Maintainer:	Steven E. Pav <shabbychef@gmail.com>
Version:	1.4.0
Date:	2024-12-17
License:	LGPL-3
Title:	Statistical Significance of the Sharpe Ratio
BugReports:	https://github.com/shabbychef/SharpeR/issues
Description:	A collection of tools for analyzing significance of assets, funds, and trading strategies, based on the Sharpe ratio and overfit of the same. Provides density, distribution, quantile and random generation of the Sharpe ratio distribution based on normal returns, as well as the optimal Sharpe ratio over multiple assets. Computes confidence intervals on the Sharpe and provides a test of equality of Sharpe ratios based on the Delta method. The statistical foundations of the Sharpe can be found in the author's Short Sharpe Course <doi:10.2139/ssrn.3036276>.
Depends:	R (≥ 3.0.0)
Imports:	matrixcalc, zoo, epsiwal, methods
Suggests:	xtable, xts, timeSeries, quantmod, MASS, TTR, testthat, sandwich, txtplot, knitr
URL:	https://github.com/shabbychef/SharpeR
VignetteBuilder:	knitr
Collate:	'SharpeR.r' 'data.r' 'utils.r' 'distributions.r' 'sr.r' 'estimation.r' 'overoptimism.r' 'sr_bias.r' 'tests.r' 'unified.r'
RoxygenNote:	7.3.2
NeedsCompilation:	no
Packaged:	2024-12-18 19:16:15 UTC; spav
Author:	Steven E. Pav [aut, cre]
Repository:	CRAN
Date/Publication:	2024-12-18 19:30:02 UTC

statistics concerning Sharpe ratio and Markowitz portfolio

Description

Inference on Sharpe ratio and Markowitz portfolio.

Sharpe Ratio

Suppose x_i are n independent draws of a normal random variable with mean \mu and variance \sigma^2. Let \bar{x} be the sample mean, and s be the sample standard deviation (using Bessel's correction). Let c_0 be the 'risk free' or 'disastrous rate' of return. Then

z = \frac{\bar{x} - c_0}{s}

is the (sample) Sharpe ratio.

The units of z are \mbox{time}^{-1/2}. Typically the Sharpe ratio is annualized by multiplying by \sqrt{d}, where d is the number of observations per year (or whatever the target annualization epoch.) It is not common practice to include units when quoting Sharpe ratio, though doing so could avoid confusion.

The Sharpe ratio follows a rescaled non-central t distribution. That is, z/K follows a non-central t-distribution with m degrees of freedom and non-centrality parameter \zeta / K, for some K, m and \zeta.

We can generalize Sharpe's model to APT, wherein we write

x_i = \alpha + \sum_j \beta_j F_{j,i} + \epsilon_i,

where the F_{j,i} are observed 'factor returns', and the variance of the noise term is \sigma^2. Via linear regression, one can compute estimates \hat{\alpha}, and \hat{\sigma}, and then let the 'Sharpe ratio' be

z = \frac{\hat{\alpha} - c_0}{\hat{\sigma}}.

As above, this Sharpe ratio follows a rescaled t-distribution under normality, etc.

The parameters are encoded as follows:

• df stands for the degrees of freedom, typically n-1, but n-J-1 in general.

• \zeta is denoted by zeta.

• d is denoted by ope. ('Observations Per Year')

• For the APT form of Sharpe, K stands for the rescaling parameter.

Optimal 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(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} Z(w),

with maximum value z_* = Z\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 variable z_* follows an Optimal Sharpe ratio distribution. For convenience, we may assume that the sample statistic has been annualized in the same manner as the Sharpe ratio, that is by multiplying by d, the number of observations per epoch.

The Optimal Sharpe Ratio distribution is parametrized by the number of assets, q, the number of independent observations, n, the noncentrality parameter,

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

the 'drag' term, c_0/R, and the annualization factor, d. The drag term makes this a location family of distributions, and by default we assume it is zero.

The parameters are encoded as follows:

• q is denoted by df1.

• n is denoted by df2.

• \zeta_* is denoted by zeta.s.

• d is denoted by ope.

• c_0/R is denoted by drag.

Spanning and Hedging

As above, let

Z(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 G be a g \times q matrix of 'hedge constraints'. Let w_* be the solution to the portfolio optimization problem:

\max_{w: 0 < w^{\top}S w \le R^2,\,G S w = 0} Z(w),

with maximum value z_* = Z\left(w_*\right). Then z_*^2 can be expressed as the difference of two squared optimal Sharpe ratio random variables. A monotonic transform takes this difference to the LRT statistic for portfolio spanning, first described by Rao, and refined by Giri.

Legal Mumbo Jumbo

SharpeR is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details.

Note

The following are still in the works:

1. Corrections for standard error based on skew, kurtosis and autocorrelation.

2. Tests on Sharpe under positivity constraint. (c.f. Silvapulle)

3. Portfolio spanning tests.

4. Tests on portfolio weights.

This package is maintained as a hobby.

Author(s)

Steven E. Pav shabbychef@gmail.com

Maintainer: Steven E. Pav shabbychef@gmail.com (ORCID)

References

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

Johnson, N. L., and Welch, B. L. "Applications of the non-central t-distribution." Biometrika 31, no. 3-4 (1940): 362-389. doi:10.1093/biomet/31.3-4.362

Lo, Andrew W. "The statistics of Sharpe ratios." Financial Analysts Journal 58, no. 4 (2002): 36-52. https://www.ssrn.com/paper=377260

Opdyke, J. D. "Comparing Sharpe Ratios: So Where are the p-values?" Journal of Asset Management 8, no. 5 (2006): 308-336. https://www.ssrn.com/paper=886728

Ledoit, O., and Wolf, M. "Robust performance hypothesis testing with the Sharpe ratio." Journal of Empirical Finance 15, no. 5 (2008): 850-859. doi:10.1016/j.jempfin.2008.03.002

Giri, N. "On the likelihood ratio test of a normal multivariate testing problem." Annals of Mathematical Statistics 35, no. 1 (1964): 181-189. doi:10.1214/aoms/1177703740

Rao, C. R. "Advanced Statistical Methods in Biometric Research." Wiley (1952).

Rao, C. R. "On Some Problems Arising out of Discrimination with Multiple Characters." Sankhya, 9, no. 4 (1949): 343-366. https://www.jstor.org/stable/25047988

Kan, Raymond and Smith, Daniel R. "The Distribution of the Sample Minimum-Variance Frontier." Journal of Management Science 54, no. 7 (2008): 1364–1380. doi:10.1287/mnsc.1070.0852

Kan, Raymond and Zhou, GuoFu. "Tests of Mean-Variance Spanning." Annals of Economics and Finance 13, no. 1 (2012) https://econpapers.repec.org/article/cufjournl/y_3a2012_3av_3a13_3ai_3a1_3akanzhou.htm

Britten-Jones, Mark. "The Sampling Error in Estimates of Mean-Variance Efficient Portfolio Weights." The Journal of Finance 54, no. 2 (1999): 655–671. https://www.jstor.org/stable/2697722

Silvapulle, Mervyn. J. "A Hotelling's T2-type statistic for testing against one-sided hypotheses." Journal of Multivariate Analysis 55, no. 2 (1995): 312–319. doi:10.1006/jmva.1995.1081

Bodnar, Taras and Okhrin, Yarema. "On the Product of Inverse Wishart and Normal Distributions with Applications to Discriminant Analysis and Portfolio Theory." Scandinavian Journal of Statistics 38, no. 2 (2011): 311–331. doi:10.1111/j.1467-9469.2011.00729.x

See Also

Useful links:

• https://github.com/shabbychef/SharpeR

• Report bugs at https://github.com/shabbychef/SharpeR/issues

News for package 'SharpeR':

Description

News for package 'SharpeR'

Changes in SharpeR Version 1.4.0 (2024-12-17)

• Add inference on group SNR via Bonferroni and MHT tests.

• Add confidence intervals on achieved SNR.

• Fix some time inference code.

• Add dependency on epsiwal and zoo packages.

Changes in SharpeR Version 1.3.0 (2021-08-15)

• Remove tests based on upsilon distribution. Also removes dependency on sadists package.

Changes in SharpeR Version 1.2.1 (2020-02-06)

• CRAN fix for warnings about ellipsis.

Changes in SharpeR Version 1.2.0 (2018-10-07)

• move github figures to location CRAN understands

• be smarter about S3 classes: do not redefine summary and print.

• add bias and variance from Bao (2009).

• support estimation of higher order moments in as.sr, and expands methods for se and confidence interval computations.

• incorporate higher order methods into one sample sr tests.

Changes in SharpeR Version 1.1.0 (2016-03-14)

• fix sr_vcov on array input.

• add SRIC method.

• add SRIC to print.sropt.

• change predint output to matrix.

Changes in SharpeR Version 1.0.0 (2015-06-18)

• sane version numbers.

• unpaired k sample test of Sharpe.

• rely on same for unpaired 2 sample test.

• prediction intervals for Sharpe based on upsilon.

• more tests.

Changes in SharpeR Version 0.1501 (2014-12-06)

• fix inference of mark frequency from e.g. xts objects.

• add rlambdap.

Changes in SharpeR Version 0.1401 (2014-01-05)

• fix second moment asymptotic covariance.

• add confidence distribution functions for sr, sr.opt.

Changes in SharpeR Version 0.1310 (2013-10-30)

• inverse second moment asymptotic covariance.

Changes in SharpeR Version 0.1309 (2013-09-20)

• spanning/hedging tests.

• sr equality test via callback variance covariance computation.

• split vignette in two.

Changes in SharpeR Version 0.1307 (2013-05-30)

• proper d.f. in sr objects with different nan fill.

• restore vignette.

