6 On the Minimization of Operational Risks

We give a risk-minimizing formula for government investments taking into account the zero intelligence law for financial markets. A problem of reducing operational risks has the specific feature that the enterprises A 1 , A 2 ,. .. , A s in which the government is willing to invest are specified in advance, possibly for political reasons, on the basis of their profitability, or in accordance with general economic strategies. Here s is sufficiently large. Moreover, the government orders the enterprises by priority: for each i, the enterprise A i is more promising than the enterprise A i−1. This means that one should not buy more stock (bonds) of the enterprise A i−1 than of A i. In other respects, one should distribute the money over these enterprises (buy stock) so as to minimize the operational risks. Governmental structures indicate the amounts of money to be invested in the highest priority enterprise A s and the least priority enterprise A 1 , and also set a budget restraint. When solving this problem, we shall adhere to the concept of zero intelligence in financial markets. This concept arose in a study of the London Stock Exchange, and since then the literature discussing the phenomenon has been steadily growing. However, the zero intelligence phenomenon is quite natural from our viewpoint [1]. The effect is that traders use stock exchange data to form their portfolios at random (" without resorting to intelligence "). However, according to Kolmogorov, randomness is none other than large complexity. For example, it does not matter if we bet on heads or tails randomly or construct a long algorithm and stick to it: in any case, the coin lands on heads (and, accordingly, tails) approximately half of the times. Let us continue the analogy. We toss a coin S times, where S is large, and repeat the series of S tossings N times. All possible outcomes, including " all heads " or " alternating heads and tails, " are equiprobable (and have a tiny probability). But a majority of outcomes have approximately equal numbers of heads and tails. In particular, this means that if we play heads and tails and bet on heads and tails half of the times each, then the risk (and also the payoff) is minimal. Consequently, returning to our problem, if we find the largest " cluster " of outcomes that are equiprobable by the …