Computing Mean–Variance Efficient Portfolios
Compute mean–variance efficient portfolios and efficient frontiers.
mvFrontier(m, var, wmin = 0, wmax = 1, n = 50, rf = NA,
groups = NULL, groups.wmin = NULL, groups.wmax = NULL)
mvPortfolio(m, var, min.return, wmin = 0, wmax = 1, lambda = NULL,
groups = NULL, groups.wmin = NULL, groups.wmax = NULL)m |
vector of expected returns |
var |
expected variance–covariance matrix |
wmin |
numeric: minimum weights |
wmax |
numeric: maximum weights |
n |
number of points on the efficient frontier |
min.return |
minimal required return |
rf |
risk-free rate |
lambda |
risk–reward trade-off |
groups |
a list of group definitions |
groups.wmin |
a numeric vector |
groups.wmax |
a numeric vector |
mvPortfolio computes a single mean–variance
efficient portfolio, using package quadprog.
It does so by minimising portfolio variance, subject
to constraints on minimum return and budget (weights
need to sum to one), and min/max constraints on the
weights.
If \code{lambda}
is specified, the function ignores the min.return
constraint and instead solves the model
min -lambda m'w - (1 - lambda) w'var w
in which \code{w} are the weights. If \code{lambda} is a vector of length 2, then the model becomes
min -lambda_1 m'w - lambda_2 w'var w
which may be more convenient (e.g. for setting \code{lambda_1} to 1).
mvFrontier computes returns, volatilities and
compositions for portfolios along an efficient frontier.
If rf is not NA, cash is included as an asset.
For mvPortfolio, a numeric vector of weights.
For mvFrontier, a list of three components:
return |
returns of portfolios |
volatility |
volatilities of portfolios |
weights |
A matrix of portfolio weights.
Each column holds the weights for one portfolio on the
frontier. If |
The i-th portfolio on the frontier corresponds
to the i-th elements of return and
volatility, and the i-th column of
portfolio.
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. (2019) Financial Optimisation with R (NMOF Manual). http://enricoschumann.net/NMOF.htm#NMOFmanual
minvar for computing the minimum-variance portfolio
na <- 4
vols <- c(0.10, 0.15, 0.20,0.22)
m <- c(0.06, 0.12, 0.09, 0.07)
const_cor <- function(rho, na) {
C <- array(rho, dim = c(na, na))
diag(C) <- 1
C
}
var <- diag(vols) %*% const_cor(0.5, na) %*% diag(vols)
wmax <- 1 # maximum holding size
wmin <- 0.0 # minimum holding size
rf <- 0.02
p1 <- mvFrontier(m, var, wmin = wmin, wmax = wmax, n = 50)
p2 <- mvFrontier(m, var, wmin = wmin, wmax = wmax, n = 50, rf = rf)
plot(p1$volatility, p1$return, pch = 19, cex = 0.5, type = "o",
xlab = "Expected volatility",
ylab = "Expected return")
lines(p2$volatility, p2$return, col = grey(0.5))
abline(v = 0, h = rf)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.