Density, distribution function, quantile function and random
generation for the Sharpe ratio distribution with
df degrees of freedom
(and optional signal-noise-ratio
1 2 3 4 5 6 7
vector of quantiles.
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
the 'signal-to-noise' parameter, zeta defined as the population mean divided by the population standard deviation, 'annualized'.
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
arguments passed on to the respective t-distribution functions, namely
vector of probabilities.
number of observations.
Suppose xi are n independent draws of a normal random variable with mean mu and variance sigma^2. Let xbar be the sample mean, and s be the sample standard deviation (using Bessel's correction). Let c0 be the 'risk free rate'. Then
z = (xbar - c0)/s
is the (sample) Sharpe ratio.
The units of z is per root time. 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)(xbar - c0)/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 = (mu - c0)/sigma, and annualization parameter d.
The parameters are encoded as follows:
n is denoted by
zeta is denoted by
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.
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.
This is a thin wrapper on the t distribution.
dt, pt, qt can accept ncp from
limited range (delta <= 37.62). Some corrections
may have to be made here for large
Steven E. Pav [email protected]
Sharpe, William F. "Mutual fund performance." Journal of business (1966): 119-138. http://ideas.repec.org/a/ucp/jnlbus/v39y1965p119.html
dt, pt, qt, rt
1 2 3 4 5 6 7 8 9
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.