qrisk {quantreg}

Function to compute Choquet portfolio weights

Description

This function solves a weighted quantile regression problem to find the optimal portfolio weights minimizing a Choquet risk criterion described in Bassett, Koenker, and Kordas (2002).

Usage

qrisk(x, alpha = c(0.1, 0.3), w = c(0.7, 0.3), mu = 0.07, 
      R = NULL, r = NULL, lambda = 10000)

Arguments

x

n by q matrix of historical or simulated asset returns

alpha

vector of alphas receiving positive weights in the Choquet criterion

w

weights associated with alpha in the Choquet criterion

mu

targeted rate of return for the portfolio

R

matrix of constraints on the parameters of the quantile regression, see below

r

rhs vector of the constraints described by R

lambda

Lagrange multiplier associated with the constraints

Details

The function calls rq.fit.hogg which in turn calls the constrained Frisch Newton algorithm. The constraints Rb=r are intended to apply only to the slope parameters, not the intercept parameters. The user is completely responsible to specify constraints that are consistent, ie that have at least one feasible point. See examples for imposing non-negative portfolio weights.

Value

pihat

the optimal portfolio weights

muhat

the in-sample mean return of the optimal portfolio

qrisk

the in-sample Choquet risk of the optimal portfolio

Author(s)

R. Koenker

References

http://www.econ.uiuc.edu/~roger/research/risk/risk.html

Bassett, G., R. Koenker, G Kordas, (2004) Pessimistic Portfolio Allocation and Choquet Expected Utility, J. of Financial Econometrics, 2, 477-492.

See Also

rq.fit.hogg, srisk

Examples

#Fig 1:  ... of Choquet paper
        mu1 <- .05; sig1 <- .02; mu2 <- .09; sig2 <- .07
        x <- -10:40/100
        u <- seq(min(c(x)),max(c(x)),length=100)
        f1 <- dnorm(u,mu1,sig1)
        F1 <- pnorm(u,mu1,sig1)
        f2 <- dchisq(3-sqrt(6)*(u-mu1)/sig1,3)*(sqrt(6)/sig1)
        F2 <- pchisq(3-sqrt(6)*(u-mu1)/sig1,3)
        f3 <- dnorm(u,mu2,sig2)
        F3 <- pnorm(u,mu2,sig2)
        f4 <- dchisq(3+sqrt(6)*(u-mu2)/sig2,3)*(sqrt(6)/sig2)
        F4 <- pchisq(3+sqrt(6)*(u-mu2)/sig2,3)
        plot(rep(u,4),c(f1,f2,f3,f4),type="n",xlab="return",ylab="density")
        lines(u,f1,lty=1,col="blue")
        lines(u,f2,lty=2,col="red")
        lines(u,f3,lty=3,col="green")
        lines(u,f4,lty=4,col="brown")
        legend(.25,25,paste("Asset ",1:4),lty=1:4,col=c("blue","red","green","brown"))
#Now generate random sample of returns from these four densities.
n <- 1000
if(TRUE){ #generate a new returns sample if TRUE
	x1 <- rnorm(n)
	x1 <- (x1-mean(x1))/sqrt(var(x1))
	x1 <- x1*sig1 + mu1
	x2 <- -rchisq(n,3)
	x2 <- (x2-mean(x2))/sqrt(var(x2))
	x2 <- x2*sig1 +mu1
	x3 <- rnorm(n)
	x3 <- (x3-mean(x3))/sqrt(var(x3))
	x3 <- x3*sig2 +mu2
	x4 <- rchisq(n,3)
	x4 <- (x4-mean(x4))/sqrt(var(x4))
	x4 <- x4*sig2 +mu2
	}
library(quantreg)
x <- cbind(x1,x2,x3,x4)
qfit <- qrisk(x)
sfit <- srisk(x)
# Try new distortion function
qfit1 <- qrisk(x,alpha = c(.05,.1), w = c(.9,.1),mu = 0.09)
# Constrain portfolio weights to be non-negative
qfit2 <- qrisk(x,alpha = c(.05,.1), w = c(.9,.1),mu = 0.09,
	       R = rbind(rep(-1,3), diag(3)), r = c(-1, rep(0,3)))