FRAPO (version 0.3-8)

DivMeasures: 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.

Value

• numeric, the value of the diversification measure.

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: $$DR(\omega) = \frac{1}{\sqrt{\rho(\omega) (1 - CR(\omega)) + CR(\omega)}}$$

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

PMD

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)