Advances in Risk Management

AMIYATOSH PURNANANDAM ET AL. 43

Therefore, although risk is defined in terms of thel^2 norm on portfo-
lio weights, it may be converted into the more traditional dollar-based
domain and coincides with the price of portfolio insurance (with or without
rebalancing).
As a consequence of equation (2.22), minimizing the distance in portfolio
weights betweenηand the acceptance set is equivalent to minimizing the
dollar-denominated amount of rebalancing. Therefore, the price of portfolio
insurance equals the amount of rebalancing, in dollars, required to ensure
the portfolioηbecomes acceptable.

2.5.4 Example revisited

Returning to the example in section 2.3, let the price vector equalq=
[1, 1.3, 0.9]. As discussed in section 2.3, the second asset’s price is obtained
by no arbitrage as−^13.^3 +^43 = 0 .90.
Existing specifications imply (Pη)−=[0, 1], andη=[1, 1, 0]along with
the payoff matrixPillustrates the results in Propositions 2.5.1 and 2.5.2. The
vectorλwoequals [0.0673, 0.1635], implying a price for portfolio insurance
ofICwo=qPλwowhich equals $0.45. Theλwoparameters are associated
with two restrictions; preventing negative terminal values and not reducing
positive terminal values.
The second optimization in equation (2.16) based onδ 0 andQyields
λw=[0.0579, 0.1405]. According to Proposition 2.5.2, the priceICwequals
$0.45, in accordance with Proposition 2.5.3.
However, the optimal amount of portfolio insurance to purchase is
xw=λw(Pη)−=(λw) 2 = 0 .1405, a quantity strictly less than one sincePis
of full row rank. Thus, with additional portfolio rebalancing, the dollar-
denominated reduction in the amount of portfolio insurance that is required
equalsICwo−xwICw=(1− 0 .1405)× 0. 45 =$0.39.

2.6 CONCLUSION

A risk measure defined on the space of portfolio holdings rather than ter-
minal values is proposed which enables diversification to reduce portfolio
risk. Consequently, derivative and insurance contracts have important roles
in risk management. Through portfolio rebalancing, our risk measure offers
firmsgreaterflexibilitythancoherentriskmeasureswhencomplyingwithan
external regulator. Indeed, our approach allows every asset in the portfolio,
including riskfree capital, to be adjusted. Thus, as in the existing literature,
risk is defined as thedistanceto an acceptance set. However, to incorporate
diversification, the concept of distance is extended to include the risky assets
as well as riskfree capital.