Marginal Portfolio Risk Contributions Depend on the Weight Constraint
Summary
The document explains why two derivatives for a two-asset portfolio's volatility differ when one asset's weight is constrained to equal one minus the other's. Roncalli's marginal risk contribution is a partial derivative: it varies one asset's weight while holding the other weights fixed. The question instead differentiates total portfolio volatility along a fully invested constraint, so the second weight changes at the same time.
The answer writes portfolio volatility as a function of two independent weights, differentiates with respect to the first while holding the second fixed, and then substitutes the fully invested weights. That recovers the marginal contribution formula. The distinction matters when interpreting risk contributions or decomposing portfolio risk: a partial derivative measures sensitivity to one weight with others fixed, whereas a constrained derivative measures the effect of reallocating weight between assets. The exchange is analytical and illustrative, but does not extend the result to more assets or other portfolio constraints.
Key ideas
- Marginal risk contribution differentiates portfolio risk with respect to one weight while holding other weights fixed.
- A fully invested portfolio links the two asset weights, so differentiating along that constraint changes both weights.
- For two assets, the independent-weight derivative yields the stated marginal contribution after substituting the fully invested weights.
- The choice of derivative reflects whether the question concerns an isolated weight change or a constrained reallocation.
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Full text
# What is the analytic value of an asset's risk contribution, if $n=2$?
# What is the analytic value of an asset's risk contribution, if $n=2$?
The marginal risk contribution of asset $i$ is defined by Roncalli in his paper on ERC as follows:
$$\frac{\partial \sigma(x)}{\partial x_i} = \frac{1}{\sigma(x)} \left( w_i \sigma_i^2 + \sum_{\substack{j=1 \\ i \neq j}}^n w_j \sigma_{i,j} \right)$$
I was thinking about computing the value for a fully invested portfolio with $n=2$ assets, i.e. $x=\{w,1-w\}$.
With $n=2$, we know that:
$$\sigma(x)=\sqrt{w^2 \sigma_1^2 + (1-w)^2 \sigma_2^2 + 2 w (1-w) \sigma_{1,2}}$$
So, we can compute the marginal contribution of asset 1 (with $x_1=w$) as:
$$\frac{d \sigma(x)}{dw} = \frac{1}{2} \frac{1}{\sigma(x)} \left[ 2w \sigma_1^2 + 2(1-w)(-1) \sigma_2^2 + 2 \sigma_{1,2} (1-2w) \right] $$
$$\frac{d \sigma(x)}{dw} = \frac{1}{\sigma(x)} \left[ w \sigma_1^2 - (1-w) \sigma_2^2 + \sigma_{1,2} (1-2w) \right] $$
If I use Roncalli's definition, I get:
$$\frac{d \sigma(x)}{dw} = \frac{1}{\sigma(x)} \left[ w \sigma_1^2 + (1-w) \sigma_{1,2} \right] $$
The two results are not the same, and I'm trying to understand why. So far, I came up with the fact that Roncalli's definition is not taking into account the fact that the portfolio is fully invested. Is that correct?
## Answer by Rustam (score 4, accepted)
https://quant.stackexchange.com/a/8296
You see, you added something new to the source formula, i.e. a dependence between weights of different assets: $w_2 = 1 - w_1$.
Let's try to forget that they are related to each other and vary them independently:
$$\sigma(x)=\sqrt{w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \sigma_{1,2}}$$
Now $\frac{dw_2}{dw_1} = 0$ and equation becomes:
$$\frac{d \sigma(x)}{dw_1} = \frac{1}{2} \frac{1}{\sigma(x)} \left[ 2w_1 \sigma_1^2 + 2 w_2 \sigma_{1,2} \right] $$
$$\frac{d \sigma(x)}{dw_1} = \frac{1}{\sigma(x)} \left[ w_1 \sigma_1^2 + w_2 \sigma_{1,2} \right] $$
Now, let's tak $w_2 = 1 - w_1$ and you receive Roncalli's formula:
$$\frac{d \sigma(x)}{dw} = \frac{1}{\sigma(x)} \left[ w_1 \sigma_1^2 + (1-w_1) \sigma_{1,2} \right]$$ I think it's a tricky question, but I believe his way is right, because when you decompose $\sigma$ into factors, it is more correct to keep weights constant.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.