Products of random matrices and investment strategies

Abstract

A simple stochastic model of investment based on communication theory is introduced and analyzed in detail. We solve it exactly in a simple case and we use a weak disorder expansion to deal with the small fluctuations of the capital between two consecutive trading periods. Some possible generalizations are also discussed. ͓S1063-651X͑96͒52011-8͔PACS number͑s͒: 02.50. Le, 05.20.Ϫy Let us consider an investor, whose aim is to increase a given capital Z by investing it in several stocks i (iϭ1, . . . ,M) with a given strategy. The total amount of capital is the sum Zϭ͚ iϭ1 M Z(i)ϩZ (0) where Z(i) is the fraction of the capital invested in the stock i and Z(0) is the capital left at the bank. The capital Z(i) is multiplied, at each trading period, by a factor a, a being a stochastic variable whose distribution is, in general, unknown. For our simple model we assume uncorrelated distributions; however, in reality, it is well known that such data can be correlated in time. We believe the following analysis can be readily generalized to the correlated case as well.The investment strategy is the following: at each ''time'' n (nϭ1, . . . ,N), one takes a fraction i ͓0,1͔ of the capital Z(i) and transfers it to the cash bank; at the same time, a fraction /M of the bank capital is taken from the bank itself and then added to Z(i). We can then write our system aswhere, as above defined, we suppose that a i are independent random variables with common distribution given by (a).The system is completely specified by a (M ϩ1)ϫ(M ϩ1) random transfer matrix A n since we can write Z ជ nϩ1 ϭA n Z ជ n . Note that the system is normalized: if a i ϭ1 for all i, i.e., if there is no gain or loss at each time step, then the total capital is conserved: Z nϩ1 ϭZ n . The first equation implies that after a trading period the amount deposited at the bank is the sum of two contributions: the part which has not been transferred and the part coming from Z(i). On the other hand, for each stock i, we have that Z(i) is incremented or diminished depending on the value of the stochastic variable a i . Note that the choice of ͑1͒ is not the only one possible: one may define Z nϩ1 (i)ϭ/M Z n (0)ϩa i (1Ϫ i )Z n (i), e.g., we may decide not to bet the fraction of the capital transferred from the bank. This choice, however, has the disadvantage of rendering calculations much harder and leaving the qualitative behavior unchanged. The goal is to optimize the capital gain by varying the set ͕͖ϭ͕, 1 , . . . , M ͖ ͓that is to have the largest Zϭ ͚ iϭ0 M Z(i) after N trading periods, in the limit N→ϱ͔.How is it possible to decide the best strategy once we know (a), and with fixed M ? We find that a special case of this problem can be mapped to a classical one in communication theory. Indeed in the 1950s Kelly considered this type optimization, albeit using information theory ͓1͔. His model is based on the concept of the so-called rate of transmission in a communication channel, as introduced by Shannon ͓2͔. Suppose one considers a channel and uses it to tra...

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