AMIYATOSH PURNANANDAM ET AL. 312.3 NUMERICAL EXAMPLE
Consider an economy with two risky assets and riskfree capital. Uncertainty
intheeconomyiscapturedbyacointoss. Forthefirstriskyasset, thepayoffis
$4ifheadsand−$2iftails, whiletheircounterpartsare$0and$2respectively
for the second risky asset. The rate of interest is assumed to be zero (r=0)
implying risk-free capital is worth $1 at timeT.
The two risky assets are negatively correlated. For emphasis, no probabil-
ity measure is required for the occurrence of the two states as the regulator
is not concerned with their likelihood. Instead, preventing insolvency in
each scenario is the regulator’s task, which is entirely independent of the
portfolio’s expected value across the scenarios. Indeed, the second risky
asset resembles a put option on the first security. Furthermore, the market
is complete since a portfolio weight of−^13 in the first asset combined with
4
3 of risk-free capital replicates the put option. Intuitively, the put option
provides negative correlation, facilitating greater diversification, by short-
ing the first asset while indirectly providing additional riskfree capital. As
illustrated in the remainder of this example, beyond serving as an effec-
tive means to hedge risk and ensure portfolio acceptability, the derivative
reduces the amount of riskfree capital required to be held by the firm.
The space of acceptable portfolio holdings whose terminal values are
non-negative in both scenarios is characterized by:
1 η 0 + 4 η 1 + 0 η 2 ≥ 0 Heads (2.7)
1 η 0 − 2 η 1 + 2 η 2 ≥ 0 Tails (2.8)Consider the portfolioη=[1, 1, 0]consisting of one unit of riskfree capital,
one unit of the first risky asset and none of the second. The portfolioηis not
acceptable since the payoff is negative if the coin toss results in tails.
In the coherent risk measure framework,ηrequires an additional unit of
riskfree capital resulting inη∗ADEH=[2, 1, 0].
Solving for our optimal portfolioη involves minimizing the distance
betweenη=[1, 1, 0]andη∗∈Aηunder thel 2 norm using quadratic pro-
gramming (QP). The portfolioη equals [1.11, 0.78, 0.22]with details
pertaining to its solution found in the next section. MATLAB code which
solves forη∗is available from the authors.
As demonstrated above, a coherent risk measure evaluates the risk ofη
as 1 due to the negative payoff when the coin toss is tails. However, the
portfolio [1.11, 0.78, 0.22]∈Aηimplies the portfolio’s risk in our frame-
work is||η∗−η|| 2 =
√
(1. 11 −1)^2 +(0. 78 −1)^2 +(0. 22 −0)^2 = 0 .33. Thus, our
proposed risk measure evaluates the risk ofηat one third that of a coherent
risk measure. However, the rebalanced portfolio has non-negative payoffs
in both scenarios and therefore satisfies the regulator.