# Function to compute Choquet portfolio weights 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). ```r 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. ## Returns - **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 ## References [http://www.econ.uiuc.edu/~roger/research/risk/risk.html](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. ## Author(s) R. Koenker ## Examples ```r #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))) ``` ## See Also `rq.fit.hogg`, `srisk`

qrisk function