  One index satisfies the duality axiom if one agent, who is uniformly more
risk-averse than another, accepts a gamble, the latter accepts any less risky
gamble under the index. Aumann and Serrano (2008) show that only one index
defined for so-called gambles satisfies the duality and positive homogeneity
axioms. We call it a duality index. This paper extends the definition of
duality index to all outcomes including all gambles, and considers a portfolio
selection problem in a complete market, in which the agent's target is to
minimize the index of the utility of the relative investment outcome. By
linking this problem to a series of Merton's optimum consumption-like problems,
the optimal solution is explicitly derived. It is shown that if the prior
benchmark level is too high (which can be verified), then the investment risk
will be beyond any agent's risk tolerance. If the benchmark level is
reasonable, then the optimal solution will be the same as that of one of the
Merton's series problems, but with a particular value of absolute risk
aversion, which is given by an explicit algebraic equation as a part of the
optimal solution. According to our result, it is riskier to achieve the same
surplus profit in a stable market than in a less-stable market, which is
consistent with the common financial intuition.
