AMIYATOSH PURNANANDAM ET AL. 35Theλparameters have interesting interpretations as each element cor-
responds to a specific regulator scenario. If the constraintPη≥0isnot
binding in scenarioiwith (Pη)i≥0, then the correspondingλiequals 0. Oth-
erwise, theoptimalλiisapositivenumberrepresentingthecostofpreventing
insolvency.
IfPη≥0, then (2.10) has an obvious solution;λ=0 andy=Pη, implying
ηis optimal. Otherwise, the general pivoting approach transforms (2.10) to
optimality. After finitely many pivots, bounded above by the number of
rows (scenarios), the vectorPη∗is non-negative. In terms of computational
complexity, a total ofMlinear equations are solved for each pivot operation.
The algorithm stops whenPη∗≥0, providing the optimal solution to (2.10).
2.4.1 Incorporating market frictions and firm preferences
In general, the objective functiongmay be defined with respect to a positive
definite matrixAas in (η∗−η)A(η∗−η). Consider a diagonal matrix of
positive elementsai:
A=
a 0
a 1
..
.
aN
representing the associated market friction (illiquidity and transaction costs)
of theithasset as well as the firm’s unwillingness to alter their position in
this asset. Largeraivalues correspond to largerpenaltiesfor altering that ele-
ment of the portfolio. Even if riskfree capital has the smallest corresponding
penalty, the addition of riskfree capital may still be sub-optimal. Indeed, a
portfolio may require a large amount of additional riskfree capital to become
acceptable, but only minor modifications to positions with largeraipenal-
ties. This issue is re-examined in the next section when pricing portfolio
insurance.
Alternatively, the price of each asset could define theaielements. In this
circumstance, the dollar-denominated amount of rebalancing is minimized
as assets with higher prices are more expensive to rebalance. However, as
alluded to earlier, inexpensive futures contracts or out-of-the-money options
often provide large future payoffs and are crucial to a firm’s investment strat-
egy, while expensive instruments such as high-coupon bonds are not. Thus,
the price of an individual security is not necessarily representative of firm
preferences towards rebalancing. Nonetheless, the potential to incorporate
prices into the solution ofη∗is apparent.
Observe that thecielements of theR(η) function in equation (2.2) are not
incorporated intoA. Indeed, solving for the optimal acceptable portfolio that
maximizesR(η) is well-beyond the scope of this paper and would require