  The aim of this paper is to study the optimal investment problem by using
coherent acceptability indices (CAIs) as a tool to measure the portfolio
performance. We call this problem the acceptability maximization. First, we
study the one-period (static) case, and propose a numerical algorithm that
approximates the original problem by a sequence of risk minimization problems.
The results are applied to several important CAIs, such as the gain-to-loss
ratio, the risk-adjusted return on capital and the tail-value-at-risk based
CAI. In the second part of the paper we investigate the acceptability
maximization in a discrete time dynamic setup. Using robust representations of
CAIs in terms of a family of dynamic coherent risk measures (DCRMs), we
establish an intriguing dichotomy: if the corresponding family of DCRMs is
recursive (i.e. strongly time consistent) and assuming some recursive structure
of the market model, then the acceptability maximization problem reduces to
just a one period problem and the maximal acceptability is constant across all
states and times. On the other hand, if the family of DCRMs is not recursive,
which is often the case, then the acceptability maximization problem ordinarily
is a time-inconsistent stochastic control problem, similar to the classical
mean-variance criteria. To overcome this form of time-inconsistency, we adapt
to our setup the set-valued Bellman's principle recently proposed in
\cite{KovacovaRudloff2019} applied to two particular dynamic CAIs - the dynamic
risk-adjusted return on capital and the dynamic gain-to-loss ratio. The
obtained theoretical results are illustrated via numerical examples that
include, in particular, the computation of the intermediate mean-risk efficient
frontiers.
