Stocks and Fundamentalists’ Behavior

noise-trader

Let us consider a simple stock market with the following characteristics. A large amount of stock is traded. In the market, there are three typical groups of traders with different strategies: fundamentalists, chartists, and noise traders. Traders can invest either in money or in stock. Since the model is designed to describe stock price movements over short periods, such as one day, the dividend from stock and the interest rate for the risk-free asset will be omitted for simplicity. Traders are myopic and bent on maximizing utility. Their utility depends on the price change they expect, and on their excess demand for stock rather than simply their demand. Their excess demand is derived from utility maximization.

Let Ytf be the amount of money that a fundamentalist holds at a time t and Xtf be the number of shares purchased by a fundamentalist at time t. Let pt be the price per share at time t. The fundamentalist’s budget constrain is given by

Ytf + ptXtf = Yt-1f + ptXt-1f —– (1)

or equivalently

ytf + ptxtf = 0 —– (2)

where

ytf = Ytf – Yt-1f

denotes the fundamentalist’s excess demand for money, and

xtf = Xtf – Xt-1f

his excess demand for stock. Suppose that the fundamentalist’s preferences are represented by the utility function,

u = α(ytf + pt+1fxtf + βxtf – (1 + βxtf) log (1 + βxtf) —– (3)

where pt+1f denotes the fundamentalist’s expectation in period t about the price in the following period t + 1. The parameters α and β are assumed to be positive. Inserting (2) into (3) the fundamentalist’s utility maximization problem becomes:

maxxtf  u =  α(pt+1– pt)xtf  βxtf – (1 + βxtf) log (1 + βxtf) —– (4)

The utility function u satisfies the standard properties: u′ (|xtf|) > 0, u′′(|xtf|) < 0 ∀ |xf|t ≤ |xf*|, where |xf*| denotes the absolute value of xf producing a maximum utility. Thus, the utility function is strictly concave. It depends on the price change expected by fundamentalists (pt+1– pt) as well as fundamentalist’s excess demand for stock xtf. The first part α(pt+1– pt)xtf implies that a rise in the expected price change increases his utility. The remaining part expresses his attitude toward risk. Even if the expected price change is positive, he does not want to invest his total wealth in the stock, and vice versa. In this sense, fundamentalists are risk averse. β is the parameter that sets the lower limitation on excess demand. All excess demand for stock derived from the utility maximization is limited to -1/β. When the expected price change (pt+1– pt) is positive, the maximum value of the utility function is also positive. This means that fundamentalists try to buy stock. By analogy, when the expected price change (pt+1– pt) is negative, the maximum value of the utility function is negative, which means that they try to sell. The utility maximization problem (4) is solved for the fundamentalist’s excess demand,

xtf = 1/β(exp(α(pt+1– pt)/β) – 1) —– (5)

Excess demand increases as the expected price change (pt+1– pt) increases. It should be noticed that the optimal value of excess supply is limited to -1/β, while the optimal value of excess demand is not restricted. Since there is little loss of generality in fixing the parameter β at unity, below, we will assume β to be constant and equal to 1. Then let us think of the fundamentalist’s expectation formation. We assume that he form his price expectation according to a simple adaptive scheme:

pt+1f = p+ ν(p* – pt) —– (6)

We see from Equation (6) that fundamentalists believe that the price moves towards the fundamental price p* by factor ν. To sum up fundamentalists’ behavior: if the price pt is below their expected price, they will try to buy stock, because they consider the stock to be undervalued. On the contrary, if the price is above the expected value, they will try to sell, because they consider the stock to be overvalued.

What Drives Investment? Or How Responsible is Kelly’s Optimum Investment Fraction?

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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 xi(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

pi(t+1) = exi(t)pi(t) —– (1)

for i = 1,…,N, where each xi(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 = ⟨exi(t)⟩ − 1 and squared volatility ∆ = ⟨e2xi(t)⟩ − ⟨exi(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 fi 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=1Nfi(exi(t) -1)] W(t) —– (2)

First, we consider the case of an unlimited investment, i.e. we put no restriction tothe value of ∑i=1Nfi. The typical growth rate

Vtyp = ⟨ln[1+  ∑i=1Nfi(exi -1)]⟩ —– (3)

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

Vtyp ≅ ∑i=1Nfi(⟨exi⟩ – 1) – fi2/2(⟨e2xi⟩ – 2⟨exi⟩ + 1 —– (4)

By solving d/df(Vtyp = 0), it easy to show that the optimal value for fi is fiopt (α, Δ) = α / (α2 + 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 αi = α + εi. The εi are drawn from the same distribution, with ⟨εi⟩ = 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 εi) about the real value of αi 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

⟨ε2i⟩ = D0 + (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. D0 is the asymptotic ignorance. All the invested fraction fopt (αi, Δ) will be different, according to the investor’s perception. Assuming that the εi are small, we expand all fi(α + εi) in equation 4 up to the 2nd order in εi, and after averaging over the distribution of εi, we obtain the mean value of the typical capital growth rate for an investor who provides a given effort E:

Vtyp = N[A − (D0 + (N/E)γ )B] —– (6)

where

A = (α (3Δ – α2))/(α2 + Δ)3 B = -(α2 – Δ)2/2(α2 + 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/dNVtyp = 0, it is easy to see that the number of optimal assets is given by

Nopt(E) = E {[(A – D0]/(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.