Advances in Risk Management

26 INCORPORATING DIVERSIFICATION INTO RISK MANAGEMENT

Define a trivial acceptable portfolioηcconsisting of $1 invested only in
riskfree capital, in other words, aN+1 vector with one as the first ele-
ment and zero in the remaining elements. This portfolio has the property
Pηc=(1+r)1>0 where 1 is aN+1 vector of ones. Given the acceptance
setAηin Definition 2.2.1, portfolio risk is defined in terms of thel 2 norm,

||x−y|| 2 equals

√∑
N
i= 0 (xi−yi)^2 ,onM. Our risk functionρ(η) maps from
the domain of portfolio holdings,M, into the non-negative real line,ρ(η):
M→R^1 +.

Definition 2.2.2 GivenAηdefined by the payoff matrixP, the risk of a

portfolioηequals

ρ(η)=inf{||η−η′|| 2 :η′∈Aη}

Observe the fundamental difference between our approach and that of
ADEH, instead of defining risk on terminal portfolio values, risk is defined
on portfolio holdings. Thus, although both measures of risk are defined by a
distancefrom an acceptance set, our concept of distance hasN+1 variables
(one for each asset) instead of only one (riskfree capital). Different objective
functions may be utilized with the important property in Definition 2.2.2
being the measurement of risk in terms of distance. Thel 2 norm is chosen
for tractability and because of its prior applications in portfolio theory.
At this stage, we state three important clarifications regarding our risk
measure. First, althoughρ(η) is Euclidean distance, it has an immediate
dollar-denominated interpretation given prices for each of the assets as dis-
cussed in section 2.5. Second, firm preferences are easily incorporated into
our risk measure. Third, the dollar-denominated value of the original port-
folio differs from its acceptable counterpart denotedη′in Definition 2.2.2.
As elaborated in section 2.4, our methodology recognizes that the firm is
not necessarily less willing to rebalance assets with higher prices. Instead,
deviations in the portfolio weights of the original portfolio are minimized
since relatively inexpensive assets such as out-of-the-money options or
futures contracts (with zero value after being market-to-market) may be
crucial to both the desirability and riskiness of a firm’s investment strategy.
Indeed, forward and swap contracts have zero initial value but potentially
large positive or negative payoffs. In contrast, the firm may be willing to alter
their holdings of expensive instruments such as Treasury bonds. Thus, our
primary objective is minimizing perturbations to the firm’s original portfo-
lioη, which is assumed to be its preferred allocation. Instead, the firm is able
to specify the cost of rebalancing each individual asset from their perspec-
tive when finding the acceptable portfolio’s solution. As seen in section 2.4,
one possibility simply has the rebalancing cost for each asset being equal
to its price. This special case minimizes the dollar-denominated amount
of rebalancing. However, such a formulation is not necessarily compati-
ble with our objective of including derivatives or other (leveraged) assets