SharpeR Initial Version 0.1306 (2013-05-21)

• put on CRAN

Compute the Sharpe ratio of a hedged Markowitz portfolio.

Description

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

Usage

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

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

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

Arguments

Details

Suppose x_i are n independent draws of a q-variate normal random variable with mean \mu and covariance matrix \Sigma. Let G be a g \times q matrix of rank g. 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,\,G S w = 0} \zeta(w),

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

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 del_sropt.

Author(s)

Steven E. Pav shabbychef@gmail.com

See Also

del_sropt, sropt, sr

Other del_sropt: del_sropt, is.del_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)
# hedge out the first one:
G <- matrix(diag(nfac)[1,],nrow=1)
asro <- as.del_sropt(Returns,G,drag=0,ope=ope)
print(asro)
G <- diag(nfac)[c(1:3),]
asro <- as.del_sropt(Returns,G,drag=0,ope=ope)
# compare to sropt on the remaining assets
# they should be close, but not exact.
asro.alt <- as.sropt(Returns[,4:nfac],drag=0,ope=ope)

# using real data.
if (require(xts)) {
  data(stock_returns)
  # hedge out SPY
  G <- diag(dim(stock_returns)[2])[3,]
  asro <- as.del_sropt(stock_returns,G=G)
}

Compute the Sharpe ratio.

Description

Computes the Sharpe ratio of some observed returns.

Usage

as.sr(x, c0 = 0, ope = 1, na.rm = FALSE, epoch = "yr", higher_order = FALSE)

## Default S3 method:
as.sr(x, c0 = 0, ope = 1, na.rm = FALSE, epoch = "yr", higher_order = FALSE)

## S3 method for class 'matrix'
as.sr(x, c0 = 0, ope = 1, na.rm = FALSE, epoch = "yr", higher_order = FALSE)

## S3 method for class 'data.frame'
as.sr(x, c0 = 0, ope = 1, na.rm = FALSE, epoch = "yr", higher_order = FALSE)

## S3 method for class 'lm'
as.sr(x, c0 = 0, ope = 1, na.rm = FALSE, epoch = "yr", higher_order = FALSE)

## S3 method for class 'xts'
as.sr(x, c0 = 0, ope = 1, na.rm = FALSE, epoch = "yr", higher_order = FALSE)

## S3 method for class 'timeSeries'
as.sr(x, c0 = 0, ope = 1, na.rm = FALSE, epoch = "yr", higher_order = FALSE)

Arguments

Details

Suppose x_i are n independent returns of some asset. Let \bar{x} be the sample mean, and s be the sample standard deviation (using Bessel's correction). Let c_0 be the 'risk free rate'. Then

z = \frac{\bar{x} - c_0}{s}

is the (sample) Sharpe ratio.

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 is not given, the converter from xts attempts to infer the observations per year, without regard to the name of the epoch given.

Value

a list containing the following components:

sr

the annualized Sharpe ratio.

df

the t-stat degrees of freedom.

c0

the risk free term.

ope

the annualization factor.

rescal

the rescaling factor.

epoch

the string epoch.

cast to class sr.

Author(s)

Steven E. Pav shabbychef@gmail.com

References

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

Lo, Andrew W. "The statistics of Sharpe ratios." Financial Analysts Journal 58, no. 4 (2002): 36-52. https://www.ssrn.com/paper=377260

See Also

reannualize

sr-distribution functions, dsr, psr, qsr, rsr

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

Examples

# Sharpe's 'model': just given a bunch of returns.
asr <- as.sr(rnorm(253*3),ope=253)
# or a matrix, with a name
my.returns <- matrix(rnorm(253*3),ncol=1)
colnames(my.returns) <- c("my strategy")
asr <- as.sr(my.returns)

# given an xts object:
if (require(xts)) {
 data(stock_returns)
 IBM <- stock_returns[,'IBM']
 asr <- as.sr(IBM,na.rm=TRUE)
}

# on a linear model, find the 'Sharpe' of the residual term
nfac <- 5
nyr <- 10
ope <- 253
set.seed(as.integer(charToRaw("determinstic")))
Factors <- matrix(rnorm(ope*nyr*nfac,mean=0,sd=0.0125),ncol=nfac)
Betas <- exp(0.1 * rnorm(dim(Factors)[2]))
Returns <- (Factors %*% Betas) + rnorm(dim(Factors)[1],mean=0.0005,sd=0.012)
APT_mod <- lm(Returns ~ Factors)
asr <- as.sr(APT_mod,ope=ope)
# try again, but make the Returns independent of the Factors.
Returns <- rnorm(dim(Factors)[1],mean=0.0005,sd=0.012)
APT_mod <- lm(Returns ~ Factors)
asr <- as.sr(APT_mod,ope=ope)

# compute the Sharpe of a bunch of strategies:
my.returns <- matrix(rnorm(253*3*4),ncol=4)
asr <- as.sr(my.returns)  # without sensible colnames?
colnames(my.returns) <- c("strat a","strat b","strat c","strat d")
asr <- as.sr(my.returns)
  

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

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 shabbychef@gmail.com

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, 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(xts)) {
 data(stock_returns)
 asro <- as.sropt(stock_returns)
}  

Confidence intervals on achieved SnR

Description

Computes approximate bounds on the achieved signal-noise ratio of the Markowitz portfolio built on sample data.

Usage

asnr_confint(z.s, level = 0.95, level.lo = (1 - level), level.hi = 1)

## S3 method for class 'sropt'
asnr_confint(z.s, level = 0.95, level.lo = (1 - level), level.hi = 1)

## S3 method for class 'del_sropt'
asnr_confint(z.s, level = 0.95, level.lo = (1 - level), level.hi = 1)

Arguments

Details

Provides an approximate bound on the achieved Signal-noise ratio of the sample Markowitz portfolio. That is if \mu and \Sigma are the unknown mean and covariance of returns, and w is the sample Markowitz portfolio, then the probability that

w^{\top}\mu / \sqrt{w^{\top}\Sigma w} \ge b

is the given probability level. See section 8.3.1 of ‘The Sharpe Ratio: Statistics and Applications’. Plugs in the \delta_2 estimator.

Value

an estimate of the non-centrality parameter, which is the maximal population Sharpe ratio.

Author(s)

Steven E. Pav shabbychef@gmail.com

References

Pav, S. E. "The Sharpe Ratio: Statistics and Applications." CRC Press, 2021.

Pav, S. E. "Inference on achieved signal noise ratio." 2020 https://arxiv.org/abs/2005.06171

See Also

Other sropt Hotelling: inference(), sric()

Examples

# generate some sropts
nfac <- 3
nyr <- 5
ope <- 253
# simulations with no covariance structure.
# under the null:
set.seed(as.integer(charToRaw("determinstic")))
Returns <- matrix(rnorm(ope*nyr*nfac,mean=0,sd=0.0125),ncol=nfac)
asro <- as.sropt(Returns,drag=0,ope=ope)
asnr_confint(asro)

# for del_sropt:
nfac <- 5
nyr <- 10
ope <- 253
set.seed(as.integer(charToRaw("fix seed")))
Returns <- matrix(rnorm(ope*nyr*nfac,mean=0.0005,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)
asnr_confint(asro)

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

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 shabbychef@gmail.com

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, sr_equality_test(), sr_test(), sr_unpaired_test(), sr_vcov(), summary.sr

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

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)

Create an 'del_sropt' object.

Description

Spawns an object of class del_sropt.

Usage

del_sropt(z.s, z.sub, df1, df2, df1.sub, drag = 0, ope = 1, epoch = "yr")

Arguments

Details

The del_sropt class contains information about the difference between two rescaled T^2-statistics, useful for spanning tests, and inference on hedged portfolios. The following are list attributes of the object:

sropt

The (optimal) Sharpe ratio statistic of the 'full' set of assets.

sropt_sub

The (optimal) Sharpe ratio statistic on some subset, or linear subspace, of the assets.

df1

The number of assets.

df2

The number of observations.

df1.sub

The number of degrees of freedom in the hedge constraint.

drag

The drag term, which is the 'risk free rate' divided by the maximum risk.

ope

The 'observations per epoch'.

epoch

The string name of the 'epoch'.

For the most part, this constructor should not be called directly, rather as.del_sropt should be called instead to compute the needed statistics.

Value

a list cast to class del_sropt, with attributes

sropt

the optimal Sharpe statistic.

sropt.sub

the optimal Sharpe statistic on the subspace.

df1

the number of assets.

df2

the number of observed vectors.

df1.sub

the input df1.sub term.

drag

the input drag term.

ope

the input ope term.

T2

the Hotelling T^2 statistic.

T2.sub

the Hotelling T^2 statistic on the subspace.

Note

WARNING: This function is not well tested, may contain errors, may change in the next package update. Take caution.

2FIX: allow rownames?

Author(s)

Steven E. Pav shabbychef@gmail.com

See Also

reannualize

as.del_sropt

Other del_sropt: as.del_sropt(), is.del_sropt()

Examples

# roll your own.
ope <- 253

set.seed(as.integer(charToRaw("be determinstic")))
n.stock <- 10
X <- matrix(rnorm(1000*n.stock),nrow=1000)
Sigma <- cov(X)
mu <- colMeans(X)
w <- solve(Sigma,mu)
z <- t(mu) %*% w
n.sub <- 6
w.sub <- solve(Sigma[1:n.sub,1:n.sub],mu[1:n.sub])
z.sub <- t(mu[1:n.sub]) %*% w.sub
df1.sub <- n.stock - n.sub

roll.own <- del_sropt(z.s=z,z.sub=z.sub,df1=10,df2=1000,
 df1.sub=df1.sub,ope=ope)
print(roll.own)

The (non-central) Sharpe ratio.

Description

Density, distribution function, quantile function and random generation for the Sharpe ratio distribution with df degrees of freedom (and optional signal-noise-ratio zeta).

