as.sropt {SharpeR}

Compute the Sharpe ratio of the Markowitz portfolio.

Description

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

Usage

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")

Arguments

X

matrix of returns, or xts object.

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 \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} \zeta(w),

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

w_* = R \frac{S^{-1}\bar{x}}{\sqrt{\bar{x}^{\top}S^{-1}\bar{x}}}

and

z_* = \sqrt{\bar{x}^{\top} S^{-1} \bar{x}} - \frac{c_0}{R}

The units of z_* are \mbox{time}^{-1/2}. Typically the Sharpe ratio is annualized by multiplying by \sqrt{\mbox{ope}}, where \mbox{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.

Value

An object of class sropt.

Author(s)

Steven E. Pav [e-mail]

See Also

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

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)
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)
}