274 Gianni Pola
In this paper we consider the Optimal Dynamic Asset Allocation (ODAA) problem
from a Control System Theory perspective. We will show that the ODAA problem
can be reformulated as a suitable optimal control problem. Given a sequence of
target sets, which represent the portfolio specifications, an optimal portfolio allocation
strategy is synthesized by maximizing the probability of fulfilling the target sets
requirements. The proposed optimal control problem has been solved by using a
Dynamic Programming [6] approach; in particular, by using recent results on the
Stochastic Invariance Problem, established in [1, 18]. The proposed approach does
not assume any particular distribution on the stochastic random variables involved and
therefore provides an appropriate framework for non–Gaussian settings. Moreover the
model does not assume stationarity in the stochastic returns dynamics. The optimal
solution is given in a closed algorithmic form.
We applied the formalism to a case study: a 2-year trade investing in the US
market. The objective of the strategy is to beat a fixed target return at the end of
the investment horizon. This study shows markedly that an (optimal) total return
fund manager should adopt acontrarianstrategy: the optimal solution requires an
increase in risky exposure in the presence of market drawdowns and a reduction in the
bull market. Indeed the strategy is aconcavedynamic strategy, thus working pretty
well in oscillating markets. We contrast the ODAA model to aconvexstrategy: the
Constant-Proportional-Portfolio-Insurance (CPPI) model.
Preliminary results on the ODAA problem can be found in [19].
The paper is organised as follows. In Section 2 we give the formal statement of
the model and show the optimal solution. Section 3 reports a case study. Section 4
contains some final remarks.
2 The model: formal statement and optimal solution
Consider an investment universe made ofmasset-classes. Givenk∈N,definethe
vector:
wk=
[
wk( 1 )wk( 2 )···wk(m)]T
∈Rm,where the entries are the returns at timek.Let
uk=[uk( 1 )uk( 2 )...uk(m)]T∈Rmbe the portfolio allocation at timek∈N. Usually some constraints are imposed on
ukin the investment process: we assume that the portfolioukis constrained to be
in a given setUk⊂Rm. The portfolio time evolution is governed by the following
stochastic dynamical control system:
xk+ 1 =xk(^1 +uTkwk+ 1 ),k∈N, (1)where:
- xk∈X=Ris the state, representing the portfolio value at timek;