Usage

dsr(x, df, zeta, ope, ...)

psr(q, df, zeta, ope, ...)

qsr(p, df, zeta, ope, ...)

rsr(n, df, zeta, ope)

Arguments

Details

Suppose x_i are n independent draws of a normal random variable with mean \mu and variance \sigma^2. Let \bar{x} be the sample mean, and s be the sample standard deviation (using Bessel's correction). Let c_0 be the 'risk free rate'. Then

z = \frac{\bar{x} - c_0}{s}

is the (sample) Sharpe ratio.

The units of z is \mbox{time}^{-1/2}. Typically the Sharpe ratio is annualized by multiplying by \sqrt{d}, where d is the number of observations per epoch (typically a year).

Letting z = \sqrt{d}\frac{\bar{x}-c_0}{s}, where the sample estimates are based on n observations, then z takes a (non-central) Sharpe ratio distribution parametrized by n 'degrees of freedom', non-centrality parameter \zeta = \frac{\mu - c_0}{\sigma}, and annualization parameter d.

The parameters are encoded as follows:

• n is denoted by df.

• \zeta is denoted by zeta.

• d is denoted by ope. ('Observations Per Year')

If the returns violate the assumptions of normality, independence, etc (as they always should in the real world), the sample Sharpe Ratio will not follow this distribution. It does provide, however, a reasonable approximation in many cases.

See ‘The Sharpe Ratio: Statistics and Applications’, section 2.2.

Value

dsr gives the density, psr gives the distribution function, qsr gives the quantile function, and rsr generates random deviates.

Invalid arguments will result in return value NaN with a warning.

Note

This is a thin wrapper on the t distribution. The functions dt, pt, qt can accept ncp from limited range (|\delta|\le 37.62). Some corrections may have to be made here for large zeta.

Author(s)

Steven E. Pav shabbychef@gmail.com

References

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

Pav, S. E. "The Sharpe Ratio: Statistics and Applications." CRC Press, 2021.

See Also

reannualize

t-distribution functions, dt, pt, qt, rt

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

Examples

rvs <- rsr(128, 253*6, 0, 253)
dvs <- dsr(rvs, 253*6, 0, 253)
pvs.H0 <- psr(rvs, 253*6, 0, 253)
pvs.HA <- psr(rvs, 253*6, 1, 253)

plot(ecdf(pvs.H0))
plot(ecdf(pvs.HA))

The (non-central) maximal Sharpe ratio distribution.

Description

Density, distribution function, quantile function and random generation for the maximal Sharpe ratio distribution with df1 and df2 degrees of freedom (and optional maximal signal-noise-ratio zeta.s).

Usage

dsropt(x, df1, df2, zeta.s, ope, drag = 0, log = FALSE)

psropt(q, df1, df2, zeta.s, ope, drag = 0, ...)

qsropt(p, df1, df2, zeta.s, ope, drag = 0, ...)

rsropt(n, df1, df2, zeta.s, ope, drag = 0, ...)

Arguments

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

Z(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} Z(w),

with maximum value z_* = Z\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 variable z_* follows an Optimal Sharpe ratio distribution. For convenience, we may assume that the sample statistic has been annualized in the same manner as the Sharpe ratio, that is by multiplying by d, the number of observations per epoch.

The Optimal Sharpe Ratio distribution is parametrized by the number of assets, q, the number of independent observations, n, the noncentrality parameter,

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

the 'drag' term, c_0/R, and the annualization factor, d. The drag term makes this a location family of distributions, and by default we assume it is zero.

The parameters are encoded as follows:

• q is denoted by df1.

• n is denoted by df2.

• \zeta_* is denoted by zeta.s.

• d is denoted by ope.

• c_0/R is denoted by drag.

See ‘The Sharpe Ratio: Statistics and Applications’, section 6.1.4.

Value

dsropt gives the density, psropt gives the distribution function, qsropt gives the quantile function, and rsropt generates random deviates.

Invalid arguments will result in return value NaN with a warning.

Note

This is a thin wrapper on the Hotelling T-squared distribution, which is a wrapper on the F distribution.

Author(s)

Steven E. Pav shabbychef@gmail.com

References

Kan, Raymond and Smith, Daniel R. "The Distribution of the Sample Minimum-Variance Frontier." Journal of Management Science 54, no. 7 (2008): 1364–1380. doi:10.1287/mnsc.1070.0852

Pav, S. E. "The Sharpe Ratio: Statistics and Applications." CRC Press, 2021.

See Also

reannualize

F-distribution functions, df, pf, qf, rf, Sharpe ratio distribution, dsr, psr, qsr, rsr.

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

Examples

# generate some variates 
ngen <- 128
ope <- 253
df1 <- 8
df2 <- ope * 10
drag <- 0
# sample
rvs <- rsropt(ngen, df1, df2, drag, ope)
hist(rvs)
# these should be uniform:
isp <- psropt(rvs, df1, df2, drag, ope)
plot(ecdf(isp))

Inference on noncentrality parameter of F-like statistic

Description

Estimates the non-centrality parameter associated with an observed statistic following an optimal Sharpe Ratio distribution.

Usage

inference(z.s, type = c("KRS", "MLE", "unbiased"))

## S3 method for class 'sropt'
inference(z.s, type = c("KRS", "MLE", "unbiased"))

## S3 method for class 'del_sropt'
inference(z.s, type = c("KRS", "MLE", "unbiased"))

Arguments

Details

Let F be an observed statistic distributed as a non-central F with \nu_1, \nu_2 degrees of freedom and non-centrality parameter \delta^2. Three methods are presented to estimate the non-centrality parameter from the statistic:

• an unbiased estimator, which, unfortunately, may be negative. This is \delta_0 of Equations (6.67) and (6.68) of ‘The Sharpe Ratio: Statistics and Applications’.

• the Maximum Likelihood Estimator, which may be zero, but not negative.

• the estimator of Kubokawa, Roberts, and Shaleh (KRS), which is a shrinkage estimator. This is \delta_2 of Equations (6.67) and (6.68) of ‘The Sharpe Ratio: Statistics and Applications’.

The sropt distribution is equivalent to an F distribution up to a square root and some rescalings.

The non-centrality parameter of the sropt distribution is the square root of that of the Hotelling, i.e. has units 'per square root time'. As such, the 'unbiased' type can be problematic!

Value

an estimate of the non-centrality parameter, which is the maximal population Sharpe ratio.

Author(s)

Steven E. Pav shabbychef@gmail.com

References

Pav, S. E. "The Sharpe Ratio: Statistics and Applications." CRC Press, 2021.

Kubokawa, T., C. P. Robert, and A. K. Saleh. "Estimation of noncentrality parameters." Canadian Journal of Statistics 21, no. 1 (1993): 45-57. doi:10.2307/3315657

Spruill, M. C. "Computation of the maximum likelihood estimate of a noncentrality parameter." Journal of multivariate analysis 18, no. 2 (1986): 216-224. doi:10.1016/0047-259X(86)90070-9

See Also

F-distribution functions, df.

Other sropt Hotelling: asnr_confint(), sric()

Examples

# generate some sropts
nfac <- 3
nyr <- 5
ope <- 253
# simulations with no covariance structure.
# under the null:
set.seed(as.integer(charToRaw("determinstic")))
Returns <- matrix(rnorm(ope*nyr*nfac,mean=0,sd=0.0125),ncol=nfac)
asro <- as.sropt(Returns,drag=0,ope=ope)
est1 <- inference(asro,type='unbiased')  
est2 <- inference(asro,type='KRS')  
est3 <- inference(asro,type='MLE')

# 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)
est1 <- inference(asro,type='unbiased')  
est2 <- inference(asro,type='KRS')  
est3 <- inference(asro,type='MLE')

# sample many under the alternative, look at the estimator.
df1 <- 3
df2 <- 512
ope <- 253
zeta.s <- 1.25
rvs <- rsropt(128, df1, df2, zeta.s, ope)
roll.own <- sropt(z.s=rvs,df1,df2,drag=0,ope=ope)
est1 <- inference(roll.own,type='unbiased')  
est2 <- inference(roll.own,type='KRS')  
est3 <- inference(roll.own,type='MLE')

# for del_sropt:
nfac <- 5
nyr <- 10
ope <- 253
set.seed(as.integer(charToRaw("fix seed")))
Returns <- matrix(rnorm(ope*nyr*nfac,mean=0.0005,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)
est1 <- inference(asro,type='unbiased')  
est2 <- inference(asro,type='KRS')  
est3 <- inference(asro,type='MLE')

Is this in the "del_sropt" class?

Description

Checks if an object is in the class 'del_sropt'

Usage

is.del_sropt(x)

Arguments

Details

To satisfy the minimum requirements of an S3 class.

Value

a boolean.

Author(s)

Steven E. Pav shabbychef@gmail.com

See Also

del_sropt

Other del_sropt: as.del_sropt(), del_sropt

Examples

roll.own <- del_sropt(z.s=2,z.sub=1,df1=10,df2=1000,df1.sub=3,ope=1,epoch="yr")
is.sropt(roll.own)

Is this in the "sr" class?

Description

Checks if an object is in the class 'sr'

Usage

is.sr(x)

Arguments

Details

To satisfy the minimum requirements of an S3 class.

Value

a boolean.

Author(s)

Steven E. Pav shabbychef@gmail.com

See Also

sr

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

Examples

rvs <- as.sr(rnorm(253*8),ope=253)
is.sr(rvs)

Is this in the "sropt" class?

Description

Checks if an object is in the class 'sropt'

Usage

is.sropt(x)

Arguments

Details

To satisfy the minimum requirements of an S3 class.

Value

a boolean.

