minvar {NMOF}

Minimum-Variance Portfolios

Description

Compute minimum-variance portfolios, subject to lower and upper bounds on weights.

Usage

minvar(var, wmin = 0, wmax = 1, method = "qp",
       groups = NULL, groups.wmin = NULL, groups.wmax = NULL)

Arguments

var

the covariance matrix: a numeric (real), symmetric matrix

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, only "qp" is supported.

groups

a list of group definitions

groups.wmin

a numeric vector

groups.wmax

a numeric vector

Details

For 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).

Value

a numeric vector (the portfolio weights) with an attribute variance (the portfolio's variance)

Author(s)

Enrico Schumann

References

Gilli, M., Maringer, D. and Schumann, E. (2019) Numerical Methods and Optimization in Finance. 2nd edition. Elsevier. doi:10.1016/C2017-0-01621-X

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

Schumann, E. (2012) Computing the global minimum-variance portfolio. http://enricoschumann.net/R/minvar.htm

See Also

TAopt

Examples

## variance-covariance matrix from daily returns, 1 Jan 2014 -- 31 Dec 2013, of
## cleaned data set at http://enricoschumann.net/data/gilli_accuracy.html

if (requireNamespace("quadprog")) {

    var <- structure(c(0.000988087100677907, -0.0000179669410403153, 0.000368923882626859,
                       0.000208303611101873, 0.000262742052359594, -0.0000179669410403153,
                       0.00171852167358765, 0.0000857467457561209, 0.0000215059246610556,
                       0.0000283532159921211, 0.000368923882626859, 0.0000857467457561209,
                       0.00075871953281751, 0.000194002299424151, 0.000188824454515841,
                       0.000208303611101873, 0.0000215059246610556, 0.000194002299424151,
                       0.000265780633005374, 0.000132611196599808, 0.000262742052359594,
                       0.0000283532159921211, 0.000188824454515841, 0.000132611196599808,
                       0.00025948420130626),
                     .Dim = c(5L, 5L),
                     .Dimnames = list(c("CBK.DE", "VOW.DE", "CON.DE", "LIN.DE", "MUV2.DE"),
                                      c("CBK.DE", "VOW.DE", "CON.DE", "LIN.DE", "MUV2.DE")))

    ##            CBK.DE     VOW.DE    CON.DE    LIN.DE   MUV2.DE
    ## CBK.DE   0.000988 -0.0000180 0.0003689 0.0002083 0.0002627
    ## VOW.DE  -0.000018  0.0017185 0.0000857 0.0000215 0.0000284
    ## CON.DE   0.000369  0.0000857 0.0007587 0.0001940 0.0001888
    ## LIN.DE   0.000208  0.0000215 0.0001940 0.0002658 0.0001326
    ## MUV2.DE  0.000263  0.0000284 0.0001888 0.0001326 0.0002595
    ##

    minvar(var, wmin = 0, wmax = 0.5)

    minvar(var,
           wmin = c(0.1,0,0,0,0), ## enforce at least 10% weight in CBK.DE
           wmax = 0.5)

    minvar(var, wmin = -Inf, wmax = Inf)   ## no bounds
    ## [1] -0.0467  0.0900  0.0117  0.4534  0.4916

    minvar(var, wmin = -Inf, wmax = 0.45)  ## no lower bounds
    ## [1] -0.0284  0.0977  0.0307  0.4500  0.4500

    minvar(var, wmin =  0.1, wmax = Inf)   ## no upper bounds
    ## [1] 0.100 0.100 0.100 0.363 0.337

    ## group constraints:
    ##   group 1 consists of asset 1 only,   and must have weight [0.25,0.30]
    ##   group 2 consists of assets 4 and 5, and must have weight [0.10,0.20]

    ##   => unconstrained
    minvar(var, wmin = 0, wmax = 0.40)
    ## [1] 0.0097 0.1149 0.0754 0.4000 0.4000

    ##   => with group constraints
    minvar(var, wmin = 0, wmax = 0.40,
           groups = list(1, 4:5),
           groups.wmin = c(0.25, 0.1),
           groups.wmax = c(0.30, 0.2))
    ## [1] 0.250 0.217 0.333 0.149 0.051
}