Risk Attribution

Overview



Risk attribution is the process of breaking down the total risk of an asset or portfolio into a sum of components. Typically, the components are factors that affect the portfolio in some statistical sense, that is, risk factors.

As an example, an equity portfolio may have its risk decomposed into components arising from

  • Value
  • Growth
  • Market Cap

Eulers Theorem



Standard risk attribution is based on Eulers theorem.

Eulers Principle, states that for a function which is homogenous of degree 1, that is
{% f(\lambda \vec{w}) = \lambda f(\vec{w}) %}
can be expressed as
{% f(\vec{w}) = \sum_i ^n w_i \frac{\partial{f}}{{\partial{w_i}}} %}
(see braga pg 45) and (see wikipedia)

If f represents a measure of risk, then the above formula represents a breakdown of contributions of each asset to the total risk, based on the marginal values of the riks.

The Axioms of Risk explicitly include homogeneity as a property.

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