When Portfolio Risk Contributions Are Additive
Summary
The document examines how to interpret asset-level contributions to portfolio variance. It describes a marginal-risk measure based on twice the covariance between an asset and the portfolio, and explains why this quantity depends on the other holdings through their correlations. As a result, assigning each manager a fixed share of total risk can be misleading when returns are correlated.
When assets are uncorrelated, the covariance of an asset with the portfolio reduces to its own variance scaled by its portfolio weight. In that special case, each asset’s contribution is more separable. The answer notes that independence implies zero correlation, but the discussion is brief and centers on variance rather than standard deviation. It does not provide a numerical example or address the broader conditions needed to interpret risk attribution across portfolios.
Key ideas
- Marginal risk depends on how an asset co-moves with the whole portfolio.
- Correlations make an asset’s attributed risk depend on the portfolio’s other holdings.
- With uncorrelated returns, each asset’s covariance with the portfolio is its variance multiplied by its weight.
- Independent returns are uncorrelated, allowing a more separable variance contribution in this special case.
Tags
Full text
# Additive portfolio risk decomposition
# Additive portfolio risk decomposition
In his paper Budgeting and Monitoring the Risk of Defined Benefit Pension Funds, Bill Sharpe writes:
> [...] the sum of the weighted marginal risks of the portfolio components will equal twice the variance of the overall portfolio. This leads some to define the risk contribution of a component as half its marginal risk (that is, its covariance with the portfolio) so that a weighted average of these values will equal the variance of the overall portfolio. [...] it sometimes leads to an incorrect view that it is possible to decompose portfolio risk into a set of additive components and to incorrect statements of the form "this manager contributed 15% to the total risk of the portfolio".
In the next paragraph, Bill Sharpe continues to say that the additive decomposition does make sense if all the asset's returns were indepdendent.
What makes the view and the statements ((highlighted in italics) incorrect in general case, but correct when the asset returns are independent? It seems either way there is an additive decomposition of the variance, but not of the standard deviation, of portfolio returns.
## Answer by vanguard2k (score 1)
https://quant.stackexchange.com/a/9049
I figure the answer is hidden in the definition of "marginal contribution to risk". I try to use the notation of the paper you linked.
Marginal contribution to risk is defined as:
$M_i = 2C_{ip}$ with $C_{ip}$ being the covariance between asset $i$ and the portfolio $p$. One can now argue that, since this covariance also depends on the other assets $j\neq i$, the value $M_i$ is implicitly connected to the other assets via the correlation structure.
On the other hand if all assets are uncorrelated, we have that $C_{ip}=C_{ii}w_i$, since $C_{ij}=0$ for $j\neq i$. Thus, the marginal contributions do not depend on the other assets. Therefore one can speak of a more "pure" contribution to variance in this case.
Finally, if the asset returns are independent, they are also uncorrelated.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.