MarkowitzR
Statistical Significance of the Markowitz Portfolio
A collection of tools for analyzing significance of Markowitz portfolios.
A number of utilities for dealing with the Markowitz portfolio.
-- Steven E. Pav, [email protected]
Installation
This package may be installed from CRAN; the latest version may be found on github via devtools:
if (require(devtools)) { # latest greatest install_github(repo = "MarkowitzR", username = "shabbychef", ref = "master")}Basic Usage
Inference on the Markowitz Portfolio
The (negative) Markowitz portfolio appears in the inverse of the uncentered second moment matrix of the 'augmented' vector of returns. Via the Central Limit Theorem and the delta method the asymptotic distribution of the Markowitz portfolio can be found. From this, Wald statistics on the individual portfolio weights can be computed.
Fake Data
First for unconditional returns:
set.seed(1001)X <- matrix(rnorm(1000 * 3), ncol = 3)ism <- mp_vcov(X, fit.intercept = TRUE)walds <- ism$W/sqrt(diag(ism$What))print(t(walds))## X1 X2 X3
## Intercept 0.83 -0.15 -1.8
Now for conditional expectation:
# generate data with given W, SigmaXgen <- function(W, Sigma, Feat) { Btrue <- Sigma %*% W Xmean <- Feat %*% t(Btrue) Shalf <- chol(Sigma) X <- Xmean + matrix(rnorm(prod(dim(Xmean))), ncol = dim(Xmean)[2]) %*% Shalf} n.feat <- 3n.ret <- 5n.obs <- 2000set.seed(101)Feat <- matrix(rnorm(n.obs * n.feat), ncol = n.feat)Wtrue <- 5 * matrix(rnorm(n.feat * n.ret), ncol = n.feat)Sigma <- cov(matrix(rnorm(100 * n.ret), ncol = n.ret))Sigma <- Sigma + diag(seq(from = 1, to = 3, length.out = n.ret))X <- Xgen(Wtrue, Sigma, Feat)ism <- mp_vcov(X, feat = Feat, fit.intercept = TRUE) walds <- ism$W/sqrt(diag(ism$What))print(t(walds))## X1 X2 X3 X4 X5
## Intercept -0.61 0.4 0.11 0.61 0.2
## Feat1 0.30 -20.7 14.56 -10.70 -24.6
## Feat2 -14.82 -4.0 12.01 29.75 -1.8
## Feat3 16.62 -5.3 -30.41 0.83 3.5
# results are not much changed when using robust# s.e.library(sandwich) ism.rse <- mp_vcov(X, feat = Feat, vcov.func = sandwich::vcovHAC, fit.intercept = TRUE)walds.rse <- ism.rse$W/sqrt(diag(ism.rse$What))print(t(walds.rse))## X1 X2 X3 X4 X5
## Intercept -0.61 0.4 0.11 0.60 0.19
## Feat1 0.29 -20.4 14.43 -10.66 -24.45
## Feat2 -14.49 -3.9 11.88 29.60 -1.80
## Feat3 16.37 -5.3 -29.85 0.83 3.48
# errors should be asymptotically normal with the# given covariance.n.feat <- 5n.ret <- 15n.obs <- 3000set.seed(101) Feat <- matrix(rnorm(n.obs * n.feat), ncol = n.feat)Wtrue <- 5 * matrix(rnorm(n.feat * n.ret), ncol = n.feat)Sigma <- cov(matrix(rnorm(100 * n.ret), ncol = n.ret))Sigma <- Sigma + diag(seq(from = 1, to = 3, length.out = n.ret))X <- Xgen(Wtrue, Sigma, Feat)ism <- mp_vcov(X, feat = Feat, fit.intercept = TRUE) Wcomp <- cbind(0, Wtrue)errs <- ism$W - Wcompdim(errs) <- c(length(errs), 1)Zerr <- solve(t(chol(ism$What)), errs)print(summary(Zerr))## V1
## Min. :-2.05
## 1st Qu.:-0.55
## Median : 0.14
## Mean : 0.20
## 3rd Qu.: 1.00
## Max. : 2.49
library(ggplot2)ph <- ggplot(data.frame(Ze = Zerr), aes(sample = Ze)) + stat_qq() + geom_abline(slope = 1, intercept = 0, colour = "red")print(ph)
# qqnorm(Zerr) qqline(Zerr,col=2)Fama French data
Now load the Fama French 3 factor portfolios.
if (!require(aqfb.data, quietly = TRUE) && require(devtools)) { # get the 10 industry data devtools::install_github("shabbychef/aqfb_data")}library(aqfb.data)# famadata(mff4) # will not matter, but convert pcts:ff.data <- 0.01 * mff4 # risk free rate:rfr <- ff.data[, "RF"] # subtract risk free from Mkt, HML and SMB:ff.ret <- ff.data[, c("Mkt", "HML", "SMB")] - rep(rfr, 2)Now analyze the Markowitz portfolio on them.
ism <- mp_vcov(ff.ret, fit.intercept = TRUE)walds <- ism$W/sqrt(diag(ism$What))print(t(walds))## Mkt HML SMB
## Intercept 4 0.32 -2
# now consider the hedging constraint: no# covariance with the market:Gmat <- matrix(c(1, 0, 0), nrow = 1)ism <- mp_vcov(ff.ret, fit.intercept = TRUE, Gmat = Gmat)walds <- ism$W/sqrt(diag(ism$What))print(t(walds))## Mkt HML SMB
## Intercept 1.5 0.32 -2