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