Fenews: Some of the most important contributions to financial engineering in recent years have come from mathematical finance, and perhaps the most important of these is the theory of coherent risk measures proposed by Artzner et al (1999, 1997). Coherent risk measures have various desirable properties – most particularly, the property of subadditivity, which says that adding individual risks together does not increase overall risk – which make them demonstrably superior risk measures to the VaR.
The area is continuing to develop, and one of the most interesting newer developments is the theory of spectral risk measures (Acerbi 2004, 2002). The distinguishing feature of spectral measures is that they explicitly relate the risk measure to a user’s risk-aversion function.
To appreciate what these are about, let us define a class of risk measures Mφ that are weighted averages of the quantiles of our loss distribution. If p is a probability and q(p) is the p-quantile of a loss distribution (i.e., so q(p) is that loss such that the probability of a loss less than or equal to it is p), then our risk measure is
where the weighting function, φ(p), also known as the risk spectrum or risk-aversion function, remains to be determined.
It is interesting to note that both the Expected Tail Loss (henceforth ETL, aka expected shortfall, conditional VaR) and the VaR are special cases of this spectral measure. The ETL gives tail losses an equal weight (equal to 1/(1-α) if quantiles are spaced at equal increments of p) and gives other losses a weight of 0. The VaR is also a special case – albeit a highly degenerate one – of Mφ. Because the VaR is just a single quantile, the spectral risk measure is made equal to the VaR if places all its weight on a loss equal to the VaR and none of its weight on any other loss, including any higher loss. So one measure places equal weight on tail losses, and the other places none at all. Click here to learn more.
You may be also interested in this column in which Kevin Dowd addresses how to estimate these risk measures using Excel.
related items:
Presentation: Spectral Measures of Risk - Coherence in Theory and Practise
Paper: Progress in Risk Measurement
The area is continuing to develop, and one of the most interesting newer developments is the theory of spectral risk measures (Acerbi 2004, 2002). The distinguishing feature of spectral measures is that they explicitly relate the risk measure to a user’s risk-aversion function.
To appreciate what these are about, let us define a class of risk measures Mφ that are weighted averages of the quantiles of our loss distribution. If p is a probability and q(p) is the p-quantile of a loss distribution (i.e., so q(p) is that loss such that the probability of a loss less than or equal to it is p), then our risk measure is
where the weighting function, φ(p), also known as the risk spectrum or risk-aversion function, remains to be determined.It is interesting to note that both the Expected Tail Loss (henceforth ETL, aka expected shortfall, conditional VaR) and the VaR are special cases of this spectral measure. The ETL gives tail losses an equal weight (equal to 1/(1-α) if quantiles are spaced at equal increments of p) and gives other losses a weight of 0. The VaR is also a special case – albeit a highly degenerate one – of Mφ. Because the VaR is just a single quantile, the spectral risk measure is made equal to the VaR if places all its weight on a loss equal to the VaR and none of its weight on any other loss, including any higher loss. So one measure places equal weight on tail losses, and the other places none at all. Click here to learn more.
You may be also interested in this column in which Kevin Dowd addresses how to estimate these risk measures using Excel.
related items:
Presentation: Spectral Measures of Risk - Coherence in Theory and Practise
Paper: Progress in Risk Measurement
Mahalanobis - am 2006-11-25 20:42 - Rubrik: Finance
