Quality & Quantity (2006) 40:457–468 © Springer 2006
Portfolio Selection under Maximum Minimum
Faculty of Business, Ningbo University, Ningbo 315211, Zhejiang, P.R.China;
of Statistics, Renmin University of China, Beijing 100872, P.R.China;
Abstract. In this paper, we studied the problem of risky portfolio selection under uncertainty.
Different from risk-return analytical methodology, we formulated a model under maximum
minimal criterion of uncertain decision-making theory. If the investor had no any distribu-
tion information of the returns and (s)he knew the variation scopes of the returns by his/her
knowledge of the market information or experts’ evaluations of the alternative risky assets,
then we showed that the optimal portfolio strategy of the model under maximal minimal
criterion could be obtained by solving linear programming. If the returns were known to be
normal distributed, the investor’s optimal portfolio strategy could be obtained by solving a
nonlinear programming. The paper also provided an algorithm to solve this programming.
At last, the paper compared this model with Markowitz’s mean-varience (M-V) model and
Young’s minmax model, and pointed out the distinctions and similarities between our model
and the other two.
Key words: Decision-making, maximum minimum criterion, portfolio selection, uncer-
tainty, maximum loss risk
There are widespread ﬁelds in practice involving optimal portfolio selec-
tion. Prominent examples include (1) asset allocation for pension plans and
insurance companies; (2) social resources allocation and management of
welfare programs; (3) security selection for stock and bond portfolio man-
agers; and (4) risk management for large public corporations. In these and
other situations, the decision maker has to consider uncertainty of out-
comes and balance between bearing risk and seeking return according to
his preference. Portfolio selection theory deals with how to form a satisfy-
ing portfolio among numerous assets.
It has been suggested that modern portfolio theory was introduced
by Markowitz (1952). In the theory of Markowitz, the return of any
Supported in part by Program for NCET, in part by the Key Project of Chinese Min-
istry of Education 104053.