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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.


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Let $\mathbb{1}$ denote a vector of ones. With the definition of risk parity in the question, we have

$$ Sw=c\mathbb{1} $$ with $c$ some constant, thus

$$ w=cS^{-1}\mathbb{1} $$

As $\mathbb{1}^Tw=1$, we have

$$ c\mathbb{1}^TS^{-1}c\mathbb{1}=1 \Rightarrow c=\frac{1}{\mathbb{1}^TS^{-1}\mathbb{1}} $$

and hence

$$ w=\frac{S^{-1}\mathbb{1}}{\mathbb{1}^TS^{-1}\mathbb{1}} $$

For risk, we get

$$ \sigma_P^2=w^TSRSw=\frac{\mathbb{1}^TS^{-1}}{\mathbb{1}^TS^{-1}\mathbb{1}}SRS\frac{S^{-1}\mathbb{1}}{\mathbb{1}^TS^{-1}\mathbb{1}}=\frac{\mathbb{1}^TR\mathbb{1}}{(\mathbb{1}^TS^{-1}\mathbb{1})^2} $$

For a single covariance we get

$$ Cov(r_i,r_p)=\frac{\sigma_iR_{i,.}\mathbb{1}}{\mathbb{1}^TS^{-1}\mathbb{1}} $$

and thus for all covariances as a vector:

$$ Cov(r,r_p)=\frac{SR\mathbb{1}}{\mathbb{1}^TS^{-1}\mathbb{1}} $$

Finally, the beta vector defined as covs over variance of the portfolio, equals

$$ \beta = \frac{\frac{SR\mathbb{1}}{\mathbb{1}^TS^{-1}\mathbb{1}}}{\frac{\mathbb{1}^TR\mathbb{1}}{(\mathbb{1}^TS^{-1}\mathbb{1})^2}}=\frac{(SR\mathbb{1})(\mathbb{1}^TS^{-1}\mathbb{1})}{\mathbb{1}^TR\mathbb{1}} $$


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