1 <- function(age, n, delta) {
  int <- function(t) { exp(-delta*t)*(t/5000) }
  output <- integrate(int, 0, n)$value
  output
}

50*contoh.3.1.1(age=50, n=100, delta=0.1)

## [1] 0.9995006

soal 3.1.2

contoh.3.1.2 <- function(age, n, delta) {
  mu <- 1/(100-age)
  int <- function(t){ mu*exp(-(delta+mu)*t) }
  output <- integrate(int, 0, 25)$value
  output 
}

contoh.3.1.2(age=50, n=25, delta=0.05)

## [1] 0.2360646

soal 3.1.4

contoh.3.1.4 <- function(m, delta, mu) {
  int <- function(t) { exp(-(delta+mu)*t)*mu }
  output <- integrate(int, m, Inf)$value
  output 
}

contoh.3.1.4(m=5, delta=0.1, mu=0.04)

## [1] 0.1418815

soal 3.1.5

Ax <- function(delta, age, benefit, B, C, w=500) {
  int <- function(t) { 
    benefit*exp(-delta*t)*
    (exp((-B/log(C))*(C^(age+t)-1)))/
    (exp((-B/log(C))*(C^age-1)))*(B*C^(age+t)) 
  }
  Ax.value <- integrate(int, 0, w-age)$value
  Ax.value 
}

mAX <- function(delta, age, benefit, B, C, m, w=500) {
  int <- function(t) { 
    benefit*exp(-delta*t)*
    (exp((-B/log(C))*(C^(age+t)-1)))/
    (exp((-B/log(C))*(C^age-1)))*(B*C^(age+t)) 
  }
  mAX.value <- integrate(int, m, w-age)$value
  mAX.value 
}

Axn <- function(delta, age, benefit, B, C, n) {
  int <- function(t) { 
    benefit*exp(-delta*t)*
    (exp((-B/log(C))*(C^(age+t)-1)))/
    (exp((-B/log(C))*(C^age-1)))*(B*C^(age+t)) 
  }
  Axn.value <- integrate(int, 0, n)$value
  Axn.value 
}

mAxn <- function(delta, age, benefit, B, C, n, m) {
  int <- function(t) { 
    benefit*exp(-delta*t)*
    (exp((-B/log(C))*(C^(age+t)-1)))/
    (exp((-B/log(C))*(C^age-1)))*(B*C^(age+t))
  }
  mAxn.value <- integrate(int, m, m+n)$value
  mAxn.value 
}

Exn <- function(delta, age, benefit, B, C, n) {
  Exn.value <- benefit*exp(-delta*n)*
    (exp((-B/log(C))*(C^(age+n)-1)))/
    (exp((-B/log(C))*(C^age-1)))
  Exn.value
}

endowment <- function(delta, age, benefit, B, C, n) {
  int <- function(t) { 
    benefit*exp(-delta*t)*
    (exp((-B/log(C))*(C^(age+t)-1)))/
    (exp((-B/log(C))*(C^age-1)))*(B*C^(age+t))
  }
  Axn.value <- integrate(int, 0, n)$value
  Exn.value <- benefit*exp(-delta*n)*
    (exp((-B/log(C))*(C^(age+n)-1)))/
    (exp((-B/log(C))*(C^age-1)))
  endowment.value = Axn.value + Exn.value
  endowment.value 
}

IAxn <- function(delta, age, benefit, B, C, n, increase) {
  int <- function(t) { 
    (benefit+increase*t)*
    exp(-delta*t)*(B*C^(age+t))*
    (exp((-B/log(C))*(C^(age+t)-1)))/
    (exp((-B/log(C))*(C^age-1))) 
  }
  IAxn.value <- integrate(int, 0, n)$value
  IAxn.value 
}

DAxn <- function(delta, age, benefit, B, C, n, decrease) {
  int <- function(t) { 
    (benefit-decrease*t)*
    exp(-delta*t)*(B*C^(age+t))*
    (exp((-B/log(C))*(C^(age+t)-1)))/
    (exp((-B/log(C))*(C^age-1))) 
  }
  DAxn.value <- integrate(int, 0, n)$value
  DAxn.value 
}

Ax(delta=0.05, age=30, benefit=1e+7, B=1e-5, C=1.09)

## [1] 456884.7

mAX(delta=0.05, age=30, benefit=1e+7, B=1e-5, C=1.09, m=5)

## [1] 449615.9

Axn(delta=0.05, age=30, benefit=1e+7, B=1e-5, C=1.09, n=20)

## [1] 38822.14

mAxn(delta=0.05, age=30, benefit=1e+7, B=1e-5, C=1.09, n=20, m=10)

## [1] 55400.56

Exn(delta=0.05, age=30, benefit=1e+7, B=1e-5, C=1.09, n=15)

## [1] 4704487

endowment(delta=0.05, age=30, benefit=1e+7, B=1e-5, C=1.09, n=30)

## [1] 2260925

IAxn(delta=0.05, age=30, benefit=1e+7, B=1e-5, C=1.09, n=4, increase=5e+6)

## [1] 11554.09

DAxn(delta=0.05, age=30, benefit=1e+7, B=1e-5, C=1.09, n=5, decrease=1e+6)

## [1] 5397.108