as.del_sropt {SharpeR}

Compute the Sharpe ratio of a hedged Markowitz portfolio.

Description

Computes the Sharpe ratio of the hedged Markowitz portfolio of some observed returns.

Usage

as.del_sropt(X, G, drag = 0, ope = 1, epoch = "yr")

## Default S3 method:
as.del_sropt(X, G, drag = 0, ope = 1, epoch = "yr")

## S3 method for class 'xts'
as.del_sropt(X, G, drag = 0, ope = 1, epoch = "yr")

Arguments

X

matrix of returns, or xts object.

G

an g \times q matrix of hedge constraints. A garden variety application would have G be one row of the identity matrix, with a one in the column of the instrument to be 'hedged out'.

drag

the 'drag' term, c_0/R. defaults to 0. It is assumed that drag has been annualized, i.e. has been multiplied by \sqrt{ope}. This is in contrast to the c0 term given to sr.

ope

the number of observations per 'epoch'. For convenience of interpretation, The Sharpe ratio is typically quoted in 'annualized' units for some epoch, that is, 'per square root epoch', though returns are observed at a frequency of ope per epoch. The default value is 1, meaning the code will not attempt to guess what the observation frequency is, and no annualization adjustments will be made.

epoch

the string representation of the 'epoch', defaulting to 'yr'.

Details

Suppose x_i are n independent draws of a q-variate normal random variable with mean \mu and covariance matrix \Sigma. Let G be a g \times q matrix of rank g. Let \bar{x} be the (vector) sample mean, and S be the sample covariance matrix (using Bessel's correction). Let

\zeta(w) = \frac{w^{\top}\bar{x} - c_0}{\sqrt{w^{\top}S w}}

be the (sample) Sharpe ratio of the portfolio w, subject to risk free rate c_0.

Let w_* be the solution to the portfolio optimization problem:

\max_{w: 0 < w^{\top}S w \le R^2,\,G S w = 0} \zeta(w),

with maximum value z_* = \zeta\left(w_*\right).

Note that if ope and epoch are not given, the converter from xts attempts to infer the observations per epoch, assuming yearly epoch.

Value

An object of class del_sropt.

Author(s)

Steven E. Pav [e-mail]

See Also

del_sropt, sropt, sr

Other del_sropt: del_sropt, is.del_sropt()

Examples

nfac <- 5
nyr <- 10
ope <- 253
# simulations with no covariance structure.
# under the null:
set.seed(as.integer(charToRaw("be determinstic")))
Returns <- matrix(rnorm(ope*nyr*nfac,mean=0,sd=0.0125),ncol=nfac)
# hedge out the first one:
G <- matrix(diag(nfac)[1,],nrow=1)
asro <- as.del_sropt(Returns,G,drag=0,ope=ope)
print(asro)
G <- diag(nfac)[c(1:3),]
asro <- as.del_sropt(Returns,G,drag=0,ope=ope)
# compare to sropt on the remaining assets
# they should be close, but not exact.
asro.alt <- as.sropt(Returns[,4:nfac],drag=0,ope=ope)

# using real data.
if (require(xts)) {
  data(stock_returns)
  # hedge out SPY
  G <- diag(dim(stock_returns)[2])[3,]
  asro <- as.del_sropt(stock_returns,G=G)
}