Author(s)

Steven E. Pav shabbychef@gmail.com

See Also

sropt

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

Compute variance covariance of Inverse 'Unified' Second Moment

Description

Computes the variance covariance matrix of the inverse unified second moment matrix.

Usage

ism_vcov(X,vcov.func=vcov,fit.intercept=TRUE)

Arguments

Details

Given p-vector x with mean \mu and covariance, \Sigma, let y be x with a one prepended. Then let \Theta = E\left(y y^{\top}\right), the uncentered second moment matrix. The inverse of \Theta contains the (negative) Markowitz portfolio and the precision matrix.

Given n contemporaneous observations of p-vectors, stacked as rows in the n \times p matrix X, this function estimates the mean and the asymptotic variance-covariance matrix of \Theta^{-1}.

One may use the default method for computing covariance, via the vcov function, or via a 'fancy' estimator, like sandwich:vcovHAC, sandwich:vcovHC, etc.

Value

a list containing the following components:

Note

By flipping the sign of X, the inverse of \Theta contains the positive Markowitz portfolio and the precision matrix on X. Performing this transform before passing the data to this function should be considered idiomatic.

This function will be deprecated in future releases of this package. Users should migrate at that time to a similar function in the MarkowitzR package.

Author(s)

Steven E. Pav shabbychef@gmail.com

References

Pav, S. E. "Asymptotic Distribution of the Markowitz Portfolio." 2013 https://arxiv.org/abs/1312.0557

See Also

sm_vcov, sr_vcov

Examples

X <- matrix(rnorm(1000*3),ncol=3)
# putting in -X is idiomatic:
ism <- ism_vcov(-X)
iSigmas.n <- ism_vcov(-X,vcov.func="normal")
iSigmas.n <- ism_vcov(-X,fit.intercept=FALSE)
# compute the marginal Wald test statistics:
ism.mu <- ism$mu[1:ism$p]
ism.Sg <- ism$Ohat[1:ism$p,1:ism$p]
wald.stats <- ism.mu / sqrt(diag(ism.Sg))

# make it fat tailed:
X <- matrix(rt(1000*3,df=5),ncol=3)
ism <- ism_vcov(X)
wald.stats <- ism$mu[1:ism$p] / sqrt(diag(ism$Ohat[1:ism$p,1:ism$p]))

if (require(sandwich)) {
 ism <- ism_vcov(X,vcov.func=vcovHC)
 wald.stats <- ism$mu[1:ism$p] / sqrt(diag(ism$Ohat[1:ism$p,1:ism$p]))
}

# add some autocorrelation to X
Xf <- filter(X,c(0.2),"recursive")
colnames(Xf) <- colnames(X)
ism <- ism_vcov(Xf)
wald.stats <- ism$mu[1:ism$p] / sqrt(diag(ism$Ohat[1:ism$p,1:ism$p]))

if (require(sandwich)) {
ism <- ism_vcov(Xf,vcov.func=vcovHAC)
 wald.stats <- ism$mu[1:ism$p] / sqrt(diag(ism$Ohat[1:ism$p,1:ism$p]))
}

The 'confidence distribution' for maximal Sharpe ratio.

Description

Distribution function and quantile function for the 'confidence distribution' of the maximal Sharpe ratio. This is just an inversion to perform inference on \zeta_* given observed statistic z_*.

Usage

pco_sropt(q,df1,df2,z.s,ope,lower.tail=TRUE,log.p=FALSE) 

qco_sropt(p,df1,df2,z.s,ope,lower.tail=TRUE,log.p=FALSE,lb=0,ub=Inf)

Arguments

Details

Suppose z_* follows a Maximal Sharpe ratio distribution (see SharpeR-package) for known degrees of freedom, and unknown non-centrality parameter \zeta_*. The 'confidence distribution' views \zeta_* as a random quantity once z_* is observed. As such, the CDF of the confidence distribution is the same as that of the Maximal Sharpe ratio (up to a flip of lower.tail); while the quantile function is used to compute confidence intervals on \zeta_* given z_*.

Value

pco_sropt gives the distribution function, and qco_sropt gives the quantile function.

Invalid arguments will result in return value NaN with a warning.

Note

When lower.tail is true, pco_sropt is monotonic increasing with respect to q, and decreasing in sropt; these are reversed when lower.tail is false. Similarly, qco_sropt is increasing in sign(as.double(lower.tail) - 0.5) * p and - sign(as.double(lower.tail) - 0.5) * sropt.

Author(s)

Steven E. Pav shabbychef@gmail.com

See Also

reannualize

dsropt,psropt,qsropt,rsropt

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

Examples

zeta.s <- 2.0
ope <- 253
ntest <- 50
df1 <- 4
df2 <- 6 * ope
rvs <- rsropt(ntest,df1=df1,df2=df2,zeta.s=zeta.s)
qvs <- seq(0,10,length.out=51)
pps <- pco_sropt(qvs,df1,df2,rvs[1],ope)

if (require(txtplot))
 txtplot(qvs,pps)

pps <- pco_sropt(qvs,df1,df2,rvs[1],ope,lower.tail=FALSE)

if (require(txtplot))
 txtplot(qvs,pps)


svs <- seq(0,4,length.out=51)
pps <- pco_sropt(2,df1,df2,svs,ope)
pps <- pco_sropt(2,df1,df2,svs,ope,lower.tail=FALSE)

pps <- pco_sropt(qvs,df1,df2,rvs[1],ope,lower.tail=FALSE)
pco_sropt(-1,df1,df2,rvs[1],ope)

qvs <- qco_sropt(0.05,df1=df1,df2=df2,z.s=rvs)
mean(qvs > zeta.s)
qvs <- qco_sropt(0.5,df1=df1,df2=df2,z.s=rvs)
mean(qvs > zeta.s)
qvs <- qco_sropt(0.95,df1=df1,df2=df2,z.s=rvs)
mean(qvs > zeta.s)
# test vectorization:
qv <- qco_sropt(0.1,df1,df2,rvs)
qv <- qco_sropt(c(0.1,0.2),df1,df2,rvs)
qv <- qco_sropt(c(0.1,0.2),c(df1,2*df1),df2,rvs)
qv <- qco_sropt(c(0.1,0.2),c(df1,2*df1),c(df2,2*df2),rvs)

The lambda-prime distribution.

Description

Distribution function and quantile function for LeCoutre's lambda-prime distribution with df degrees of freedom and the observed t-statistic, tstat.

Usage

plambdap(q, df, tstat, lower.tail = TRUE, log.p = FALSE)

qlambdap(p, df, tstat, lower.tail = TRUE, log.p = FALSE)

rlambdap(n, df, tstat)

Arguments

Details

Let t be distributed as a non-central t with \nu degrees of freedom and non-centrality parameter \delta. We can view this as

t = \frac{Z + \delta}{\sqrt{V/\nu}}.

where Z is a standard normal, \delta is the non-centrality parameter, V is a chi-square RV with \nu degrees of freedom, independent of Z. We can rewrite this as

\delta = t\sqrt{V/\nu} + Z.

Thus a 'lambda-prime' random variable with parameters t and \nu is one expressable as a sum

t\sqrt{V/\nu} + Z

for Chi-square V with \nu d.f., independent from standard normal Z

See ‘The Sharpe Ratio: Statistics and Applications’, section 2.4.

Value

dlambdap gives the density, plambdap gives the distribution function, qlambdap gives the quantile function, and rlambdap generates random deviates.

Invalid arguments will result in return value NaN with a warning.

Note

plambdap should be an increasing function of the argument q, and decreasing in tstat. qlambdap should be increasing in p

Author(s)

Steven E. Pav shabbychef@gmail.com

References

Pav, S. E. "The Sharpe Ratio: Statistics and Applications." CRC Press, 2021.

Lecoutre, Bruno. "Another look at confidence intervals for the noncentral t distribution." Journal of Modern Applied Statistical Methods 6, no. 1 (2007): 107–116. https://eris62.eu/telechargements/Lecoutre_Another_look-JMSAM2007_6(1).pdf

Lecoutre, Bruno. "Two useful distributions for Bayesian predictive procedures under normal models." Journal of Statistical Planning and Inference 79 (1999): 93–105.

See Also

t-distribution functions, dt,pt,qt,rt

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

Examples

rvs <- rnorm(128)
pvs <- plambdap(rvs, 253*6, 0.5)
plot(ecdf(pvs))
pvs <- plambdap(rvs, 253*6, 1)
plot(ecdf(pvs))
pvs <- plambdap(rvs, 253*6, -0.5)
plot(ecdf(pvs))
# test vectorization:
qv <- qlambdap(0.1,128,2)
qv <- qlambdap(c(0.1),128,2)
qv <- qlambdap(c(0.2),128,2)
qv <- qlambdap(c(0.2),253,2)
qv <- qlambdap(c(0.1,0.2),128,2)
qv <- qlambdap(c(0.1,0.2),c(128,253),2)
qv <- qlambdap(c(0.1,0.2),c(128,253),c(2,4))
qv <- qlambdap(c(0.1,0.2),c(128,253),c(2,4,8,16))
# random generation
rv <- rlambdap(1000,252,2)

Power calculations for Sharpe ratio tests

Description

Compute power of test, or determine parameters to obtain target power.

Usage

power.sr_test(n=NULL,zeta=NULL,sig.level=0.05,power=NULL,
              alternative=c("one.sided","two.sided"),ope=NULL)

Arguments

Details

Suppose you perform a single-sample test for significance of the Sharpe ratio based on the corresponding single-sample t-test. Given any three of: the effect size (the population SNR, \zeta), the number of observations, and the type I and type II rates, this function computes the fourth.

See ‘The Sharpe Ratio: Statistics and Applications’, section 2.5.8.

This is a thin wrapper on power.t.test.

