SharpeR (version 0.1306)

as.sropt: 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")
  ## S3 method for class 'default':
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

epoch

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

Value

• A list with containing the following components:
• w
  the optimal portfolio.
• mu
  the estimated mean return vector.
• Sigma
  the estimated covariance matrix.
• df1
  the number of assets.
• df2
  the number of observed vectors.
• T2
  the Hotelling $T^2$ statistic.
• sropt
  the optimal Sharpe statistic.
• drag
  the input drag term.
• ope
  the input ope term.

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.

See Also

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

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

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(quantmod)) {
  get.ret <- function(sym,...) {
    OHLCV <- getSymbols(sym,auto.assign=FALSE,...)
    lrets <- diff(log(OHLCV[,paste(c(sym,"Adjusted"),collapse=".",sep="")]))
    # chomp first NA!
    lrets[-1,]
  }
  get.rets <- function(syms,...) { some.rets <- do.call("cbind",lapply(syms,get.ret,...)) }
  some.rets <- get.rets(c("IBM","AAPL","A","C","SPY","XOM"))
  asro <- as.sropt(some.rets)
}