Package 'SharpeR'

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 [email protected]

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

---

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

X	matrix of returns, or xts object.
G	an $g \times q$ matrix of hedge constraints. A garden variety application would have G be one row of the identity matrix, with a one in the column of the instrument to be 'hedged out'.
drag	the 'drag' term, $c_0/R$. defaults to 0. It is assumed that drag has been annualized, i.e. has been multiplied by $\sqrt{ope}$. This is in contrast to the c0 term given to sr.
ope	the number of observations per 'epoch'. For convenience of interpretation, The Sharpe ratio is typically quoted in 'annualized' units for some epoch, that is, 'per square root epoch', though returns are observed at a frequency of ope per epoch. The default value is 1, meaning the code will not attempt to guess what the observation frequency is, and no annualization adjustments will be made.
epoch	the string representation of the 'epoch', defaulting to 'yr'.

Details

Suppose $x_i$ are $n$ independent draws of a $q$-variate normal random variable with mean $\mu$ and covariance matrix $\Sigma$. Let $G$ 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 [email protected]

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

x	vector of returns, or object of class data.frame, xts, or lm.
c0	the 'risk-free' or 'disastrous' rate of return. this is assumed to be given in the same units as x, not in 'annualized' terms.
ope	the number of observations per 'epoch'. For convenience of interpretation, The Sharpe ratio is typically quoted in 'annualized' units for some epoch, that is, 'per square root epoch', though returns are observed at a frequency of ope per epoch. The default value is 1, meaning the code will not attempt to guess what the observation frequency is, and no annualization adjustments will be made.
na.rm	logical. Should missing values be removed?
epoch	the string representation of the 'epoch', defaulting to 'yr'.
higher_order	a Boolean. If true, we compute cumulants of the returns to leverage higher order accuracy formulae when possible.

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 [email protected]

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_equality_test(), sr_test(), sr_unpaired_test(), sr_vcov(), sr, 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

X	matrix of returns, or xts object.
drag	the 'drag' term, $c_0/R$. defaults to 0. It is assumed that drag has been annualized, i.e. has been multiplied by $\sqrt{ope}$. This is in contrast to the c0 term given to sr.
ope	the number of observations per 'epoch'. For convenience of interpretation, The Sharpe ratio is typically quoted in 'annualized' units for some epoch, that is, 'per square root epoch', though returns are observed at a frequency of ope per epoch. The default value is 1, meaning the code will not attempt to guess what the observation frequency is, and no annualization adjustments will be made.
epoch	the string representation of the 'epoch', defaulting to 'yr'.

Details

Suppose $x_i$ are $n$ independent draws of a $q$-variate normal random variable with mean $\mu$ and covariance matrix $\Sigma$. Let $\bar{x}$ be the (vector) sample mean, and $S$ be the sample covariance matrix (using Bessel's correction). Let

$\zeta(w) = \frac{w^{\top}\bar{x} - c_0}{\sqrt{w^{\top}S w}}$

be the (sample) Sharpe ratio of the portfolio $w$, subject to risk free rate $c_0$.

Let $w_*$ be the solution to the portfolio optimization problem:

$\max_{w: 0 < w^{\top}S w \le R^2} \zeta(w),$

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

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

and

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

The units of $z_*$ are $\mbox{time}^{-1/2}$. Typically the Sharpe ratio is annualized by multiplying by $\sqrt{\mbox{ope}}$, where $\mbox{ope}$ is the number of observations per year (or whatever the target annualization epoch.)

Note that if ope and epoch are not given, the converter from xts attempts to infer the observations per epoch, assuming yearly epoch.

Value

An object of class sropt.

Author(s)

Steven E. Pav [email protected]

See Also

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

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

Examples

nfac <- 5
nyr <- 10
ope <- 253
# simulations with no covariance structure.
# under the null:
set.seed(as.integer(charToRaw("be determinstic")))
Returns <- matrix(rnorm(ope*nyr*nfac,mean=0,sd=0.0125),ncol=nfac)
asro <- as.sropt(Returns,drag=0,ope=ope)
# under the alternative:
Returns <- matrix(rnorm(ope*nyr*nfac,mean=0.0005,sd=0.0125),ncol=nfac)
asro <- as.sropt(Returns,drag=0,ope=ope)
# generating correlated multivariate normal data in a more sane way
if (require(MASS)) {
  nstok <- 10
  nfac <- 3
  nyr <- 10
  ope <- 253
  X.like <- 0.01 * matrix(rnorm(500*nfac),ncol=nfac) %*% 
    matrix(runif(nfac*nstok),ncol=nstok)
  Sigma <- cov(X.like) + diag(0.003,nstok)
  # under the null:
  Returns <- mvrnorm(ceiling(ope*nyr),mu=matrix(0,ncol=nstok),Sigma=Sigma)
  asro <- as.sropt(Returns,ope=ope)
  # under the alternative
  Returns <- mvrnorm(ceiling(ope*nyr),mu=matrix(0.001,ncol=nstok),Sigma=Sigma)
  asro <- as.sropt(Returns,ope=ope)
}

