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Here's an interesting puzzle. Below is an R snippet that identifies the tangency point of a quadratic function with respect to a line drawn from the point (0,rf) on the y-axis. For those familiar with portfolio theory, this point is in return and risk space and the solution is set of weights that define the tangency portfolio (max sharpe ratio). The snippet allows for negative weights (i.e. shorts) and there is one equality weight constraint which requires the sum of the weights = 1. require(quadprog) # create artifical data nO <- 100 # number of observations nA <- 10 # number of assets mData <- array(rnorm(nO * nA, mean = 0.001, sd = 0.01), dim = c(nO, nA)) rf <- 0.0001 # riskfree rate (2.5% pa) mu <- apply(mData, 2, mean) # means mu2 <- mu - rf # excess means # qp aMat <- as.matrix(mu2) bVec <- 1 # set expectation of portfolio excess return to 1 zeros <- array(0, dim = c(nA,1)) solQP <- solve.QP(cov(mData), zeros, aMat, bVec, meq = 1) # rescale variables to obtain weights w <- as.matrix(solQP$solution/sum(solQP$solution)) # compute sharpe ratio SR <- t(w) %*% mu2 / sqrt(t(w) %*% cov(mData) %*% w) My question -- how to adapt the code to solve for the optimal set of weights such that the sum of weights sum to an arbitrary number (including the corner case of a self-financing portfolio where the sum of weights = 0) as opposed to unity? Alternatively, you might consider adding an element 'cash' to the covariance matrix with variance-covariance of 0, and add an equality constraint requiring the weight on cash = 1. However this matrix would be not be positive semi-definite. Also I suspect the non-cash weights might be trivially zero.

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