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

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 %"

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.

Author

Steven E. Pav shabbychef@gmail.com

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

References

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

Examples

Run this code


# 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)

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