# using real data.
if (require(xts)) {
 data(stock_returns)
 asro <- as.sropt(stock_returns)
}

---

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 [email protected]

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)

---

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

z.s	an optimum Sharpe ratio statistic, on some set of assets.
z.sub	an optimum Sharpe ratio statistic, on a linear subspace of the assets. If larger than z.s an error is thrown.
df1	the number of assets in the portfolio.
df2	the number of observations.
df1.sub	the rank of the linear subspace of the hedge constraint. by restricting attention to the subspace.
drag	the 'drag' term, $c_0/R$. defaults to 0. It is assumed that drag has been annualized, i.e. has been multiplied by $\sqrt{ope}$. This is in contrast to the c0 term given to sr.
ope	the number of observations per 'epoch'. For convenience of interpretation, The Sharpe ratio is typically quoted in 'annualized' units for some epoch, that is, 'per square root epoch', though returns are observed at a frequency of ope per epoch. The default value is 1, meaning the code will not attempt to guess what the observation frequency is, and no annualization adjustments will be made.
epoch	the string representation of the 'epoch', defaulting to 'yr'.

Details

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 [email protected]

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

x, q	vector of quantiles.
df	the number of observations the statistic is based on. This is one more than the number of degrees of freedom in the corresponding t-statistic, although the effect will be small when df is large.
zeta	the 'signal-to-noise' parameter, $\zeta$ defined as the population mean divided by the population standard deviation, 'annualized'.
ope	the number of observations per 'epoch'. For convenience of interpretation, The Sharpe ratio is typically quoted in 'annualized' units for some epoch, that is, 'per square root epoch', though returns are observed at a frequency of ope per epoch. The default value is 1, meaning the code will not attempt to guess what the observation frequency is, and no annualization adjustments will be made.
...	arguments passed on to the respective t-distribution functions, namely lower.tail with default TRUE, log with default FALSE, and log.p with default FALSE.
p	vector of probabilities.
n	number of observations.

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 [email protected]

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_equality_test(), sr_test(), sr_unpaired_test(), sr_vcov(), sr, 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

x, q	vector of quantiles.
df1	the number of assets in the portfolio.
df2	the number of observations.
zeta.s	the non-centrality parameter, defined as $\zeta_* = \sqrt{\mu^{\top}\Sigma^{-1}\mu},$ for population parameters. defaults to 0, i.e. a central maximal Sharpe ratio distribution.
ope	the number of observations per 'epoch'. For convenience of interpretation, The Sharpe ratio is typically quoted in 'annualized' units for some epoch, that is, 'per square root epoch', though returns are observed at a frequency of ope per epoch. The default value is 1, meaning the code will not attempt to guess what the observation frequency is, and no annualization adjustments will be made.
drag	the 'drag' term, $c_0/R$. defaults to 0. It is assumed that drag has been annualized, i.e. is given in the same units as x and q.
log	logical; if TRUE, densities $f$ are given as $\mbox{log}(f)$.
p	vector of probabilities.
n	number of observations.
...	arguments passed on to the respective Hotelling $T^2$ functions.

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 [email protected]

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_test(), sropt

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

z.s	an object of type sropt, or del_sropt
type	the estimator type. one of c("KRS", "MLE", "unbiased")

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.

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

• the estimator of Kubokawa, Roberts, and Shaleh (KRS), which is a shrinkage estimator.

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 [email protected]

References

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

Spruill, M. C. "Computation of the maximum likelihood estimate of a noncentrality parameter." Journal of multivariate analysis 18, no. 2 (1986): 216-224. https://www.sciencedirect.com/science/article/pii/0047259X86900709

See Also

F-distribution functions, df.

Other sropt Hotelling: 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

x	an object of some kind.

Details

To satisfy the minimum requirements of an S3 class.

Value

a boolean.

Author(s)

Steven E. Pav [email protected]

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

x	an object of some kind.

Details

To satisfy the minimum requirements of an S3 class.

Value

a boolean.

Author(s)

Steven E. Pav [email protected]

See Also

sr

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

x	an object of some kind.

Details

To satisfy the minimum requirements of an S3 class.

Value

a boolean.

Author(s)

Steven E. Pav [email protected]

See Also

sropt

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

Examples

nfac <- 5
nyr <- 10
ope <- 253
# simulations with no covariance structure.
# under the null:
set.seed(as.integer(charToRaw("be determinstic")))
Returns <- matrix(rnorm(ope*nyr*nfac,mean=0,sd=0.0125),ncol=nfac)
asro <- as.sropt(Returns,drag=0,ope=ope)
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

X	an $n \times p$ matrix of observed returns.
vcov.func	a function which takes an object of class lm, and computes a variance-covariance matrix. If equal to the string "normal", we assume multivariate normal returns.
fit.intercept	a boolean controlling whether we add a column of ones to the data, or fit the raw uncentered second moment.

