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Most of the investments in practice are carried out without certain horizons. There are many factors to drive investment to a stop. In this paper, we consider a portfolio selection policy with market-related stopping time. Particularly, we assume that the investor exits the market once his wealth reaches a given investment target or falls below a bankruptcy threshold. Our objective is to minimize the expected time when the investment target is obtained, at the same time, we guarantee the probability that bankruptcy happens is no larger than a given level. We formulate the problem as a mix integer linear programming model and make analysis of the model by using a numerical example.

Portfolio theory deals with the
question of how to find an optimal policy to invest among various assets. The mean-variance
analysis of Markowitz [

An important assumption of the previous portfolio selection model is that the investment horizon is definite. That means an investor knows with certainty the exit time at the beginning of the investment. However, most of the investments in practice are carried out without certain horizons. There are many factors, related to the market or not, which can drive the investment stop. For example, sudden huge consumption, serious illness, and retirement are market-unrelated reasons. Also, those market-related reasons may more strongly affect the investment horizon. A natural example is that the investor may exit the market once his wealth reaches an investment target, which is closely related to the market and also the investment policy itself. Because of the disparity between theory and practice, it seems sympathetic to relax the restrictive assumption that the investment horizon is preknown with certainty.

Research on this subject has been investigated in continuous
setting. Yaari [

In this paper, we consider a portfolio selection problem with endogenous stopping time in discrete framework, which has not been well discussed in literatures. Specially, we assume that the investor exits the market once his wealth hits an investment target or he is bankrupt. This assumption actually reflects most investors’ investment behavior in real life. Our objective is to minimize the expected time that the investment target is obtained, at the same time we guarantee that the probability of which bankruptcy happens is no larger than a given threshold. The investment process is represented by a multistage scenarios tree, in which the discrete stages and notes denote the decision time points and the market states, respectively.

The rest part of the paper is organized as follows. In
Section

Consider the following investment problem. We distribute the investment budget among a set of assets, and the portfolio can be adjusted at several discrete decision time points during the investment process. At the beginning of the investment, we assign a target wealth and also a bankruptcy threshold. Our objective is to obtain this target wealth and stop the investment as soon as possible. At the same time, we also need to avoid that the bankruptcy occurs before the target wealth obtained.

The problem is
based on a finite multistage scenarios tree structure. Our portfolio includes a
set of

We consider an objective related to the
achievement of performance goal and bankruptcy. The investment stops once the
goal is reached or the bankruptcy occurs,
the related stopping time is
denoted as

In this section, we will derive
the deterministic formulation of the problem (

Based on the previously given
notations on the scenarios tree, we first have the allocation of the initial investment
wealth
represented as

Also, for a self-financing process that we are
considering here, the realized
wealth will be reinvested at this decision point, which means

Therefore, we
conclude the budget constraints at scenario

We come to the formulation of the
objective function and the probability constraint. Let us consider the
investment process. There are basically three different outputs at a given scenario

Now, we define two 0-1 variables to
describe the investment story. On scenario

Parallel to

Reading
the definitions,

We consider again the indicator
variables

For the current state, we define

Combine the above definitions and review

If we replace these nonlinear constraints by a set of linear
constraints, then the problem can be hopefully formulated
as a linear programming problem,
which will benefit for the further research on solution methods and
applications. Since the indicator variables are all defined as binary 0-1 variables,
we derive the transformation

It is direct to check that for given values of

Therefore, we now
replace (

Up to now, we have almost derived out the
formulation of the model based on a series of indicator variables, including

Consider the constraints of

We
combine the constraints of

Next,
let us focus on the dynamics of

The dynamic equation holds for the following reasons.

First, suppose
the investment has been continued to the scenario

Second, if the investment has stopped, either on
the parent scenario

Now, we have derived all the constraints of
the indicator variables by (

Next, we construct an example to analyze the model and illustrate the solving process. The problem is input by an MATLAB program, and numerically solved by using Cplex software.

The
investment process is represented by a 3-stage triple tree, noted from time 1
to time 4, as showed in Figure

The scenario tree of example.

Cplex takes 0.41 second to optimize the problem. Reading the
solution file, we find that there are chances to obtain the payoff target
before the investment horizon, as clearly as in the third stage on the scenarios of

The selected solutions of example.

Time 1 | Time 2 | Time 3 | Time 4 | ||||
---|---|---|---|---|---|---|---|

Payoff target obtained | Scenario | None | None | None | |||

Solution | |||||||

Bankruptcy happens | Scenario | None | None | ||||

Solution | |||||||

For two-stage problem, there are well-known algorithms such as branch-and-bound, Lagrangian relaxation, or cutting plane methods for solving it. When we extend it into the multistage case, as we are doing now, the problem becomes much more complex. As the size of the problem increases, the existing solution methods become less efficient. We will further investigate on more applicable solution methodologies. In addition to the solution methodology, another relevant research topic is to compare the investment policies under different objectives and risk constraints.

This research was supported by Guangdong Natural Science Foundation (no. 06300496), China, National Social Science Foundation (no. 08CJY064), China, and the National Science Foundation for Distinguished Young Scholars (no. 70825002), China.