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Risk Parity Portfolio Betas from Volatilities and Correlations

Article Quant Q&A · Author: Celine
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Summary

The document derives asset betas relative to a risk parity portfolio from asset volatilities and the correlation matrix. Risk parity is defined here as weights whose products with their respective volatilities are equal, with weights normalized to sum to one. This implies weights proportional to inverse volatility.

Using the covariance decomposition into a diagonal volatility matrix and a correlation matrix, the derivation obtains the portfolio variance and each asset’s covariance with the portfolio. Dividing those covariances by portfolio variance gives a beta vector proportional to asset volatility times its correlation-matrix row sum, scaled by the sum of inverse volatilities and normalized by the sum of all correlations. The result depends on this specific equal-weighted-risk definition and full investment constraint; it is not a general formula for other risk parity variants or constraints.

Key ideas

  • Under the stated risk parity condition, portfolio weights are proportional to inverse asset volatility.
  • The covariance matrix can be expressed as the product of volatility, correlation, and volatility matrices.
  • Portfolio variance under these weights depends on the sum of all entries in the correlation matrix.
  • Each asset’s covariance with the portfolio is determined by its volatility and its correlation row sum.
  • The beta expression depends on the stated normalization and risk parity definition.

Tags

Portfolio constructionStatisticsRisk management
Full text
# Express the covariance in terms of the standard deviations and correlations


# 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.

## Answer by Kermittfrog (score 1, accepted)

https://quant.stackexchange.com/a/68611

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

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

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.