minvar function

Minimum-Variance Portfolios

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

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).

Returns

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

References

Gilli, M., Maringer, D. and Schumann, E. (2019) Numerical Methods and Optimization in Finance. 2nd edition. Elsevier. 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. (2012) Computing the global minimum-variance portfolio. https://enricoschumann.net/R/minvar.htm

Author(s)

Enrico Schumann

See Also

TAopt

Examples

## variance-covariance matrix from daily returns, 1 Jan 2014 -- 31 Dec 2013, of
## cleaned data set at https://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
}