Deriving Marginal Risk Contribution from Portfolio Volatility
Summary
This note derives marginal risk contribution for a two-asset portfolio by differentiating portfolio volatility with respect to an asset weight. Applying the chain rule to the square root of portfolio variance leaves portfolio volatility in the denominator; multiplying the derivative by the asset weight gives that asset’s risk contribution.
It then shows how the expression can be written using covariance between the asset and the portfolio. Portfolio returns are a weighted sum of asset returns, so covariance’s linearity makes the asset-to-portfolio covariance equal to the weighted sum of its covariances with all assets. Dividing this quantity by portfolio variance gives the asset’s beta to the portfolio, yielding an equivalent beta-based expression for risk contribution. The explanation is algebraic and provides no empirical example; it assumes the usual variance-based volatility measure and does not discuss estimation choices or broader portfolio constraints.
Key ideas
- The derivative of portfolio volatility with respect to an asset weight follows from the chain rule applied to portfolio variance.
- The square root remains represented through portfolio volatility in the derivative’s denominator.
- Multiplying the volatility derivative by the asset weight gives that asset’s marginal risk contribution.
- Covariance linearity expresses covariance between an asset and the portfolio as a weighted sum of asset covariances.
- The covariance form of marginal risk contribution can be rewritten using the asset’s beta to the portfolio.
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# Marginal Risk Contribution Formula
# Marginal Risk Contribution Formula
I am trying to understand and implement the standard 'marginal risk contribution' approach to portfolio risk and hoping to reconcile the formulae provided for its calculation in different sources. Specifically I am trying to understand the difference of these two papers:
- Page 4 of this paper by Roncalli (http://thierry-roncalli.com/download/erc.pdf)
- Pages 2-3 of this paper by Kazemi (http://people.umass.edu/kazemi/An%20Introduction%20to%20Risk%20Parity.pdf)
What I'd really like more help understanding is:
1) on page 2 of the Kazemi PDF, where MC1 is defined, how is this partial derivative of the portfolio vol with respect to w1 taken? (how does the square root of the portfolio vol disappear and how does the portfolio vol appear in the denominator)
2) how does the alternative formulation of MC in terms of the covariance between the asset and the portfolio (page 3 of Kazemi - "beta") get derived? ideally, looking for a step-by-step guide which illuminates how MC can be thought of in this way
## Answer by Cettt (score 8, accepted)
https://quant.stackexchange.com/a/37879
concerning your first question: the derivative does not disappear: $\sigma(R_p)$ contains the square root. To be more precise, set $$ \sigma(R_p) = \sqrt{w_1^2\cdot\sigma(R_1)^2 + w_2^2\cdot\sigma(R_2)^2 + 2w_1w_2\text{Cov}(R_1, R_2)}. $$ Then we get using the chain rule: \begin{align} \frac{\partial\sigma(R_p)}{\partial w_1} &= \frac 12 \cdot \biggl(\sqrt{w_1^2\cdot\sigma(R_1)^2 + w_2^2\cdot\sigma(R_2)^2 + 2w_1w_2\text{Cov}(R_1, R_2)}\biggr)^{-1} \cdot\Bigl(2w_1\cdot\sigma(R_1)^2 + 2w_2\text{Cov}(R_1, R_2)\Bigr) = \\ &= \frac{1}{\sigma(R_p)} \cdot\Bigl(w_1\cdot\sigma(R_1)^2 + w_2\text{Cov}(R_1, R_2)\Bigr). \end{align} So you see, the square root is still there, it is just hidden in $\sigma(R_p)$. $MC_1$ can be obtained from this by just multiplying the derivative with $w_1$.
Concerning you second question: notice two things. First, $$ R_p = \sum_{j =1}^N w_j R_j $$ and second the covariance function is bilinear. This implies that $$ \text{Cov}(R_i, R_p) = \text{Cov}\Bigl(R_i, \sum_{j =1}^N w_j R_j\Bigr) = \sum_{j = 1}^N w_j \text{Cov}(R_i, R_j). $$ From this you can easily derive the alternative representation: \begin{align} MC_1 &= w_1 \cdot \frac{\sum_{j = 1}^N w_j \text{Cov}(R_1, R_j)}{\sigma(R_p)} = w_1\sigma(R_p) \cdot \frac{\sum_{j = 1}^N w_j \text{Cov}(R_1, R_j)}{\sigma(R_p)^2} \\ &= w_1\sigma(R_p) \cdot \frac{\text{Cov}(R_1, R_p)}{\sigma(R_p)^2} = w_1\sigma(R_p) \cdot \beta_1. \end{align} I hope this helps a little.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.