The guaranteed allocation of financial investments under uncertainty

In this paper the problem of deposit allocation between the assets is considered. The sum of the individual deposit between the three assets (unrisky and two risky) increased within a year one can offer in the form
$$f(x, y) = r(1 − x_1 − x_2) + x_1y_1 + x_2y_2,$$
where r – profitability of unrisky assets; $y_i$ – profitability of risky assets i; action $x = (x_1, x_2)$ and $x_i (i = 1, 2)$ – part of investing funds in assets i.
The profit of the decision maker is defined both by the plan of diversification $(1−x_1 −x_2, x_1, x_2)$ and by uncertaintes (value of profitability of risky assets) $y = (y_1, y_2) ∈ Y$ . It should be noted that between profitabilities of risky assets some kind of dependence may exist. In this case
$$Y = \{y = (y_1, y_2)|y_1 ∈ [a, b], ψ_1(y_1) \leqslant y_2 \leqslant ψ_2(y_1)\}.$$
The profitability of portfolio is defined by the sum $f(x, y)$.
It is demanded for the depositor to define parts of one ruble $x_i (i = 1, 2)$ under which final cash $f(x, y)$ will be as much as possible.
So one can suppose the mathematical model on deposit diversification between three assets as ordered the three
$$\langle X, Y, f(x, y)\rangle,$$
where criterion$ f(x, y)$ is defined over. Set $X$ actions $x$ of the decision maker is
$$X = \{x = (x_1, x_2)|x_1 + x_2 \leqslant 1 ∧ x_i \geqslant 0(i = 1, 2)\} ⊂ \mathbb{R}^2.$$
So the decision maker does not know the real profitability of risky assets, and has to look at some «corridor change» for them. Any characteristics of probability are absent.
The depositor wants to have the greatest possible profit or possible less risks. To estimate risk we use Savage functions of regret (from principle of minimax regret).
In this paper a guaranteed on outcome solution and a guaranteed on risk solution to this problem is given. This solution is based on the principle of guaranteed result and function of regret used by Savage.

Key words: uncertainty, outcome, risk, deposit diversification, guaranteed result.