Simulating financial strategies with given value-at-risk (VaR) and mean

Question

I would like to generate thousands of distributions with respect to: VaR@5%=-7% and Median=0%.

A first idea:

1. Choose a family, say the Gaussian one.

2. Estimate its parameters. This is straightforward for the Gaussian but let's write the general algorithm (see mpiktas' answer on Estimating a distribution based on three percentiles):

   library(nleqslv)
   VaR_Threshold <- c(0.05,0.5)
   VaR_Value <- c(-0.07,0)
   
   ufn <- function(x,q) q-pnorm(VaR_Value, x[1],x[2])
   usol <- nleqslv(c(0,1), ufn,q=VaR_Threshold)
   usol$x
   
   [1] -1.262016e-11  4.255698e-02 (estimated parameters)
   
   plot(x<-seq(-1,1,by=0.01),dnorm(x,usol$x[1],usol$x[2]),type="l",col=2)
   points(p<-VaR_Value,dnorm(p,usol$x[1],usol$x[2]))
   
   pnorm(VaR_Value,usol$x[1],usol$x[2])
   
   [1] 0.05 0.50 (check)
   

So this is for the Gaussian family: one unique solution.

My question: is there a parametric family flexible enough to generate thousands of distributions with a given VaR and median? I tried with the stable distributions (4 parameters : 2 manually fixed + 2 optimized), in vain, probably because I don't master it...

Answer 1

Almost any family with more than two parameters will do: you just have to fix all the parameters but two. Here is an example with stable distributions.

library(fBasics)
VaR_Threshold <- c( 0.05, 0.5)
VaR_Value     <- c(-0.07, 0)
alpha <- 1.2  # in (0,2]
beta  <- 0
# The third argument, gamma, should be positive.
f <- function(x) {
  r <- sum(( 
    VaR_Threshold - 
    pstable(VaR_Value, alpha, beta, abs(x[1]), x[2], pm=2)
  )^2)
  cat(r, "\n")  # for debugging
  r
}
p <- optim(c(1,0),f)$par
pstable(VaR_Value, alpha, beta, abs(p[1]), p[2], pm=2)