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#' Monte Carlo VaR for DB pension
#'
#' Generates Monte Carlo VaR for DB pension in Chapter 6.7.
#'
#' @param mu Expected rate of return on pension-fund assets
#' @param sigma Volatility of rate of return of pension-fund assets
#' @param p Probability of unemployment in any period
#' @param life.expectancy Life expectancy
#' @param number.trials Number of trials
#' @param cl VaR confidence level
#' @return VaR for DB pension
#' @references Dowd, Kevin. Measuring Market Risk, Wiley, 2007.
#'
#' @author Dinesh Acharya
#' @examples
#'
#' # Estimates the price of an American Put
#' DBPensionVaR(.06, .2, .05, 80, 100, .95)
#'
#' @export
DBPensionVaR <- function(mu, sigma, p, life.expectancy, number.trials, cl){
# Parameter Setting
contribution.rate <- .15
initial.income <- 25
income.growth.rate <- .02
M <- number.trials
L <- life.expectancy
# r is return on investment
# Asset Side
# Initialization
r <- matrix(0, 40, M)
fund <- matrix(0, 40, M)
employment.state <- matrix(0, 40, M)
actual.income <- matrix(0, 40, M)
contribution <- matrix(0, 40, M)
years.contributed <- matrix(0, 40, M)
terminal.fund <- double(M)
terminal.return <- double(M)
years.contributed <- matrix(0, 40, M)
employment.income <- matrix(0, 40, M)
total.years.contributed <- double(M)
for (j in 1:M) {
fund[1, j] <- contribution.rate * initial.income
years.contributed[1, j] <- 1
for (i in 2:40) {
r[i, j] <- rnorm(1, mu, sigma)
employment.state[i, j] <- rbinom(1,1,1-p)
employment.income[i, j] <- initial.income * exp(income.growth.rate*(i-1))
actual.income[i, j] <- employment.state[i, j] * employment.income[i, j]
contribution[i, j] <- contribution.rate * actual.income[i, j]
fund[i, j] <- contribution[i, j] + fund[i - 1, j] * (1 + r[i, j])
terminal.fund[j] <- fund[i, j]
terminal.return[j] <- r[i, j]
years.contributed[i, j] <- employment.state[i, j] + years.contributed[i - 1, j]
total.years.contributed[j] <- years.contributed[i,j]
}
}
mean.terminal.fund <- mean(terminal.fund)
std.terminal.fund <- sd(terminal.fund)
terminal.employment.income <- (1 - p) * initial.income * exp(income.growth.rate * 39)
pension <- double(M)
annuity.rate <- double(M)
implied.fund <- double(M)
for (j in 1:M){
pension[j] <- (total.years.contributed[j] / 40) * terminal.employment.income
annuity.rate[j] <- .04
implied.fund[j] <- pvfix(annuity.rate[j], L-65, pension[j])
}
mean.terminal.employment.income <- mean(terminal.employment.income)
mean.t.years.contributed <- mean(total.years.contributed)
mean.pension <- mean(pension)
mean.implied.func <- mean(implied.fund)
std.implied.fund <- sd(implied.fund)
# Profit Loss and VaR
profit.or.loss <- terminal.fund - implied.fund
mean.profit.or.loss <- mean(profit.or.loss)
std.profit.or.loss <- sd(profit.or.loss)
hist(-profit.or.loss, 20)
y <- HSVaR(profit.or.loss, cl)
return(y)
}
# Accessory functions
pvfix<-function(r, n, c){
# pvfix computes the present value of a series of future cashflows (e.g. savings)
# parameters:
# r interest rate per period (constant throughout the period)
# n number of periods
# c cashflow each month (assumed to be fixed)
s <- (c/r)*(1-(1/(1+r)^n))
return(s)
}
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