Advances in Risk Management

AMIYATOSH PURNANANDAM ET AL. 25

to terminal portfolio values. Thus, the portfolio weights of the underlying
assets cannot be altered to exploit the benefits of diversification. Instead,
according to the stricter version of ADEH’s acceptance set which invokes
Axiom 2.2′, ADEH focus solely on the amount of riskfree capital required
to ensure the portfolio has non-negative terminal values in the scenarios
considered relevant by the regulator. This exclusive focus on riskfree capi-
tal is overcome by our methodology which operates on a different domain.
Specifically, defineM⊂RN+^1 as the space of portfolio holdings with the
subset of acceptable portfolios denotedAη⊂M.
As in ADEH, the definition of acceptable portfolios ensures the firm can-
not become insolvent in any of the regulator’sMscenarios. However, firms
may supplement this set of scenarios to obtain additional protection against
insolvency.

Definition2.2.1 ThesetofacceptableportfolioholdingsAη⊂Mcontains

all portfolios that have non-negative outcomes,Pη≥0, in allMscenarios

evaluated by the regulator.

Clearly, the acceptance setAηdepends on the payoff matrixPwith the reg-
ulator controlling the number of scenarios (rows). Moreover, the regulator
focuses on preventing insolvency but does not considervariabilityin a firm’s
portfolio value when measuring risk. With respect to risk factors such as
market, interest rate or foreign exchange movements, these variables may
define theMscenarios. For example, the Standard Portfolio Analysis of
Risk (abbreviated SPAN) risk management system employed by most inter-
national exchanges such as the CBOT, CME, NYBOT, NYMEX and LIFFE,
investigates 16 scenarios defined by changes in price and volatility. In each
of these scenarios, the firm’s portfolio is required to be non-negative.

Proposition 2.2.1 The acceptance setAηhas the following two properties:

1. Closed under multiplication byγ≥0.


2. Convexity.

Proof: First, it must be shown that ifη∈Aη, thenγη∈Aηforγ≥0. This
property follows fromη∈Aηbeing equivalent toPη≥0 and the property
P[γη]=γPηwhich is non-negative since bothγandPηare non-negative.
Second, ifη 1 ,η 2 ∈Aη, implyingPη 1 ≥0 andPη 2 ≥0, thenγη 1 +(1−γ)η 2 ∈
Aηfor 0≤γ≤1 sinceP[γη 1 +(1−γ)η 2 ]=γPη 1 +(1−γ)Pη 2 ≥0.
Therefore, as in the ADEH framework, unless each element ofPηis
non-negative, the portfolioηis unacceptable. In this instance, an opti-
mal acceptableη is found based on itsproximitytoηas we assume firms
prefer to engage in as little portfolio rebalancing as possible given their
initial preference forη. Quadratic programming solves for the portfolioη
in section 2.4.