Exactly one of the parameters n, zeta, power, and sig.level must be passed as NULL, and that parameter is determined from the others. Notice that sig.level has non-NULL default, so NULL must be explicitly passed if you want to compute it.

Value

Object of class power.htest, a list of the arguments (including the computed one) augmented with method, note and n.epoch elements, the latter is the number of epochs under the given annualization (ope), NA if none given.

Author(s)

Steven E. Pav shabbychef@gmail.com

References

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

Johnson, N. L., and Welch, B. L. "Applications of the non-central t-distribution." Biometrika 31, no. 3-4 (1940): 362-389. doi:10.1093/biomet/31.3-4.362

Pav, S. E. "The Sharpe Ratio: Statistics and Applications." CRC Press, 2021.

Lehr, R. "Sixteen S-squared over D-squared: A relation for crude sample size estimates." Statist. Med., 11, no 8 (1992): 1099–1102. doi:10.1002/sim.4780110811

See Also

reannualize

power.t.test, sr_test

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

Examples

anex <- power.sr_test(253,1,0.05,NULL,ope=253) 
anex <- power.sr_test(n=253,zeta=NULL,sig.level=0.05,power=0.5,ope=253) 
anex <- power.sr_test(n=NULL,zeta=0.6,sig.level=0.05,power=0.5,ope=253) 
# Lehr's Rule 
zetas <- seq(0.1,2.5,length.out=51)
ssizes <- sapply(zetas,function(zed) { 
  x <- power.sr_test(n=NULL,zeta=zed,sig.level=0.05,power=0.8,
       alternative="two.sided",ope=253)
  x$n / 253})
# should be around 8.
print(summary(ssizes * zetas * zetas))
# e = n z^2 mnemonic approximate rule for 0.05 type I, 50% power
ssizes <- sapply(zetas,function(zed) { 
  x <- power.sr_test(n=NULL,zeta=zed,sig.level=0.05,power=0.5,ope=253)
  x$n / 253 })
print(summary(ssizes * zetas * zetas - exp(1)))

Power calculations for optimal Sharpe ratio tests

Description

Compute power of test, or determine parameters to obtain target power.

Usage

power.sropt_test(df1=NULL,df2=NULL,zeta.s=NULL,
                 sig.level=0.05,power=NULL,ope=1)

Arguments

Details

Suppose you perform a single-sample test for significance of the optimal Sharpe ratio based on the corresponding single-sample T^2-test. Given any four of: the effect size (the population optimal SNR, \zeta_*), the number of assets, the number of observations, and the type I and type II rates, this function computes the fifth.

See ‘The Sharpe Ratio: Statistics and Applications’, section 6.3.3.

Exactly one of the parameters df1, df2, zeta.s, power, and sig.level must be passed as NULL, and that parameter is determined from the others. Notice that sig.level has non-NULL default, so NULL must be explicitly passed if you want to compute it.

Value

Object of class power.htest, a list of the arguments (including the computed one) augmented with method, note and n.epoch elements, the latter is the number of epochs under the given annualization (ope), NA if none given.

Author(s)

Steven E. Pav shabbychef@gmail.com

References

Pav, S. E. "The Sharpe Ratio: Statistics and Applications." CRC Press, 2021.

See Also

reannualize

power.t.test, sropt_test

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

Examples

anex <- power.sropt_test(8,4*253,1,0.05,NULL,ope=253) 

prediction interval for Sharpe ratio

Description

Computes the prediction interval for Sharpe ratio.

Usage

predint(
  x,
  oosdf,
  oosrescal = 1/sqrt(oosdf + 1),
  ope = NULL,
  level = 0.95,
  level.lo = (1 - level)/2,
  level.hi = 1 - level.lo,
  type = c("t", "Z", "Mertens", "Bao")
)

Arguments

Details

Given n_0 observations x_i from a normal random variable, with mean \mu and standard deviation \sigma, computes an interval [y_1,y_2] such that with a fixed probability, the sample Sharpe ratio over n future observations will fall in the given interval. The coverage is over repeated draws of both the past and future data, thus this computation takes into account error in both the estimate of Sharpe and the as yet unrealized returns. Coverage is approximate. Prediction intervals are computed by inflating a confidence interval by an amount which depends on the sample sizes.

See ‘The Sharpe Ratio: Statistics and Applications’, sections 2.5.9 and 3.5.2.

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 %"

Note

if level.lo < 0 or level.hi > 1, NaN will be returned.

Author(s)

Steven E. Pav shabbychef@gmail.com

References

Pav, S. E. "The Sharpe Ratio: Statistics and Applications." CRC Press, 2021.

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

See Also

confint.sr.

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

Examples

# should reject null
set.seed(1234)
etc <- predint(rnorm(1000,mean=0.5,sd=0.1),oosdf=127,ope=1)
etc <- predint(matrix(rnorm(1000*5,mean=0.05),ncol=5),oosdf=63,ope=1)

# check coverage
mu <- 0.0005
sg <- 0.013
n1 <- 512
n2 <- 256
p  <- 100
x1 <- matrix(rnorm(n1*p,mean=mu,sd=sg),ncol=p)
x2 <- matrix(rnorm(n2*p,mean=mu,sd=sg),ncol=p)
sr1 <- as.sr(x1)
sr2 <- as.sr(x2)
# check coverage of prediction interval
etc1 <- predint(sr1,oosdf=n2-1,level=0.95)
is.ok <- (etc1[,1] <= sr2$sr) & (sr2$sr <= etc1[,2])
covr <- mean(is.ok)

Print values.

Description

Displays an object, returning it invisibly, (via invisible(x).)

Usage

## S3 method for class 'sr'
print(x, ...)

## S3 method for class 'sropt'
print(x, ...)

## S3 method for class 'del_sropt'
print(x, ...)

Arguments

Value

the object, wrapped in invisible.

Author(s)

Steven E. Pav shabbychef@gmail.com

References

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

See Also

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

Examples

# compute a 'daily' Sharpe
mysr <- as.sr(rnorm(253*8),ope=1,epoch="day")
print(mysr)
# roll your own.
ope <- 253
zeta <- 1.0
n <- 6 * ope
rvs <- rsr(1,n,zeta,ope=ope)
roll.own <- sr(sr=rvs,df=n-1,ope=ope,rescal=sqrt(1/n))
print(roll.own)
# put a bunch in. naming becomes a problem.
rvs <- rsr(5,n,zeta,ope=ope)
roll.own <- sr(sr=rvs,df=n-1,ope=ope,rescal=sqrt(1/n))
print(roll.own)
# for sropt objects:
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)
print(asro)

Change the annualization of a Sharpe ratio.

Description

Changes the annualization factor of a Sharpe ratio statistic, or the rate at which observations are made.

Usage

reannualize(object, new.ope = NULL, new.epoch = NULL)

## S3 method for class 'sr'
reannualize(object, new.ope = NULL, new.epoch = NULL)

## S3 method for class 'sropt'
reannualize(object, new.ope = NULL, new.epoch = NULL)

Arguments

Value

the input object with the annualization and/or epoch updated.

Author(s)

Steven E. Pav shabbychef@gmail.com

See Also

sr

sropt

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

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

Examples

# compute a 'daily' Sharpe
mysr <- as.sr(rnorm(253*8),ope=1,epoch="day")
# turn into annual 
mysr2 <- reannualize(mysr,new.ope=253,new.epoch="yr")

# for sropt
ope <- 253
zeta.s <- 1.0  
df1 <- 10
df2 <- 6 * ope
rvs <- rsropt(1,df1,df2,zeta.s,ope,drag=0)
roll.own <- sropt(z.s=rvs,df1,df2,drag=0,ope=ope,epoch="yr")
# make 'monthly'
roll.monthly <- reannualize(roll.own,new.ope=21,new.epoch="mo.")
# make 'daily'
roll.daily <- reannualize(roll.own,new.ope=1,new.epoch="day")

Standard error computation

Description

Estimates the standard error of the Sharpe ratio statistic.

Usage

se(z, type)

## S3 method for class 'sr'
se(z, type = c("t", "Lo", "Mertens", "Bao"))

Arguments

Details

For an observed Sharpe ratio, estimate the standard error. The following methods are recognized:

t

The default, based on Johnson & Welch, with a correction for small sample size. Also known as 'Lo'.

Mertens

An approximation to the standard error taking into skewness and kurtosis of the returns distribution.

Bao

An even higher accuracty approximation using higher order moments.

There should be very little difference between these except for very small sample sizes.

See ‘The Sharpe Ratio: Statistics and Applications’, sections 2.5.1 and 3.2.3.

Value

an estimate of standard error.

Note

The units of the standard error are consistent with those of the input sr object.

Author(s)

Steven E. Pav shabbychef@gmail.com

References

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

Johnson, N. L., and Welch, B. L. "Applications of the non-central t-distribution." Biometrika 31, no. 3-4 (1940): 362-389. doi:10.1093/biomet/31.3-4.362

Lo, Andrew W. "The statistics of Sharpe ratios." Financial Analysts Journal 58, no. 4 (2002): 36-52. https://www.ssrn.com/paper=377260

Bao, Yong. "Estimation Risk-Adjusted Sharpe Ratio and Fund Performance Ranking Under a General Return Distribution." Journal of Financial Econometrics 7, no. 2 (2009): 152-173. doi:10.1093/jjfinec/nbn022

Opdyke, J. D. "Comparing Sharpe Ratios: So Where are the p-values?" Journal of Asset Management 8, no. 5 (2006): 308-336. https://www.ssrn.com/paper=886728

Pav, S. E. "The Sharpe Ratio: Statistics and Applications." CRC Press, 2021.

Walck, C. "Hand-book on STATISTICAL DISTRIBUTIONS for experimentalists." 1996. https://www.stat.rice.edu/~dobelman/textfiles/DistributionsHandbook.pdf

See Also

sr-distribution functions, dsr, sr_variance.

