A minimax policy for optimal portfolio choice

Abstract

The optimal decision problem in portfolio theory considers a decision-maker (DM) who has to optimally invest a proportion of his wealth,x _i, in a risky asseti, wherex _i ⩾ 0 fori = 1, 2,…,m and ∑ ^m _i=1 x _i =l and the return$\mathop {\tilde r}\nolimits_i$ is random. Any choice of vectorx′ =(x ₁,x ₂,…,x _m ) that is feasible i.e.

$$x \geqslant 0,e'x = 1$$

((1))

wheree is anm ×l vector with each element unity and prime denotes transpose, is called a portfolio policy with a random return${\tilde z}$:

$$\tilde z = \tilde r'x,\tilde r' = (\mathop {\tilde r}\nolimits_1 ,\mathop {\tilde r}\nolimits_2 ,...\mathop {\tilde r}\nolimits_m$$

((2))

With$E(\tilde r) = \mathop {(\mu }\nolimits_1 ,\mathop \mu \nolimits_2 ,...\mathop \mu \nolimits_m )'$ and$\operatorname{cov} (\tilde r) = V(\mathop u\nolimits_{ij} ),i,j = 1,2,...,m$denoting the mean vector and the variance-covariance matrix of${\tilde r}$ respectively, the mean and variance of return${\tilde z}$ on a feasible portfolio policy are, respectively given by

$$E(\tilde z) = \mu 'x,\operatorname{var} (\tilde z) = x'Vx$$

((3))

Denoting the set of parameters byθ = (µ,V), the traditional optimal solution of the portfolio theory solves the following quadratic programming problem (QP):

$$\begin{gathered}\min h = x'Vx \hfill \\x \in R \hfill \\\end{gathered}$$

((4))

where

$$R:\left\{ {x\left| {e'x = 1 \geqslant M,x} \right. \geqslant 0} \right\}$$

((5))

M: a fixed positive number By varyingM parametrically, the whole set of optimal or efficient portfolios is determined according to Markowitz’s theory and its various extensions (Szego 1980).