Separating Portfolio Market and Residual Risk with a Covariance Matrix
Summary
The document explains how to decompose portfolio risk into the component associated with a world equity index and residual risk when only expected returns and a covariance matrix are available. For each asset, its beta to the world index can be obtained from its covariance with the world index divided by the world index variance. Combining those betas gives the portfolio’s market exposure.
The market component of the asset covariance matrix is formed from the beta vector and world-index variance. Subtracting it from the full covariance matrix gives residual covariance, and the portfolio’s residual variance is the portfolio weights applied on both sides of that matrix. The residual covariance need not be diagonal, so assets can retain correlated risks beyond their world-market exposure. The method assumes arithmetic returns and covariances measured at the relevant horizon; it does not require estimating a separate regression error from return time series.
Key ideas
- Asset betas to the world index can be calculated from index covariances and world-index variance.
- Portfolio beta is the weighted sum of its constituent asset betas.
- The market-related covariance component is constructed from the beta vector and world-index variance.
- Subtracting market covariance from total covariance yields a residual covariance matrix that may have off-diagonal terms.
- Portfolio residual variance is calculated by applying portfolio weights to that residual covariance matrix.
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# Unsystematic and systematic risk of a portfolio
# Unsystematic and systematic risk of a portfolio
I have 8 country stock indexes and 1 world stock index. I do not actually have time series data but I'm given the following data:
- $\mu$, the vector of expected future returns for all 8 country indexes and world index (9 indexes).
- $\Omega$, the variance covariance matrix of all 9 indexes.
I'm forming a MV efficient and Michaud resampling portfolio over the 8 country indexes - the world index is not considered an investable asset class. I want to compare the two portfolios by looking at the systematic risk and unsystematic risk of both portfolios w.r.t. the world market index. So we have the two weights vectors produced by the two methodologies:
- $_1w$ (MV)
- $_2w$ (REF, Resampled Efficient Frontier).
We can calculate the betas of both portfolios by going $_j\beta_p = \sum_{i=1}^8 (_jw_i )\frac{\sigma_{i,world}}{\sigma^2_{world}}$ for $j = 1,2$. Being able to sum the coefficients like this follows from OLS.
How do I get from here to the unsystematic and systematic risk of the portfolios? I can't get the error from the specification that generates the betas so it seems I'm stuck?
## Answer by John (score 3)
https://quant.stackexchange.com/a/4143
Assuming those are arithmetic returns and covariances at the horizon, calculate a $9\times1$ vector containing the betas with respect to the world index using the covariance matrix, call it $\beta$. The covariance resulting from the world index can be described as $\beta\sigma_{world}^{2}\beta'$. The matrix $\Sigma_{residual}\equiv\Omega-\beta\sigma_{world}^{2}\beta'$ will then reflect the residual covariance. Note that this residual covariance matrix is not necessarily a diagonal matrix, as some CAPM-like models would require. To get a measure of the residual risk of the portfolio, you would then calculate $w'\Sigma_{residual}w$.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.