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

Examples

asr <- as.sr(rnorm(128,0.2))
anse <- se(asr,type="t")
anse <- se(asr,type="Lo")

Compute variance covariance of 'Unified' Second Moment

Description

Computes the variance covariance matrix of sample mean and second moment.

Usage

sm_vcov(X,vcov.func=vcov,fit.intercept=TRUE)

Arguments

Details

Given p-vector x, the 'unified' sample is the p(p+3)/2 vector of x stacked on top of \mbox{vech}(x x^{\top}). Given n contemporaneous observations of p-vectors, stacked as rows in the n \times p matrix X, this function computes the mean and the variance-covariance matrix of the 'unified' sample.

One may use the default method for computing covariance, via the vcov function, or via a 'fancy' estimator, like sandwich:vcovHAC, sandwich:vcovHC, etc.

Value

a list containing the following components:

Note

This function will be deprecated in future releases of this package. Users should migrate at that time to a similar function in the MarkowitzR package.

Author(s)

Steven E. Pav shabbychef@gmail.com

References

Pav, S. E. "Asymptotic Distribution of the Markowitz Portfolio." 2013 https://arxiv.org/abs/1312.0557

See Also

ism_vcov, sr_vcov

Examples

X <- matrix(rnorm(1000*3),ncol=3)
Sigmas <- sm_vcov(X)
Sigmas.n <- sm_vcov(X,vcov.func="normal")
Sigmas.n <- sm_vcov(X,fit.intercept=FALSE)

# make it fat tailed:
X <- matrix(rt(1000*3,df=5),ncol=3)
Sigmas <- sm_vcov(X)

if (require(sandwich)) {
 Sigmas <- sm_vcov(X,vcov.func=vcovHC)
}

# add some autocorrelation to X
Xf <- filter(X,c(0.2),"recursive")
colnames(Xf) <- colnames(X)
Sigmas <- sm_vcov(Xf)

if (require(sandwich)) {
Sigmas <- sm_vcov(Xf,vcov.func=vcovHAC)
}

Create an 'sr' object.

Description

Spawns an object of class sr.

Usage

sr(
  sr,
  df,
  c0 = 0,
  ope = 1,
  rescal = sqrt(1/(df + 1)),
  epoch = "yr",
  cumulants = NULL
)

Arguments

Details

The sr class contains information about a rescaled t-statistic. The following are list attributes of the object:

sr

The Sharpe ratio statistic.

df

The d.f. of the equivalent t-statistic.

c0

The drag 'risk free rate' used.

ope

The 'observations per epoch'.

rescal

The rescaling parameter.

epoch

The string name of the 'epoch'.

The stored Sharpe statistic, sr is equal to the t-statistic times rescal * sqrt{ope}.

For the most part, this constructor should not be called directly, rather as.sr should be called instead to compute the Sharpe ratio.

Value

a list cast to class sr.

Note

2FIX: allow rownames?

Author(s)

Steven E. Pav shabbychef@gmail.com

References

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

See Also

reannualize

as.sr

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

Examples

# roll your own.
ope <- 253
zeta <- 1.0
n <- 3 * ope
rvs <- rsr(1,n,zeta,ope=ope)
roll.own <- sr(sr=rvs,df=n-1,ope=ope,rescal=sqrt(1/n))
# put a bunch in. naming becomes a problem.
rvs <- rsr(5,n,zeta,ope=ope)
roll.own <- sr(sr=rvs,df=n-1,ope=ope,rescal=sqrt(1/n))

sr_bias .

Description

Computes the asymptotic bias of the sample Sharpe ratio based on moments.

Usage

sr_bias(snr, n, cumulants, type = c("simple", "second_order"))

Arguments

Details

The sample Sharpe ratio has bias of the form

B = \left(\frac{3}{4n} + 3 \frac{\gamma_2}{8n}\right) \zeta - \frac{1}{2n} \gamma_1 + o\left(n^{-3/2}\right),

where \zeta is the population Signal Noise ratio, n is the sample size, \gamma_1 is the population skewness, and \gamma_2 is the population excess kurtosis. This form of the bias appears as Equation (5) in Bao, which claims an accuracy of only o\left(n^{-1}\right). The author believes this approximation is slightly more accurate.

A more accurate form is given by Bao (Equation (3)) as

B = \frac{3\zeta}{4n}\zeta + \frac{49\zeta}{32n^2} - \gamma_1 \left(\frac{1}{2n} + \frac{3}{8n^2}\right) + \gamma_2 \zeta \left(\frac{3}{8n} - \frac{15}{32n^2}\right) + \frac{3\gamma_3}{8n^2} - \frac{5\gamma_4 \zeta}{16n^2} - \frac{5\gamma_1^2\zeta}{4n^2} + \frac{105\gamma_2^2 \zeta}{128 n^2} - \frac{15 \gamma_1 \gamma_2}{16n^2} + o\left(n^{-2}\right),

where \gamma_3 through \gamma_5 are the fifth through seventh cumulants of the error term.

See ‘The Sharpe Ratio: Statistics and Applications’, section 3.2.3.

Value

the approximate bias of the Sharpe ratio. The bias is the expected value of the sample Sharpe minus the Signal Noise ratio.

Note

much of the code is adapted from Gauss code provided by Yong Bao.

Author(s)

Steven E. Pav shabbychef@gmail.com

References

Bao, Yong. "Estimation Risk-Adjusted Sharpe Ratio and Fund Performance Ranking Under a General Return Distribution." Journal of Financial Econometrics 7, no. 2 (2009): 152-173. doi:10.1093/jjfinec/nbn022

Pav, S. E. "The Sharpe Ratio: Statistics and Applications." CRC Press, 2021.

See Also

sr_variance

Examples

# bias under normality:
sr_bias(1, 100, rep(0,2), type='simple')
sr_bias(1, 100, rep(0,5), type='second_order')

# plugging in sample estimates
x <- rnorm(1000)
n <- length(x)
mu <- mean(x)
sdv <- sd(x)
snr <- mu / sdv
# these are not great estimates, but close enough:
sku <- mean((x-mu)^3) / sdv^3
kur <- (mean((x-mu)^4) / sdv^4) - 4
sr_bias(snr, n, c(sku,kur), type='simple')

conditional test for maximum Sharpe ratios.

Description

Performs tests for the hypothesis

\zeta_{(k)} \le \zeta_0

against the alternative

\zeta_{(k)} > \zeta_0,

where \zeta_{(k)} is the signal-noise ratio of the asset selected because it has the largest Sharpe ratio. The test is conditional on having selected the maximum. Testing is via the polyhedral lemma of Lee et al.

Usage

sr_conditional_test(
  srs,
  df,
  ope = 1,
  R = NULL,
  Rmax = NULL,
  zeta_0 = 0,
  conf.level = 0.95,
  alternative = c("two.sided", "less", "greater")
)

Arguments

Details

Performs the conditional estimation procedure as outlined in Section 4.1.5 of The Sharpe Ratio: Statistics and Applications.

Value

A list with class "htest" containing the following components:

Author(s)

Steven E. Pav shabbychef@gmail.com

References

Pav, S. E. "The Sharpe Ratio: Statistics and Applications." CRC Press, 2021.

Pav, S. E. "Conditional inference on the asset with maximum Sharpe ratio." 2019 https://arxiv.org/abs/1906.00573

Lee, J. D., Sun, D. L., Sun, Y. and Taylor, J. E. "Exact post-selection inference, with application to the Lasso." Ann. Statist. 44, no. 3 (2016): 907-927. doi:10.1214/15-AOS1371. https://arxiv.org/abs/1311.6238

See Also

reannualize

sr_max_test

Examples

# generate some fake data
ope <- 252
zeta0 <- 1.0
set.seed(1234)
zetas <- rsr(50, zeta=zeta0, df=ope*2, ope=ope)
sr_conditional_test(zetas,df=ope*2,ope=ope,R=diag(length(zetas)))

Paired test for equality of Sharpe ratio

Description

Performs a hypothesis test of equality of Sharpe ratios of p assets given paired observations.

Usage

sr_equality_test(X,type=c("chisq","F","t"),
                 alternative=c("two.sided","less","greater"),
                 contrasts=NULL,
                 vcov.func=vcov)

Arguments

Details

Given n i.i.d. observations of the excess returns of p strategies, we test

H_0: \frac{\mu_i}{\sigma_i} = \frac{\mu_j}{\sigma_j}, 1 \le i < j \le p

using the method of Wright, et. al.

More generally, a matrix of constrasts, E can be given, and we can test

H_0: E s = 0,

where s is the vector of Sharpe ratios of the p strategies.

When E consists of a single row (a single contrast), as is the case when the default contrasts are used and only two strategies are compared, then an approximate t-test can be performed against the alternative hypothesis H_a: E s > 0

Both chi-squared and F- approximations are supported; the former is described by Wright. et. al., the latter by Leung and Wong.

See ‘The Sharpe Ratio: Statistics and Applications’, section 3.3.1.

Value

Object of class htest, a list of the test statistic, the size of X, and the method noted.

Author(s)

Steven E. Pav shabbychef@gmail.com

References

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

Wright, J. A., Yam, S. C. P., and Yung, S. P. "A note on the test for the equality of multiple Sharpe ratios and its application on the evaluation of iShares." J. Risk. to appear. https://www.risk.net/journal-risk/2340067/test-equality-multiple-sharpe-ratios

Leung, P.-L., and Wong, W.-K. "On testing the equality of multiple Sharpe ratios, with application on the evaluation of iShares." J. Risk 10, no. 3 (2008): 15–30. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=907270

Memmel, C. "Performance hypothesis testing with the Sharpe ratio." Finance Letters 1 (2003): 21–23.

