Atkinson Index

From Open Risk Manual

Definition

For the purpose of measuring credit portfolio or market portfolio Concentration Risk, income inequality or diversity, the Atkinson Index is a parametric family of indexes which, given an index parameter epsilon is defined as follows

Details

If we have n exposures (alternatively values / income measurements) E_i summing up to a total value of


E_T = \sum^{n}_{i=1} E_{i}

where each observation's fraction is defined as


w_{i} = \frac{E_i}{E_T}

Then the Atkinson index is given by[1]

A_\varepsilon=
\begin{cases}
1-   n^{\varepsilon/(\varepsilon-1)}   \left(\sum_{i=1}^{n} w_{i}^{1-\varepsilon}\right)^{1/(1-\varepsilon)}
& \mbox{for}\ 0 \leq \epsilon \neq 1 \\
1- n e^{\left(\frac{1}{n} \sum_{i=1}^{n} \log w_{i}\right)}
& \mbox{for}\ \varepsilon=1,
\end{cases}
  • The index parameter epsilon can take any positive value
  • The index varies between zero (homogeneous observations) and one (perfect concentration)

Usage

None

Variations

None

Issues and Challenges

None

Implementations

Open Source implementations of the Atkinson index are available in

See Also

References

  1. A. Atkinson, "On the measurement of inequality", Journal of Economic Theory 2, 244-263 (1970)

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