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Document Details :
Title: Marginal Decomposition of Risk Measures
Author(s): VENTER, Gary G. , MAJOR, John A. , KREPS, Rodney E.
Journal: ASTIN Bulletin
Volume: 36 Issue: 2 Date: 2006
The marginal approach to risk and return analysis compares the marginal return
from a business decision to the marginal risk imposed. Allocation distributes
the total company risk to business units and compares the profit/risk ratio of
the units. These approaches coincide when the allocation actually assigns the
marginal risk to each business unit, i.e., when the marginal impacts add up to
the total risk measure. This is possible for one class of risk measures (scalable
measures) under the assumption of homogeneous growth and by a subclass
(transformed probability measures) otherwise. For homogeneous growth, the
allocation of scalable measures can be accomplished by the directional derivative.
The first well known additive marginal allocations were the Myers-Read method
from Myers and Read (2001) and co-Tail Value at Risk, discussed in Tasche (2000).
Now we see that there are many others, which allows the choice of risk measure
to be based on economic meaning rather than the availability of an allocation
method. We prefer the term “decomposition” to “allocation” here because of
the use of the method of co-measures, which quantifies the component composition
of a risk measure rather than allocating it proportionally to something.
Risk adjusted profitability calculations that do not rely on capital allocation
still may involve decomposition of risk measures. Such a case is discussed.
Calculation issues for directional derivatives are also explored.