as.sropt: Compute the Sharpe ratio of the Markowitz portfolio.
In SharpeR: Statistical Significance of the Sharpe Ratio

Description Usage Arguments Details Value Author(s) See Also Examples

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

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

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

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

`X`	matrix of returns, or `xts` object.
`drag`	the 'drag' term, c0/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'.

Suppose xi are n independent draws of a q-variate normal random variable with mean mu and covariance matrix Sigma. Let xbar be the (vector) sample mean, and S be the sample covariance matrix (using Bessel's correction). Let

zeta(w) = (w'xbar - c0)/sqrt(w'Sw)

be the (sample) Sharpe ratio of the portfolio w, subject to risk free rate c0.

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

max {zeta(w) | 0 < w'Sw <= R^2},

with maximum value z* = zeta(w*). Then

w* = R S^-1 xbar / sqrt(xbar' S^-1 xbar)

and

z* = sqrt(xbar' S^-1 xbar) - c0/R

The units of z* are per root time. Typically the Sharpe ratio is annualized by multiplying by sqrt(ope), where ope is the number of observations per year (or whatever the target annualization epoch.)

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

An object of class sropt.

Steven E. Pav shabbychef@gmail.com

sropt, sr, sropt-distribution functions, dsropt, psropt, qsropt, rsropt

Other sropt: confint.sr(), dsropt(), is.sropt(), pco_sropt(), power.sropt_test(), reannualize(), sropt_test(), sropt

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)
asro <- as.sropt(Returns,drag=0,ope=ope)
# under the alternative:
Returns <- matrix(rnorm(ope*nyr*nfac,mean=0.0005,sd=0.0125),ncol=nfac)
asro <- as.sropt(Returns,drag=0,ope=ope)
# generating correlated multivariate normal data in a more sane way
if (require(MASS)) {
  nstok <- 10
  nfac <- 3
  nyr <- 10
  ope <- 253
  X.like <- 0.01 * matrix(rnorm(500*nfac),ncol=nfac) %*% 
    matrix(runif(nfac*nstok),ncol=nstok)
  Sigma <- cov(X.like) + diag(0.003,nstok)
  # under the null:
  Returns <- mvrnorm(ceiling(ope*nyr),mu=matrix(0,ncol=nstok),Sigma=Sigma)
  asro <- as.sropt(Returns,ope=ope)
  # under the alternative
  Returns <- mvrnorm(ceiling(ope*nyr),mu=matrix(0.001,ncol=nstok),Sigma=Sigma)
  asro <- as.sropt(Returns,ope=ope)
}

# using real data.
if (require(xts)) {
 data(stock_returns)
 asro <- as.sropt(stock_returns)
}