Ledoit, O., and Wolf, M. "Robust performance hypothesis testing with the Sharpe ratio." Journal of Empirical Finance 15, no. 5 (2008): 850-859. doi:10.1016/j.jempfin.2008.03.002

Lo, Andrew W. "The statistics of Sharpe ratios." Financial Analysts Journal 58, no. 4 (2002): 36-52. https://www.ssrn.com/paper=377260

Pav, S. E. "The Sharpe Ratio: Statistics and Applications." CRC Press, 2021.

See Also

sr_test

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

Examples

# under the null 
set.seed(1234)
rv <- sr_equality_test(matrix(rnorm(500*5),ncol=5))

# under the alternative (but with identity covariance)
ope <- 253
nyr <- 10
nco <- 5
set.seed(909)
rets <- 0.01 * sapply(seq(0,1.7/sqrt(ope),length.out=nco),
  function(mu) { rnorm(ope*nyr,mean=mu,sd=1) })
rv <- sr_equality_test(rets)

# using real data
if (require(xts)) {
 data(stock_returns)
 pvs <- sr_equality_test(stock_returns)
}

# test for uniformity
pvs <- replicate(1024,{ x <- sr_equality_test(matrix(rnorm(400*5),400,5),type="chisq")
                       x$p.value })
plot(ecdf(pvs))
abline(0,1,col='red') 


if (require(sandwich)) {
  set.seed(as.integer(charToRaw("0b2fd4e9-3bdf-4e3e-9c75-25c6d18c331f")))
  n.manifest <- 10
  n.latent <- 4
  n.day <- 1024
  snr <- 0.95
  la_A <- matrix(rnorm(n.day*n.latent),ncol=n.latent)
  la_B <- matrix(runif(n.latent*n.manifest),ncol=n.manifest)
  latent.rets <- la_A %*% la_B
  noise.rets <- matrix(rnorm(n.day*n.manifest),ncol=n.manifest)
  some.rets <- snr * latent.rets + sqrt(1-snr^2) * noise.rets
  # naive vcov
  pvs0 <- sr_equality_test(some.rets)
  # HAC vcov
  pvs1 <- sr_equality_test(some.rets,vcov.func=vcovHAC)
  # more elaborately:
  pvs <- sr_equality_test(some.rets,vcov.func=function(amod) {
	vcovHAC(amod,prewhite=TRUE) })
}

test for multiple Sharpe ratios.

Description

Performs tests for the hypothesis

\forall i \zeta_i \le \zeta_0

against the alternative

\exists i zeta_i > \zeta_0

Multiple methods are supported for the test, including Bonferroni correction, a chi-bar-square test, and Follman's test.

It is assumed that returns have a compound symmetric correlation structure. That is, the correlation matrix has \rho on all off-diagonal elements. Returns are assumed to follow an elliptical distribution with kurtosis factor \kappa, which equals 1 in the case of Gaussian returns. The kurtosis factor is one third the kurtosis of marginal returns.

Usage

sr_max_test(
  srs,
  df,
  ope = 1,
  kappa = 1,
  rho = 0,
  zeta_0 = 0,
  conf.level = 0.95,
  type = c("Bonferroni", "chi-bar-square", "Follman"),
  loglog = TRUE
)

Arguments

Details

A few test methodologies are supported. These are described in more detail in Section 4.1 of The Sharpe Ratio: Statistics and Applications.

• Performs the Bonferroni correction as described in equation (4.8).

• The chi-bar-square test described in section 4.1.3.

• Follman's test, given in equation (4.20). This test does not yet support Hansen's asymptotic correction and may not produce confidence intervals.

Moreover, Hansen's ‘log-log’ adjustment is also optionally applied.

Value

A list with class "htest" containing the following components:

Author(s)

Steven E. Pav shabbychef@gmail.com

References

Pav, S. E. "The Sharpe Ratio: Statistics and Applications." CRC Press, 2021.

Pav, S. E. "Conditional inference on the asset with maximum Sharpe ratio." 2019 https://arxiv.org/abs/1906.00573

Follman, D. "A Simple Multivariate Test for One-Sided Alternatives." JASA, 91, no 434 (1996): 854-861. doi:10.2307/2291680

Hansen, P. R. "A Test for Superior Predictive Ability." J. Bus. Ec. Stats, 23, no 4 (2005). doi:10.1198/073500105000000063

See Also

reannualize

sr_conditional_test

Examples

# generate some fake data
ope <- 252
zeta0 <- 1.0
set.seed(1234)
zetas <- rsr(50, zeta=zeta0, df=ope*2, ope=ope)
sr_max_test(zetas,df=ope*2,ope=ope,type='Bonferroni')
sr_max_test(zetas,zeta_0=zeta0,df=ope*2,ope=ope,type='Bonferroni')
sr_max_test(zetas,zeta_0=zeta0,df=ope*2,ope=ope,type='chi-bar-square')

test for Sharpe ratio

Description

Performs one and two sample tests of Sharpe ratio on vectors of data.

Usage

sr_test(
  x,
  y = NULL,
  alternative = c("two.sided", "less", "greater"),
  zeta = 0,
  ope = 1,
  paired = FALSE,
  conf.level = 0.95,
  type = c("exact", "t", "Z", "Mertens", "Bao"),
  ...
)

Arguments

Details

Given n observations x_i from a normal random variable, with mean \mu and standard deviation \sigma, tests

H_0: \frac{\mu}{\sigma} = S

against two or one sided alternatives.

Can also perform two sample tests of Sharpe ratio. For paired observations x_i and y_i, tests

H_0: \frac{\mu_x}{\sigma_x} = \frac{\mu_u}{\sigma_y}

against two or one sided alternative, via sr_equality_test.

For unpaired (and independent) observations, tests

H_0: \frac{\mu_x}{\sigma_x} - \frac{\mu_u}{\sigma_y} = S

against two or one-sided alternatives via an asymptotic approximation.

The one sample test admits a number of different methods:

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.

See confint.sr for more information on these types

See ‘The Sharpe Ratio: Statistics and Applications’, section 3.2.1, 3.2.2, and 3.3.1.

Value

A list with class "htest" containing the following components:

Author(s)

Steven E. Pav shabbychef@gmail.com

References

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

Pav, S. E. "The Sharpe Ratio: Statistics and Applications." CRC Press, 2021.

See Also

reannualize

sr_equality_test, sr_unpaired_test, t.test.

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

Examples

# should reject null
x <- sr_test(rnorm(1000,mean=0.5,sd=0.1),zeta=2,ope=1,alternative="greater")
x <- sr_test(rnorm(1000,mean=0.5,sd=0.1),zeta=2,ope=1,alternative="two.sided")
# should not reject null
x <- sr_test(rnorm(1000,mean=0.5,sd=0.1),zeta=2,ope=1,alternative="less")

# test for uniformity
pvs <- replicate(128,{ x <- sr_test(rnorm(1000),ope=253,alternative="two.sided")
                        x$p.value })
plot(ecdf(pvs))
abline(0,1,col='red') 
# testing an object of class sr
asr <- as.sr(rnorm(1000,1 / sqrt(253)),ope=253)
checkit <- sr_test(asr,zeta=0)

test for equation on unpaired Sharpe ratios

Description

Performs hypothesis tests on a single equation on k independent samples of Sharpe ratio.

Usage

sr_unpaired_test(
  srs,
  contrasts = NULL,
  null.value = 0,
  alternative = c("two.sided", "less", "greater"),
  ope = NULL,
  conf.level = 0.95
)

Arguments

Details

For 1 \le j \le k, suppose you have n_j observations of a normal random variable with mean \mu_j and standard deviation \sigma_j, with all observations independent. Given constants a_j and value b, this code tests the null hypothesis

H_0: \sum_j a_j \frac{\mu_j}{\sigma_j} = b

against two or one sided alternatives.

See ‘The Sharpe Ratio: Statistics and Applications’, section 3.3.1.

Value

A list with class "htest" containing the following components:

Author(s)

Steven E. Pav shabbychef@gmail.com

References

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

Pav, S. E. "The Sharpe Ratio: Statistics and Applications." CRC Press, 2021.

See Also

sr_equality_test, sr_test, t.test.

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

Examples

# basic usage
set.seed(as.integer(charToRaw("set the seed")))
# default contrast is 1,-1,1,-1,1,-1
etc <- sr_unpaired_test(as.sr(matrix(rnorm(1000*6,mean=0.02,sd=0.1),ncol=6)))
print(etc)

etc <- sr_unpaired_test(as.sr(matrix(rnorm(1000*4,mean=0.0005,sd=0.01),ncol=4)),
  alternative='greater')
print(etc)

etc <- sr_unpaired_test(as.sr(matrix(rnorm(1000*4,mean=0.0005,sd=0.01),ncol=4)),
  contrasts=c(1,1,1,1),null.value=-0.1,alternative='greater')
print(etc)

inp <- list(as.sr(rnorm(500)),as.sr(runif(200)-0.5),
            as.sr(rnorm(30)),as.sr(rnorm(100)))
etc <- sr_unpaired_test(inp)

inp <- list(as.sr(rnorm(500)),as.sr(rnorm(100,mean=0.2,sd=1)))
etc <- sr_unpaired_test(inp,contrasts=c(1,1),null.value=0.2)
etc$conf.int

sr_variance .

Description

Computes the variance of the sample Sharpe ratio.

