Advances in Risk Management

AMIYATOSH PURNANANDAM ET AL. 33

To clarify, portfolio theory selects portfolio weights to exploit diversifica-
tionbeforechoosing the desired amount of riskfree capital. These decisions
are independent and sequential since the risky portfolio is assumed to
already be fully diversified. However, this assumption is not present in our
methodology. As demonstrated in the next section, we recognize that a fully
diversified portfolio results in our risk measure being identical to that of
ADEH. Furthermore, to minimize the assumptions and structure imposed
on firm preferences, our risk measure’s objective is to perturb the firm’s
original portfolio by the least amount possible while complying with the
regulator.
Section 2.4 also reveals that the firm may ignore all the flexibility offered
by our approach and preserve their original risky asset allocation.As alluded
to earlier, we provide the firm withN+1 degrees of freedom to satisfy
the regulator in contrast to ADEH who only allow the portfolio weight of
riskfree capital to be manipulated. Thus, fixing the original positions in the
risky assets or focusing one’s attention on the division between the risky
portfolio and riskfree capital eliminates any possibility of diversification
since portfolio theory requires an optimal solution to be expressed in terms
of portfolio weights.
Finding more general solutions forη∗that incorporate market frictions
into the rebalancing decision is addressed in section 2.4. In our previous
example, the positive portfolio payoff in the heads scenario was reduced.
Section 2.5 computes the value of a portfolio insurance contract which elimi-
nates negative terminal values without reducing their positive counterparts.
To summarize, this section offers an illustration of how firms may comply
with the demands of a regulator while holding less riskfree capital. Indeed,
regulators may adopt our risk measure without compromising their original
role of preventing insolvency in each scenario.

2.4 IMPLEMENTATION

IfPηhas any negative elements, then the regulator deems the portfolio to be
unacceptable. This section is concerned with implementing our risk measure
by solving for the portfolioη∗∈Aηsuch thatPη∗≥0 andη∗is “as close as
possible” to the firm’s original portfolioη.

Definition 2.4.1 Allowinggto represent thel 2 norm, the portfolioη∗∈Aη

is the solution to the optimization problem:

minη∗∈RN+ 1 g(η∗−η) (2.9)

subject to Pη∗≥ 0

Equation (2.9) solves for the minimum amount of portfolio rebalancing,
which is not a dollar-denominated quantity. Indeed, the properties of our
risk measure described in Proposition 2.2.2 apply to portfolio weights.