Advances in Risk Management

AMIYATOSH PURNANANDAM ET AL. 27

capableofofferingnegativecorrelation. Irrespectiveofthiscomplication, the
dollar-denominated risk measure implied by Definition 2.2.2 is presented in
subsection 2.5.3.
In section 2.4, an enhanced methodology which accounts for firm prefer-
ences also becomes explicit, and addresses the second issue. This extension
is identical to the incorporation of market frictions such as transaction costs
and illiquidity into portfolio rebalancing decisions. The third observation
also applies to the ADEH risk measure as the addition of riskfree capital
increases the dollar-denominated value of the original portfolio. Indeed, no
existing risk management system reduces portfolio risk while preserving its
original value.
Ifηalready comprises an acceptable portfolio, then its associated risk
equals zero. For example, the portfolioηchas zero risk,ρ(ηc)=0. Other-
wise, portfolio risk is determined by the amount of rebalancing a portfolio
requires to become acceptable. This illustrates a major advantage of our
risk measure. A firm may rebalance their portfolio by purchasing derivative
instruments, insurance contracts, or simply reducing their exposure to cer-
tain risky assets. In summary, portfolio rebalancing may include, but is not
limited to, increasing the amount of riskfree capital.
As a final observation, generalized scenarios considered in ADEH are
also addressed in our methodology. A generalized scenario is defined as
a combination of multiple scenarios aggregated by a specified probability
measure. For example, one element of the SPAN procedure considers a
30 percent chance of an extreme scenario in conjunction with a 70 percent
chance of another base scenario. In our context, the {0.3, 0.7} probability
forms the generalized scenario:

0. 3 ×Payoff vector in extreme scenario

+ 0. 7 ×Payoff vector in the base scenario

=Payoff vector of generalized scenario

which is no different than any other row of thePmatrix. However, whether
or not the scenarios underlying the generalized scenario are themselves
included as distinct rows ofPis immaterial to our analysis. For example,
the extreme scenario in SPAN is not evaluated as an individual scenario.

2.2.1 Properties of risk measure

The next proposition summarizes the properties of our risk measure. Inter-
estingly, all but one of ADEH’s coherence axioms are preserved. However,
removal of the translation invariance axiom results in an important gener-
alization by eliminating the strict dependence on riskfree capital to reduce
risk. To clarify, the operationsη 1 ±η 2 are applied componentwise to signify
operations on two vectors representing portfolio holdings.