trackingPortfolio: Compute a Tracking Portfolio

In NMOF: Numerical Methods and Optimization in Finance

Compute a Tracking Portfolio

Description

Computes a portfolio similar to a benchmark, e.g. for tracking the benchmark's performance or identifying factors.

Usage

trackingPortfolio(var, wmin = 0, wmax = 1,
                  method = "qp", objective = "variance", R,
                  ls.algo = list())

Arguments

var	the covariance matrix: a numeric (real), symmetric matrix. The first asset is the benchmark.
R	a matrix of returns: each colums holds the returns of one asset; each rows holds the returns for one observation. The first asset is the benchmark.
wmin	numeric: a lower bound on weights. May also be a vector that holds specific bounds for each asset.
wmax	numeric: an upper bound on weights. May also be a vector that holds specific bounds for each asset.
method	character. Currently, "qp" and "ls" are supported.
objective	character. Currently, "variance" and "sum.of.squares" are supported.
ls.algo	a list of named elements, for settings for method ‘ls’; see Details

Details

With method "qp", the function uses solve.QP from package quadprog. Because of the algorithm that solve.QP uses, var has to be positive definite (i.e. must be of full rank).

With method "ls", the function uses LSopt. Settings can be passed via ls.algo, which corresponds to LSopt's argument algo. Default settings are 2000 iterations and printBar, printDetail set to FALSE.

R is needed only when objective is "sum.of.squares" or method is ‘ls’. (See Examples.)

Value

a numeric vector (the portfolio weights)

Author(s)

Enrico Schumann

References

Gilli, M., Maringer, D. and Schumann, E. (2019) Numerical Methods and Optimization in Finance. 2nd edition. Elsevier. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/C2017-0-01621-X")}

Schumann, E. (2023) Financial Optimisation with R (NMOF Manual). https://enricoschumann.net/NMOF.htm#NMOFmanual

Schumann, E. (2020) Return-based tracking portfolios. https://enricoschumann.net/notes/return-based-tracking-portfolios.html

Sharpe, W. F. (1992) Asset Allocation: Management Style and Performance Measurement. Journal of Portfolio Management. 18 (2), 7–19. https://web.stanford.edu/~wfsharpe/art/sa/sa.htm

See Also

minvar

Examples

if (requireNamespace("quadprog")) {
    ns <- 120
    R <- randomReturns(na = 1 + 20,
                       ns = ns,
                       sd = 0.03,
                       mean = 0.005,
                       rho = 0.7)

    var <- cov(R)

    sol.qp <- trackingPortfolio(var, wmax = 0.4)
    sol.ls <- trackingPortfolio(var = var, R = R, wmax = 0.4, method = "ls")
    data.frame(QP = round(100*sol.qp, 1),
               LS = round(100*sol.ls, 1))

    sol.qp <- trackingPortfolio(var, R = R, wmax = 0.4,
                                objective = "sum.of.squares")
    sol.ls <- trackingPortfolio(var = var, R = R, wmax = 0.4, method = "ls",
                                objective = "sum.of.squares")
    data.frame(QP = round(100*sol.qp, 1),
               LS = round(100*sol.ls, 1))


    ## same as 'sol.qp' above
    sol.qp.R   <- trackingPortfolio(R = R,
                                    wmax = 0.4,
                                    objective = "sum.of.squares")
    sol.qp.var <- trackingPortfolio(var = crossprod(R),
                                    wmax = 0.4,
                                    objective = "variance")
    ## ==> should be the same
    all.equal(sol.qp.R, sol.qp.var)
}

NMOF documentation built on Nov. 5, 2025, 6:38 p.m.