# Portfolio optimization which finds an optimal portfolio with the smallest distance to a benchmark. PortfolioOptimProjection is a linear program for financial portfolio optimization. The function finds an optimal portfolio which has the smallest distance to a benchmark portfolio given by `bvec`. Solution is by the algorithm due to Zhao and Li modified to account for the fact that the benchmark portfolio `bvec` has the dimension of portfolio weights and the solved linear program has a much higher dimension since the solution vector to the LP problem consists of a set of primal variables: financial portfolio weights, auxiliary variables coming from the reduction of the mean-risk problem to a linear program and also a set of dual variables depending on the number of constrains in the primal problem (see Palczewski). ```r PortfolioOptimProjection (dat, portfolio_return, risk=c("CVAR","DCVAR","LSAD","MAD"), alpha=0.95, bvec, Aconstr=NULL, bconstr=NULL, LB=NULL, UB=NULL, maxiter=500, tol=1e-7) ``` ## Arguments - `dat`: Time series of returns data; dat = cbind(rr, pk), where $rr$ is an array (time series) of asset returns, for $n$ returns and $k$ assets it is an array with $\dim(rr) = (n, k)$, $pk$ is a vector of length $n$ containing probabilities of returns. - `portfolio_return`: Target portfolio return. - `risk`: Risk measure chosen for optimization; one of "CVAR", "DCVAR", "LSAD", "MAD", where "CVAR" -- denotes Conditional Value-at-Risk (CVaR), "DCVAR" -- denotes deviation CVaR, "LSAD" -- denotes Lower Semi Absolute Deviation, "MAD" -- denotes Mean Absolute Deviation. - `alpha`: Value of alpha quantile used to compute portfolio VaR and CVaR; used also as quantile value for risk measures CVAR and DCVAR. - `bvec`: Benchmark portfolio, a vector of length k; function `PortfolioOptimProjection` finds an optimal portfolio with the smallest distance to `bvec`. - `Aconstr`: Matrix defining additional constraints, $\dim (Aconstr) = (m,k)$, where $k$ -- number of assets, $m$ -- number of constraints. - `bconstr`: Vector defining additional constraints, length ($bconstr$) $ = m$. - `LB`: Vector of length k, lower bounds of portfolio weights $\theta$; warning: condition LB = NULL is equivalent to LB = rep(0, k) (lower bound zero). - `UB`: Vector of length k, upper bounds for portfolio weights $\theta$. - `maxiter`: Maximal number of iterations. - `tol`: Accuracy of computations, stopping rule. ## Returns PortfolioOptimProjection returns a list with items: ||||| |:--|:--|:--|:--| |`return_mean`|vector of asset returns mean values.| |`mu`|realized portfolio return.| |`theta`|portfolio weights.| |`CVaR`|portfolio CVaR.| |`VaR`|portfolio VaR.| |`MAD`|portfolio MAD.| |`risk`|portfolio risk measured by the risk measure chosen for optimization.| |`new_portfolio_return`|modified target portfolio return; when the original target portfolio return| |` `|is to high for the problem, the optimization problem is solved for| |` `|new_portfolio_return as the target return.| ## Examples ```r library(mvtnorm) k = 3 num =100 dat <- cbind(rmvnorm (n=num, mean = rep(0,k), sigma=diag(k)), matrix(1/num,num,1)) # a data sample with num rows and (k+1) columns for k assets; w_m <- rep(1/k,k) # benchmark portfolio, a vector of length k, port_ret = 0.05 # portfolio target return alpha_optim = 0.95 # minimal constraints set: \sum theta_i = 1 # has to be in two inequalities: 1 - \epsilon <= \sum theta_i <= 1 +\epsilon a0 <- rep(1,k) Aconstr <- rbind(a0,-a0) bconstr <- c(1+1e-8, -1+1e-8) LB <- rep(0,k) UB <- rep(1,k) res <- PortfolioOptimProjection(dat, port_ret, risk="MAD", alpha=alpha_optim, w_m, Aconstr, bconstr, LB, UB, maxiter=200, tol=1e-7) cat ( c("Projection optimal portfolio:\n\n")) cat(c("weights \n")) print(res$theta) cat (c ("\n mean = ", res$mu, " risk = ", res$risk, "\n CVaR = ", res$CVaR, " VaR = ", res$VaR, "\n MAD = ", res$MAD, "\n\n")) ``` ## References Palczewski, A., LP Algorithms for Portfolio Optimization: The PortfolioOptim Package, R Journal, 10(1) (2018), 308--327. DOI:10.32614/RJ-2018-028. Zhao, Y-B., Li, D., Locating the least 2-norm solution of linear programs via a path-following method, SIAM Journal on Optimization, 12 (2002), 893--912. DOI:10.1137/S1052623401386368.

PortfolioOptimProjection function