I really need help on this problem. Any suggestion is greatly appreciated!
Suppose there are $n$ assets with $n\times n $ covariance matrix $C=SRS$, where $S$ is a matrix with standard deviations $\sigma_i$ on its diagonal and zeroes off-diagonal, and $R$ is a correlation matrix. Let $w_p$ be the risk parity portfolio, defined so that $w_P,_i\sigma_i = w_P,_j\sigma_j,\forall 1 \leq i, j \leq n$. As usual $w_P^Tu = 1$, $u$ is the unit vector. Express $\beta_i$ (the covariance of asset $i$ to the risk parity portfolio, divided by the risk parity portfolio's variance) in terms of the standard deviations and correlations.