A reasonable way to describe assets price variations (on a given time-scale) is to assume them to be multiplicative random walks with log-normal step. This comes from the assumption that growth rates of prices are more significant than their absolute variations. So, we describe the price of a financial assets as a time-dependent multiplicative random process. We introduce a set of N Gaussian random variables x_{i}(t) depending on a time parameter t. By this set, we define N independent multiplicative Gaussian random walks, whose assigned discrete time evolution is given by

p_{i}(t+1) = e^{xi}(t)p_{i}(t) —– (1)

for i = 1,…,N, where each x_{i}(t) is not correlated in time. To optimize an investment, one can choose different risk-return strategies. Here, by optimization we will mean the maximization of the typical capital growth rate of a portfolio. A capital W(t), invested into different financial assets who behave as multiplicative random walks, grows almost certainly at an exponential rate ⟨ln W (t+1)/W (t)⟩, where one must average over the distribution of the single multiplicative step. We assume that an investment is diversified according to the Kelly’s optimum investment fraction, in order to maximize the typical capital growth rate over N assets with identical average return α = ⟨e^{xi}(t)⟩ − 1 and squared volatility ∆ = ⟨e^{2xi}(t)⟩ − ⟨e^{xi}(t)⟩^{2}. It should be noted that Kelly capital growth criterion, which maximizes the expected log of final wealth, provides the strategy that maximizes long run wealth growth asymptotically for repeated investments over time. However, one drawback is found in its very risky behavior due to the log’s essentially zero risk aversion; consequently it tends to suggest large concentrated investments or bets that can lead to high volatility in the short-term. Many investors, hedge funds, and sports bettors use the criterion and its seminal application is to a long sequence of favorable investment situations. On each asset, the investor will allocate a fraction f_{i} of his capital, according to the return expected from that asset. The time evolution of the total capital is ruled by the following multiplicative process

W(t+1) = [1 + ∑_{i=1}^{N}f_{i}(e^{xi}(t) -1)] W(t) —– (2)

First, we consider the case of an unlimited investment, i.e. we put no restriction tothe value of ∑_{i=1}^{N}f_{i}. The typical growth rate

V_{typ} = ⟨ln[1+ ∑_{i=1}^{N}f_{i}(e^{xi} -1)]⟩ —– (3)

of the investor’s capital can be calculated through the following 2^{nd}-order expansion in e^{xi} -1, if we assume that fluctuations of prices are small and uncorrelated, that seems to be quite reasonable

V_{typ} ≅ ∑_{i=1}^{N}f_{i}(⟨e^{xi}⟩ – 1) – f_{i}^{2}/2(⟨e^{2xi}⟩ – 2⟨e^{xi}⟩ + 1 —– (4)

By solving d/df(V_{typ} = 0), it easy to show that the optimal value for f_{i} is f_{i}^{opt} (α, Δ) = α / (α^{2} + Δ) ∀ i. We assume that the investor has little ignorance about the real value of α, that we represent by a Gaussian fluctuation around the real value of α. In the investor’s mind, each asset is different, because of this fluctuation α_{i} = α + ε_{i}. The ε_{i} are drawn from the same distribution, with ⟨ε_{i}⟩ = 0 as errors are normally distributed around the real value. We suppose that the investor makes an effort E to investigate and get information about the statistical parameters of the N assets upon which he will spread his capital. So, his ignorance (i.e. the width of the distribution of the ε_{i}) about the real value of α_{i} will be a decreasing function of the effort “per asset” E ; more, we suppose that an even infinite effort will not make N this ignorance vanish. In order to plug these assumptions in the model, we write the width of the distribution of ε as

⟨ε^{2}_{i}⟩ = D_{0} + (N/E)^{γ} —– (5)

with γ > 0. As one can see, the greater is E, the more exact is the perception, and better is the investment. D_{0} is the asymptotic ignorance. All the invested fraction f^{opt} (α_{i}, Δ) will be different, according to the investor’s perception. Assuming that the ε_{i} are small, we expand all f_{i}(α + ε_{i}) in equation 4 up to the 2^{nd} order in ε_{i}, and after averaging over the distribution of ε_{i}, we obtain the mean value of the typical capital growth rate for an investor who provides a given effort E:

V_{typ}^{–} = N[A − (D_{0} + (N/E)^{γ} )B] —– (6)

where

A = (α (3Δ – α^{2}))/(α^{2} + Δ)^{3} B = -(α^{2} – Δ)^{2}/2(α^{2} + Δ)^{3} —– (7)

We are now able to find the optimal number of assets to be included in the portfolio (i.e., for which the investment is more advantageous, taken into account the effort provided to get information), by solving d/dNV_{typ}^{–} = 0, it is easy to see that the number of optimal assets is given by

N^{opt}(E) = E {[(A – D_{0}]/(1 + γ)B}^{1/γ} —– (8)

that is an increasing function of the effort E. If the investor has no limit in the total capital fraction invested in the portfolio (so that it can be greater than 1, i.e. the investor can invest more money than he has, borrowing it from an external source), the capital can take negative values, if the assets included in the portfolio encounter a simultaneous negative step. So, if the total investment fraction is greater than 1, we should take into account also the cost of refunding loss to the bank, to predict the typical growth rate of the capital.