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:

mu	a $q = p(p+3)/2$ vector of the negative Markowitz portfolio, then the vech'd precision matrix of the sample data
Ohat	the $q \times q$ estimated variance covariance matrix.
n	the number of rows in X.
p	the number of assets.

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 [email protected]

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

q	vector of quantiles.
df1	the number of assets in the portfolio.
df2	the number of observations.
z.s	an observed Sharpe ratio statistic, annualized.
ope	the number of observations per 'epoch'. For convenience of interpretation, The Sharpe ratio is typically quoted in 'annualized' units for some epoch, that is, 'per square root epoch', though returns are observed at a frequency of ope per epoch. The default value is 1, meaning the code will not attempt to guess what the observation frequency is, and no annualization adjustments will be made.
lower.tail	logical; if TRUE (default), probabilities are $P[X \le x]$, otherwise, $P[X > x]$.
log.p	logical; if TRUE, probabilities p are given as $\mbox{log}(p)$.
p	vector of probabilities.
lb	the lower bound for the output of qco_sropt.
ub	the upper bound for the output of qco_sropt.

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 [email protected]

See Also

reannualize

dsropt,psropt,qsropt,rsropt

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

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

q	vector of quantiles.
df	the degrees of freedom of the t-statistic.
tstat	the observed (non-central) t-statistic.
lower.tail	logical; if TRUE (default), probabilities are $P[X \le x]$, otherwise, $P[X > x]$.
log.p	logical; if TRUE, probabilities p are given as $\mbox{log}(p)$.
p	vector of probabilities.
n	number of observations. If 'length(n) > 1', the length is taken to be the number required.

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 [email protected]

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_equality_test(), sr_test(), sr_unpaired_test(), sr_vcov(), sr, 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

n	Number of observations
zeta	the 'signal-to-noise' parameter, defined as the population mean divided by the population standard deviation, 'annualized'.
sig.level	Significance level (Type I error probability).
power	Power of test (1 minus Type II error probability).
alternative	One- or two-sided test.
ope	the number of observations per 'epoch'. For convenience of interpretation, The Sharpe ratio is typically quoted in 'annualized' units for some epoch, that is, 'per square root epoch', though returns are observed at a frequency of ope per epoch. The default value is 1, meaning the code will not attempt to guess what the observation frequency is, and no annualization adjustments will be made.

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 [email protected]

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_equality_test(), sr_test(), sr_unpaired_test(), sr_vcov(), sr, 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

df1	the number of assets in the portfolio.
df2	the number of observations.
zeta.s	the 'signal-to-noise' parameter, defined as ...
sig.level	Significance level (Type I error probability).
power	Power of test (1 minus Type II error probability).
ope	the number of observations per 'epoch'. For convenience of interpretation, The Sharpe ratio is typically quoted in 'annualized' units for some epoch, that is, 'per square root epoch', though returns are observed at a frequency of ope per epoch. The default value is 1, meaning the code will not attempt to guess what the observation frequency is, and no annualization adjustments will be made.

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 [email protected]

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_test(), sropt

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

x	a (non-empty) numeric vector of data values, or an object of class sr.
oosdf	the future (or 'out of sample', thus 'oos') degrees of freedom. In the vanilla Sharpe case, this is the number of future observations minus one.
oosrescal	the rescaling parameter for the future Sharpe ratio. The default value holds for the case of unattributed models ('vanilla Shape'), but can be set to some other value to deal with the magnitude of attribution factors in the future period.
ope	the number of observations per 'epoch'. For convenience of interpretation, The Sharpe ratio is typically quoted in 'annualized' units for some epoch, that is, 'per square root epoch', though returns are observed at a frequency of ope per epoch. The default value is to take the same ope from the input x object, if it is unambiguous.
level	the confidence level required.
level.lo	the lower confidence level required.
level.hi	the upper confidence level required.
type	which method to apply. Only methods based on an approximate standard error are supported.

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 [email protected]

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_equality_test(), sr_test(), sr_unpaired_test(), sr_vcov(), sr, 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

x	an object of class sr or sropt.
...	further arguments to be passed to or from methods.

Value

the object, wrapped in invisible.

Author(s)

Steven E. Pav [email protected]

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_equality_test(), sr_test(), sr_unpaired_test(), sr_vcov(), sr, 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

object	an object of class sr or sropt.
new.ope	the new observations per epoch. If none given, it is not updated.
new.epoch	a string representation of the epoch. If none given, it is not updated.

Value

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

Author(s)

Steven E. Pav [email protected]

See Also

sr

sropt

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

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

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

z	an observed Sharpe ratio statistic, of class sr.
type	estimator type. one of "t", "Lo", "Mertens", "Bao"

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 [email protected]

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. http://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_equality_test(), sr_test(), sr_unpaired_test(), sr_vcov(), sr, summary.sr

Examples

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

---