# Express the covariance in terms of the standard deviations and correlations

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.

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

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