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)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 |
groups |
a list of group definitions |
groups.wmin |
a numeric vector |
groups.wmax |
a numeric vector |
a numeric vector (the portfolio weights) with an attribute
variance (the portfolio's variance)
Enrico Schumann
Gilli, M., Maringer, D. and Schumann, E. (2019) Numerical Methods and Optimization in Finance. 2nd edition. Elsevier. https://www.elsevier.com/books/numerical-methods-and-optimization-in-finance/gilli/978-0-12-815065-8
Schumann, E. (2012) Computing the global minimum-variance portfolio. http://enricoschumann.net/R/minvar.htm
Schumann, E. (2019) Financial Optimisation with R (NMOF Manual). http://enricoschumann.net/NMOF.htm#NMOFmanual
## 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
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