Usage

sr_variance(snr, n, cumulants)

Arguments

Details

The sample Sharpe ratio has variance of the form

V = \frac{1}{n}\left(1 + \frac{\zeta^2}{2}\right) +\frac{1}{n^2}\left(\frac{19\zeta^2}{8} + 2\right) -\gamma_1\zeta\left(\frac{1}{n} + \frac{5}{2n^2}\right) +\gamma_2\zeta^2\left(\frac{1}{4n} + \frac{3}{8n^2}\right) +\frac{5\gamma_3\zeta}{4n^2} +\gamma_1^2\left(\frac{7}{4n^2} - \frac{3\zeta^2}{2n^2}\right) +\frac{39\gamma_2^2\zeta^2}{32n^2} -\frac{15\gamma_1\gamma_2\zeta}{4n^2} +o\left(n^{-2}\right),

where \zeta is the population Signal Noise ratio, n is the sample size, \gamma_1 is the population skewness, and \gamma_2 is the population excess kurtosis, and \gamma_3 through \gamma_5 are the fifth through seventh cumulants of the error term. This form of the variance appears as Equation (4) in Bao.

See ‘The Sharpe Ratio: Statistics and Applications’, section 3.2.3.

Value

the variance of the sample statistic.

Author(s)

Steven E. Pav shabbychef@gmail.com

References

Bao, Yong. "Estimation Risk-Adjusted Sharpe Ratio and Fund Performance Ranking Under a General Return Distribution." Journal of Financial Econometrics 7, no. 2 (2009): 152-173. doi:10.1093/jjfinec/nbn022

Pav, S. E. "The Sharpe Ratio: Statistics and Applications." CRC Press, 2021.

See Also

sr_bias.

Examples

# variance under normality:
sr_variance(1, 100, rep(0,5))

Compute variance covariance of Sharpe Ratios.

Description

Computes the variance covariance matrix of sample Sharpe ratios.

Usage

sr_vcov(X,vcov.func=vcov,ope=1)

Arguments

Details

Given n contemporaneous observations of p returns streams, this function estimates the asymptotic variance covariance matrix of the vector of sample Sharpes, \left[\zeta_1,\zeta_2,\ldots,\zeta_p\right]

One may use the default method for computing covariance, via the vcov function, or via a 'fancy' estimator, like sandwich:vcovHAC, sandwich:vcovHC, etc.

This code first estimates the covariance of the 2p vector of the vector x stacked on its Hadamard square, x^2. This is then translated back to a variance covariance on the vector of sample Sharpe ratios via the Delta method.

Value

a list containing the following components:

Author(s)

Steven E. Pav shabbychef@gmail.com

References

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

Lo, Andrew W. "The statistics of Sharpe ratios." Financial Analysts Journal 58, no. 4 (2002): 36-52. https://www.ssrn.com/paper=377260

See Also

reannualize

sr-distribution functions, dsr

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

Examples

X <- matrix(rnorm(1000*3),ncol=3)
colnames(X) <- c("ABC","XYZ","WORM")
Sigmas <- sr_vcov(X)
# make it fat tailed:
X <- matrix(rt(1000*3,df=5),ncol=3)
Sigmas <- sr_vcov(X)

if (require(sandwich)) {
Sigmas <- sr_vcov(X,vcov.func=vcovHC)
}

# add some autocorrelation to X
Xf <- filter(X,c(0.2),"recursive")
colnames(Xf) <- colnames(X)
Sigmas <- sr_vcov(Xf)

if (require(sandwich)) {
Sigmas <- sr_vcov(Xf,vcov.func=vcovHAC)
}

# should run for a vector as well
X <- rnorm(1000)
SS <- sr_vcov(X)

Sharpe Ratio Information Coefficient

Description

Computes the Sharpe Ratio Information Coefficient of Paulsen and Soehl, an asymptotically unbiased estimate of the out-of-sample Sharpe of the in-sample Markowitz portfolio.

Usage

sric(z.s)

Arguments

Details

Let X be an observed T \times k matrix whose rows are i.i.d. normal. Let \mu and \Sigma be the sample mean and sample covariance. The Markowitz portfolio is

w = \Sigma^{-1}\mu,

which has an in-sample Sharpe of \zeta = \sqrt{\mu^{\top}\Sigma^{-1}\mu}.

The Sharpe Ratio Information Criterion is defined as

SRIC = \zeta - \frac{k-1}{T\zeta}.

The expected value (over draws of X and of future returns) of the SRIC is equal to the expected value of the out-of-sample Sharpe of the (in-sample) portfolio w (again, over the same draws.)

Value

The Sharpe Ratio Information Coefficient.

Author(s)

Steven E. Pav shabbychef@gmail.com

References

Paulsen, D., and Soehl, J. "Noise Fit, Estimation Error, and Sharpe Information Criterion." arxiv preprint (2016): https://arxiv.org/abs/1602.06186

See Also

Other sropt Hotelling: asnr_confint(), inference()

Examples

# generate some sropts
nfac <- 3
nyr <- 5
ope <- 253
# simulations with no covariance structure.
# under the null:
set.seed(as.integer(charToRaw("fix seed")))
Returns <- matrix(rnorm(ope*nyr*nfac,mean=0,sd=0.0125),ncol=nfac)
asro <- as.sropt(Returns,drag=0,ope=ope)
srv <- sric(asro)

Create an 'sropt' object.

Description

Spawns an object of class sropt.

Usage

sropt(z.s, df1, df2, drag = 0, ope = 1, epoch = "yr", T2 = NULL)

Arguments

Details

The sropt class contains information about a rescaled T^2-statistic. The following are list attributes of the object:

sropt

The (optimal) Sharpe ratio statistic.

df1

The number of assets.

df2

The number of observations.

drag

The drag term, which is the 'risk free rate' divided by the maximum risk.

ope

The 'observations per epoch'.

epoch

The string name of the 'epoch'.

For the most part, this constructor should not be called directly, rather as.sropt should be called instead to compute the needed statistics.

Value

a list cast to class sropt, with the following attributes:

sropt

the optimal Sharpe statistic.

df1

the number of assets.

df2

the number of observed vectors.

drag

the input drag term.

ope

the input ope term.

epoch

the input epoch term.

T2

the Hotelling T^2 statistic.

Note

2FIX: allow rownames?

Author(s)

Steven E. Pav shabbychef@gmail.com

See Also

reannualize

as.sropt

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

Examples

# roll your own.
ope <- 253
zeta.s <- 1.0
df1 <- 10
df2 <- 6 * ope
set.seed(as.integer(charToRaw("fix seed")))
rvs <- rsropt(1,df1,df2,zeta.s,ope,drag=0)
roll.own <- sropt(z.s=rvs,df1,df2,drag=0,ope=ope)
print(roll.own)
# put a bunch in. naming becomes a problem.
rvs <- rsropt(5,df1,df2,zeta.s,ope,drag=0)
roll.own <- sropt(z.s=rvs,df1,df2,drag=0,ope=ope)
print(roll.own)

test for optimal Sharpe ratio

Description

Performs one sample tests of Sharpe ratio of the Markowitz portfolio.

Usage

sropt_test(X,alternative=c("greater","two.sided","less"),
            zeta.s=0,ope=1,conf.level=0.95)

Arguments

Details

Suppose x_i are n independent draws of a q-variate normal random variable with mean \mu and covariance matrix \Sigma. This code tests the hypothesis

H_0: \mu^{\top}\Sigma^{-1}\mu = \delta_0^2

The default alternative hypothesis is the one-sided

H_1: \mu^{\top}\Sigma^{-1}\mu > \delta_0^2

but this can be set otherwise.

Note there is no 'drag' term here since this represents a linear offset of the population parameter.

See ‘The Sharpe Ratio: Statistics and Applications’, section 6.3.2.

Value

A list with class "htest" containing the following components:

Author(s)

Steven E. Pav shabbychef@gmail.com

References

Pav, S. E. "The Sharpe Ratio: Statistics and Applications." CRC Press, 2021.

See Also

reannualize

sr_test, t.test.

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

Examples

# test for uniformity
pvs <- replicate(128,{ x <- sropt_test(matrix(rnorm(1000*4),ncol=4),alternative="two.sided")
                        x$p.value })
plot(ecdf(pvs))
abline(0,1,col='red') 

# input a sropt objects:
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)
stest <- sropt_test(asro,alternative="two.sided")

Stock Returns Data

Description

Nineteen years of daily log returns on three stocks and an ETF.

Usage

data(stock_returns)

Format

An xts object with 4777 observations and 4 columns.

The columns are the daily log returns for the tickers IBM, AAPL, SPY and XOM, as sourced from Yahoo finance using the quantmod package. Daily returns span from January, 2000 through December, 2018. Returns are ‘log returns’, which are the differences of the logs of daily adjusted closing price series, as defined by Yahoo finance (thus presumably including adjustments for splits and dividends). Dates of observations are the date of the second close defining the return, not the first.

Note

The author makes no guarantees regarding correctness of this data.

Author(s)

Steven E. Pav shabbychef@gmail.com

Source

Data were collected on October 2, 2019, from Yahoo finance using the quantmod package.

Examples

if (require(xts)) {
 data(stock_returns)
 as.sr(stock_returns)
}

Summarize a Sharpe, or (delta) optimal Sharpe object.

Description

Computes a ‘summary’ of an object, adding in some statistics.

Usage

## S3 method for class 'sr'
summary(object, ...)

## S3 method for class 'sropt'
summary(object, ...)

Arguments

Details

Enhances an object of class sr, sropt or del_sropt to also include t- or T-statistics, p-values, and so on.

Value

When an sr object is input, the object cast to class summary.sr with some additional fields:

tval

the equivalent t-statistic.

pval

the p-value under the null.

serr

the standard error of the Sharpe ratio.

When an sropt object is input, the object cast to class summary.sropt with some additional fields:

pval

the p-value under the null.

SRIC

the SRIC value, see sric.

Author(s)

Steven E. Pav shabbychef@gmail.com

References

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

See Also

print.sr.

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

Examples

# Sharpe's 'model': just given a bunch of returns.
set.seed(1234)
asr <- as.sr(rnorm(253*3),ope=253)
summary(asr)