confint.sr {SharpeR}

Confidence Interval on (optimal) Signal-Noise Ratio

Description

Computes approximate confidence intervals on the (optimal) Signal-Noise ratio given the (optimal) Sharpe ratio. Works on objects of class sr and sropt.

Usage

## S3 method for class 'sr'
confint(
  object,
  parm,
  level = 0.95,
  level.lo = (1 - level)/2,
  level.hi = 1 - level.lo,
  type = c("exact", "t", "Z", "Mertens", "Bao"),
  ...
)

## S3 method for class 'sropt'
confint(
  object,
  parm,
  level = 0.95,
  level.lo = (1 - level)/2,
  level.hi = 1 - level.lo,
  ...
)

## S3 method for class 'del_sropt'
confint(
  object,
  parm,
  level = 0.95,
  level.lo = (1 - level)/2,
  level.hi = 1 - level.lo,
  ...
)

Arguments

object

an observed Sharpe ratio statistic, of class sr or sropt.

parm

ignored here, but required for the general method.

level

the confidence level required.

level.lo

the lower confidence level required.

level.hi

the upper confidence level required.

type

which method to apply.

...

further arguments to be passed to or from methods.

Details

Constructs confidence intervals on the Signal-Noise ratio given observed Sharpe ratio statistic. The available methods are:

exact

The default, which is only exact when returns are normal, based on inverting the non-central t distribution.

t

Uses the Johnson Welch approximation to the standard error, centered around the sample value.

Z

Uses the Johnson Welch approximation to the standard error, performing a simple correction for the bias of the Sharpe ratio based on Miller and Gehr formula.

Mertens

Uses the Mertens higher order approximation to the standard error, centered around the sample value.

Bao

Uses the Bao higher order approximation to the standard error, performing a higher order correction for the bias of the Sharpe ratio.

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

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

Given observations of z_*, compute confidence intervals on the population analogue, defined as

\zeta_* = \sqrt{\mu^{\top} \Sigma^{-1} \mu}

Value

A matrix (or vector) with columns giving lower and upper confidence limits for the parameter. These will be labelled as level.lo and level.hi in %, e.g. "2.5 %"

Author(s)

Steven E. Pav [e-mail]

References

Sharpe, William F. "Mutual fund performance." Journal of business (1966): 119-138. https://ideas.repec.org/a/ucp/jnlbus/v39y1965p119.html

See Also

confint, se, predint

Other sr: as.sr(), dsr(), is.sr(), plambdap(), power.sr_test(), predint(), print.sr(), reannualize(), se(), sr_equality_test(), sr_test(), sr_unpaired_test(), sr_vcov(), sr, summary.sr

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

Examples

# using "sr" class:
ope <- 253
df <- ope * 6
xv <- rnorm(df, 1 / sqrt(ope))
mysr <- as.sr(xv,ope=ope)
confint(mysr,level=0.90)
# using "lm" class
yv <- xv + rnorm(length(xv))
amod <- lm(yv ~ xv)
mysr <- as.sr(amod,ope=ope)
confint(mysr,level.lo=0.05,level.hi=1.0)
# rolling your own.
ope <- 253
df <- ope * 6
zeta <- 1.0
rvs <- rsr(128, df, zeta, ope)
roll.own <- sr(sr=rvs,df=df,c0=0,ope=ope)
aci <- confint(roll.own,level=0.95)
coverage <- 1 - mean((zeta < aci[,1]) | (aci[,2] < zeta))
# using "sropt" class
ope <- 253
df1 <- 4
df2 <- ope * 3
rvs <- as.matrix(rnorm(df1*df2),ncol=df1)
sro <- as.sropt(rvs,ope=ope)
aci <- confint(sro)
# on sropt, rolling your own.
zeta.s <- 1.0
rvs <- rsropt(128, df1, df2, zeta.s, ope)
roll.own <- sropt(z.s=rvs,df1,df2,drag=0,ope=ope)
aci <- confint(roll.own,level=0.95)
coverage <- 1 - mean((zeta.s < aci[,1]) | (aci[,2] < zeta.s))
# using "del_sropt" class
nfac <- 5
nyr <- 10
ope <- 253
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)
aci <- confint(asro,level=0.95)
# under the alternative
Returns <- matrix(rnorm(ope*nyr*nfac,mean=0.001,sd=0.0125),ncol=nfac)
asro <- as.del_sropt(Returns,G,drag=0,ope=ope)
aci <- confint(asro,level=0.95)