24 INCORPORATING DIVERSIFICATION INTO RISK MANAGEMENTThe organization of this chapter is as follows. Section 2.2 details the
properties of our proposed risk measure, while a simple example which
illustrates our approach is given in section 2.3. Section 2.4 focuses on the
implementation of our risk measure and demonstrates that coherent risk
measures are contained in our framework. Section 2.5 considers the pricing
of portfolio insurance while section 2.6 concludes.
2.2 RISK MEASURE WITH DIVERSIFICATION
Consider the time horizon [0,T] and a finite numberNof risky assets
denotedxifori=0, 1, 2,...,Nwithx 0 representing riskfree capital. LetP
denote aM×(N+1) payoff matrix withMrows indexed byj=1, 2,...,M
corresponding to the regulator’s set of scenarios andN+1 columns corre-
sponding to the available assets. Elements ofPare individual asset payoffs
in a given scenario.
P=
(1+r) P 1 (ω 1 ) ... PN(ω 1 )
(1+r) P 1 (ω 2 ) ... PN(ω 2 )
..
...
...
.
(1+r) P 1 (ωM) ... PN(ωM)
A vector of portfolio holdingsη=[η 0 ,η 1 ,...,ηN]represents the number
of units, not dollar amounts or fractions of a portfolio, invested in the various
assets. Portfolio valuesPηin theMscenarios determine whether a portfolio
complies with the demands of an external regulator.
Coherent risk measures evaluate a portfolio’s risk according to its value
in the worst possible scenario or under the probability measure that pro-
duces the largest negative outcome. Mathematically, these risk measures
are defined in terms of terminal portfolio values,X=Pη,as
ρ(X)=max
jEPj[−X|Pj∈P]
1 +r(2.1)withPrepresenting a set of scenarios andrthe riskfree rate of interest. In
our framework,EPj[−X] is replaced byPη−j, thejthrow ofPη−=−min{0,
Pη} as each row ofPηcorresponds to a regulator’s scenario. Note that each
scenario represents a probability measure. However, the expected value
of the portfolio across multiple scenarios is not computed by the regulator.
Instead, the worst outcome across the scenarios defines the risk of a coherent
risk measure.
It is important to emphasize that coherent risk measures do not account
for diversification. Although ADEH have a subadditivity axiom that paral-
lels a property implied by our risk measure, their definition of risk applies