DivMeasures {FRAPO} | R Documentation |
Diversification Measures
Description
These functions compute the diversification ratio, the volatility weighted average correlation and concentration ratio of a portfolio.
Usage
dr(weights, Sigma)
cr(weights, Sigma)
rhow(weights, Sigma)
Arguments
weights |
Vector: portfolio weights. |
Sigma |
Matrix: Variance-covariance matrix of portfolio assets. |
Details
The diversification ratio of a portfolio is defined as:
DR(\omega) = \frac{\sum_{i = 1}^N \omega_i \sigma_i}{\sqrt{\omega'
\Sigma \omega}}
for a portfolio of N
assets and \omega_i
signify the
weight of the i-th asset and \sigma_i
its standard deviation and
\Sigma
the variance-covariance matrix of asset returns. The
diversification ratio is therefore the weighted average of the assets'
volatilities divided by the portfolio volatility.
The concentration ration is defined as:
CR = \frac{\sum_{i = 1}^N (\omega_i \sigma_i)^2}{(\sum_{i = 1}^N
\omega_i \sigma_i)^2}
and the volatility-weighted average correlation of the assets as:
\rho(\omega) = \frac{\sum_{i > j}^N (\omega_i \sigma_i \omega_j
\sigma_j)\rho_{ij}}{\sum_{i > j}^N (\omega_i \sigma_i \omega_j
\sigma_j)}
The following equation between these measures does exist:
DR(\omega) = \frac{1}{\sqrt{\rho(\omega) (1 - CR(\omega)) +
CR(\omega)}}
Value
numeric
, the value of the diversification measure.
Author(s)
Bernhard Pfaff
References
Choueifaty, Y. and Coignard, Y. (2008): Toward Maximum Diversification, Journal of Portfolio Management, Vol. 34, No. 4, 40–51.
Choueifaty, Y. and Coignard, Y. and Reynier, J. (2011): Properties of the Most Diversified Portfolio, Working Paper, http://papers.ssrn.com
See Also
Examples
data(MultiAsset)
Rets <- returnseries(MultiAsset, method = "discrete", trim = TRUE)
w <- Weights(PMD(Rets))
V <- cov(Rets)
DR <- dr(w, V)
CR <- cr(w, V)
RhoW <- rhow(w, V)
test <- 1 / sqrt(RhoW * (1 - CR) + CR)
all.equal(DR, test)