qrisk {quantreg} | R Documentation

## 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)))

[Package quantreg version 5.24 Index]