PHYSICAL REVIEW E 96, 022305 (2017) Effects of correlations and fees in random multiplicative environments: Implications for portfolio management Ofer Alper, Anelia Somekh-Baruch, Oz Pirvandy, Malka Schaps, and Gur Yaari Bar-Ilan University, Ramat Gan 5290002, Israel (Received 28 May 2016; revised manuscript received 22 April 2017; published 7 August 2017) Geometric Brownian motion (GBM) is frequently used to model price dynamics of ﬁnancial assets, and a weighted average of multiple GBMs is commonly used to model a ﬁnancial portfolio. Diversiﬁed portfolios can lead to an increased exponential growth compared to a single asset by effectively reducing the effective noise. The sum of GBM processes is no longer a log-normal process and has a complex statistical properties. The nonergodicity of the weighted average process results in constant degradation of the exponential growth from the ensemble average toward the time average. One way to stay closer to the ensemble average is to maintain a balanced portfolio: keep the relative weights of the different assets constant over time. To keep these proportions constant, whenever assets values change, it is necessary to rebalance their relative weights, exposing this strategy to fees (transaction costs). Two strategies that were suggested in the past for cases that involve fees are rebalance the portfolio periodically and rebalance it in a partial way. In this paper, we study these two strategies in the presence of correlations and fees. We show that using periodic and partial rebalance strategies, it is possible to maintain a steady exponential growth while minimizing the losses due to fees. We also demonstrate how these redistribution strategies perform in a phenomenal way on real-world market data, despite the fact that not all assumptions of the model hold in these real-world systems. Our results have important implications for stochastic dynamics in general and to portfolio management in particular, as we show that there is a superior alternative to the common buy-and-hold strategy, even in the presence of correlations and fees. DOI: 10.1103/PhysRevE.96.022305 I. INTRODUCTION Multiplicative stochastic processes are commonly used to model assets prices as these are strongly affected by relative ﬂuctuations. The most popular framework used in ﬁnance to model stock prices is GBM. It is used in the classic Black- Scholes model [1] as well as in many other models [2,3]. It was noted that in random multiplicative environments the median represents the typical path much better than the mean. Hence, it is not recommended to simply maximize the expectation of the wealth [4] but rather to maximize the expectation of the log of the wealth [5]. In fact, utilizing the log space of wealth can be traced back to Daniel Bernoulli [6]. Inspired by the St. Petersburg paradox, he invented the log utility function in 1708 [7], which reﬂects a declining marginal utility of the wealth, and later also led to the concept of risk aversion [8]. In gambling and repeated investments, Kelly was the ﬁrst to suggest the usage of logarithmic utility [9,10]. Finite sum of multiplicative stochastic processes is not ergodic [11], which implies that the time average is not the same as the ensemble average. There is a growing interest in stochastic ergodicity breaking in a wide range of physical systems [12,13]. The nonergodicity of this process implies that systems that are naïvely expected to ﬂourish (arithmetic mean larger than 1), in reality may be doomed to extinction (geometric mean lower than 1). A possible workaround for this unfortunate outcome can be achieved by diversiﬁcation and cooperation [14,15]. The dominance of the geometric mean for the typical path in multiplicative processes was demonstrated by proving that the price trajectory behaves almost surely as [16] lim S (t )(t →∞) = ⎧ ⎪ ⎨ ⎪ ⎩ ∞, E{ln[S (1)]} > 0 oscillation, E{ln[S (1)]}= 0 0, E{ln[S (1)]} < 0. (1) Also, for assets with price dynamics that follow GBM, the price will be nowhere near its expected value for large times [17]. The noise ﬂuctuations in GBM process have a negative effect on the growth, as can be seen in the Ito correction term [18]. This correction term is of σ 2 magnitude and reﬂects the difference between the arithmetic mean and the geometric mean. A useful way to reduce the noise is by diversiﬁcation in which N GBM processes are summed together. The sum of GBM processes is not ergodic, thus there is a difference between the time average, lim t →∞, and the ensemble average, lim N →∞, where the ensemble average > time average. In reality, the interesting dynamics is when N and t are ﬁnite. One of the results derived from the nonergodicity of the process is that for small t the growth is close to the ensemble average but decreases over time toward the time average as described in Ref. [11]. By inﬁnite diversiﬁcation (i.e., lim N →∞), the stochastic noise is removed and the ensemble average is achieved. One way to recover from the nonergodicity without inﬁnite diversiﬁcation and to gain a steady growth over time is by keeping constant weight for each GBM in the weighted sum as described in Ref. [19]. The constant rebalanced portfolios are also called balanced portfolios. Balanced portfolios should not be confused with buy-and-hold (passive) portfolios in which the number of shares held in each asset are kept ﬁxed, hence when the wealth changes, wealth fractions also change. The wealth of a balanced portfolio composed of GBM assets is a log-normal process. It was shown that balanced portfolios, in the absence of transaction costs, have steady expected growth [17]. In practice, the constant rebalance required for a balanced portfolio exposes it to transaction costs. The effects of transaction costs on a Kelly portfolio with one risky asset and one risk-free asset with zero growth were studied in Ref. [20]. It was shown in Ref. [20] that there is an optimal rebalance 2470-0045/2017/96(2)/022305(12) 022305-1 ©2017 American Physical Society