Testing Whether Portfolio Risk Contributions Identify Weights Uniquely

Summary

The document asks whether a portfolio’s vector of total risk contributions uniquely determines its asset weights. It defines each contribution as the asset weight multiplied by the derivative of portfolio volatility with respect to that weight, and frames the question as injectivity of the mapping from weights to contributions. The motivation is to know whether an inverse portfolio can be unique when it exists.

The author rewrites the problem as a system of nonlinear equations and reports a proof for two assets under a fully invested constraint, but cannot extend the argument to an arbitrary number of assets. No general theorem, counterexample, or numerical evidence is provided. The discussion also does not specify additional restrictions on the covariance matrix or portfolio domain that might affect uniqueness, so the question remains open within the stated setup. It is a mathematical inquiry into risk attribution and portfolio construction rather than a complete solution.

Key ideas

• Risk contribution is defined using portfolio weights and derivatives of portfolio volatility.
• The central question is whether the risk contribution mapping is injective.
• The author reports a two-asset result under a full-investment constraint.
• A general proof for more assets is not supplied.
• Uniqueness may depend on assumptions about the portfolio domain and covariance structure.

Tags

# How to show that the risk contribution function is or is not injective?


# How to show that the risk contribution function is or is not injective?












Assume a portoflio $w \in \mathbb{R}^n$, you can get the total risk contribution $\psi_i$ of asset $i$ by doing:

$$\psi_i = w_i \frac{\partial \sigma(w)}{\partial w_i}= \frac{1}{\sigma(w)} \left[ w_i^2 \sigma_i^2 + \sum_{j=1}^n w_i w_j \sigma_{i,j} \right]$$

So I can define a function $\Psi(x): \mathbb{R}^n \rightarrow \mathbb{R}^n$ which computes the contributions of a given portfolio $x$, and I will always find a unique answer.

My equation is, assume I have a set of risk contributions $\overrightarrow{\psi}=\{\psi_1, ... ,\psi_n\}$ , I'm looking to see whether $w^*=\Psi^{-1} \left( \overrightarrow{\psi} \right)$ is unique if it exists. In other words, I'm trying to see whether $\Psi(x)$ is injective.

Do you know if the proof of such statement exists, or how would you tackle the problem because the only thing I can think about now is to look at the $n$ nonlinear equations system with $n$ variable.

EDIT

I worked a bit on the problem and managed to formulate it as follows:

I have to show that $\nexists u,v \in \mathbb{R}^n$ such that $u \neq v$ and:

$$ \sigma_i^2 (u_i^2-v_i^2) + \sum_{j=1, i\neq j}^n \sigma_{i,j} (u_i-vi) = 0 \quad \forall i$$

I managed to prove this for $n=2$ and $u_1+u_2=v_1+v_2=1$, but I'm struggling to prove it for $n$ assets. I tried by recurrence but the fully invested constraint prevents me from doing so...

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.