Classes of preferences of portfolio investors for multi-period case and their asymptotic properties

В работе излагаются особенности влияния длины инвестиционного горизонта на выбор инвестором оптимального для себя портфеля. Приводятся условия, при которых неэффективные портфели для однопериодного случая становятся эффективными с увеличением инвестиционного горизонта. Сравниваются два различных подхода к описанию предпочтений инвестора при выборе портфеля. Один из подходов основывается на кривых безразличия, а другой – на критерии "допустимых потерь". The methods of the definition of preferences of portfolio investors for the multi-period investment horizon are considered and the dependence of the investor behavior on the horizon length is studied. It is assumed that the capitalization share of each portfolio security doesn't vary in time. Hence the portfolio restructuring on each step of the investment process is necessary to allocate the whole portfolio value between the component securities in a proportion chosen by investor. It is supposed that the restructuring transaction costs are equal to zero. In portfolio theory, different approaches are used. In this paper, three of them are considered. The first one involves the definition of the effective portfolio set, the second one involves the concept of the indifference curves and the third one involves the drawdown criteria. To illustrate all these concepts, the example of the simple portfolio is applied, which consists only of two securities #1 and #2 with m 1 >m 2 , s 1 >s 2 , where m – the expected relative return (1 plus the period yield) of a security and s 2 – its variation. The relative returns of each security for different periods are supposed to be stochastically mutual independent and to have equal lognormal probability distributions. It is worth to note that the supposition about lognormality of distributions is to some extent excessive when solving the optimal portfolio problem for large investment horizon. It is because of validity of central limit theorem of probability theory in accordance with which under wide conditions the production of independent random values has asymptotic lognormal distribution. Strictly speaking, this consideration is of validity to less extent for the first statement below. The following properties can be formulated: 1. If () 1 2 2 1 2 1 2 1 2 s s s m m m s s s < < − − ρ where ρ is a correlation coefficient between two security's relative returns, then (1) there exist non-effective one-period portfolios and (2) all these portfolios become effective when T is sufficiently large. That is, there exist portfolios …