Description Sharpe Ratio Optimal Sharpe Ratio Spanning and Hedging Legal Mumbo Jumbo Note Author(s) References
Inference on Sharpe ratio and Markowitz portfolio.
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' or 'disastrous rate' of return. Then
z = (xbar - c0)/s
is the (sample) Sharpe ratio.
The units of z are per root time. 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 ζ / 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 alpha, and sigma, and then let the 'Sharpe ratio' be
z = (alpha - c0)/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.
Suppose xi are n independent draws of a q-variate normal random variable with mean mu and covariance matrix Sigma. Let xbar be the (vector) sample mean, and S be the sample covariance matrix (using Bessel's correction). Let
Z(w) = (w'xbar - c0)/sqrt(w'Sw)
be the (sample) Sharpe ratio of the portfolio w, subject to risk free rate c0.
Let w* be the solution to the portfolio optimization problem:
max {Z(w) | 0 < w'Sw <= R^2},
with maximum value z* = Z(w*). Then
w* = R S^-1 xbar / sqrt(xbar' S^-1 xbar)
and
z* = sqrt(xbar' S^-1 xbar) - c0/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' Sigma^-1 mu),
the 'drag' term, c0/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
.
As above, let
Z(w) = (w'xbar - c0)/sqrt(w'Sw)
be the (sample) Sharpe ratio of the portfolio w, subject to risk free rate c0.
Let G be a g x q matrix of 'hedge constraints'. Let w* be the solution to the portfolio optimization problem:
max {Z(w) | 0 < w'Sw <= R^2, G S w = 0},
with maximum value z* = Z(w*). 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.
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.
The following are still in the works:
Corrections for standard error based on skew, kurtosis and autocorrelation.
Tests on Sharpe under positivity constraint. (c.f. Silvapulle)
Portfolio spanning tests.
Tests on portfolio weights.
This package is maintained as a hobby.
Steven E. Pav [email protected]
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Kan, Raymond and Zhou, GuoFu. "Tests of Mean-Variance Spanning." Annals of Economics and Finance 13, no. 1 (2012) http://www.aeconf.net/Articles/May2012/aef130105.pdf
Britten-Jones, Mark. "The Sampling Error in Estimates of Mean-Variance Efficient Portfolio Weights." The Journal of Finance 54, no. 2 (1999): 655–671. http://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. http://dx.doi.org/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. http://dx.doi.org/10.1111/j.1467-9469.2011.00729.x
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