dsropt: 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 epoc

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 $\mbox{log}(f)$.

vector of probabilities.

number of observations.

...

arguments passed on to the respective Hotelling $T^2$ functions.

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.

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 bydf1.
$n$is denoted bydf2.
$\zeta_*$is denoted byzeta.s.
$d$is denoted byope.
$c_0/R$is denoted bydrag.

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. http://mansci.journal.informs.org/cgi/doi/10.1287/mnsc.1070.0852

Examples

Run this code

# 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))

Run the code above in your browser using DataLab