John Doyle and Tielman Van Vleck
The prediction of stock prices is an imperfect art. Prediction of human behavior will never be an exact science. Short of finding a way to predict where the price of a stock will go, the best way (other than insider trading) to profit from a stock is to predict it's current value and hope that others will realize this value at a later date bringing the value of the shares to what you had estimated. Estimating what people will pay for the stock in the future is extremely valuable information.
The value of a stock is the sum of all future dividends from the stock. The price individuals are willing to pay is a function of the present value of a stock plus the risk of the investment. Risk refers to the stock's volatility. Since risk is a disincentive to purchasing a stock, investors require a higher rate of return when making riskier investments. The goal of this project is to develop a model which will estimate the risk of an investment using chaos theories.
The risk of a stock can be divided into two parts: systematic risk and unsystematic risk. Systematic risk comes from national disasters and changes in our economy which affect the entire country such as going to war or changes in the interest rate. These risks can not be avoided when buying American stocks. Unsystematic risks are those risks that affect only certain firms. Unsystematic risk may be nearly eliminated by dividing one's total investment between several firms. In finance circles, the systematic risk is reported as the beta coefficient (b) of the stock. Hundreds of unfortunate souls have dedicated their lives to estimating betas for various firms.
The Security Market Line is a financial model showing a positive linear relationship between the expected return on investments and their risk. This line traces back to the risk-free rate which is considered to be that of treasury-bills. If a stock lies above this line than it is less risky than investors believe and is therefore overvalued. Conversely, if it is found below the line the stock is undervalued because investors feel that the stock is less risky than it really is. An accurate estimate of risk would give one the ability to identify under valued stocks which could be bought for less than their true value.
While fractals can never be expected to predict a certain outcome, they can create graphs which look a great deal like actual plots of stock prices. Though the outcome will be quite random, the volatility depicted could be a useful estimator of risk. A model must be constructed to predicts possible trends for each stock. In order to estimate the risk of individual stocks, the model of possible outcomes will be run many times. The risk of the stock will be derived from the deviation in outcomes provided by this model.
There are two chaos models which we will be considering to estimate the risk of investments. The first, developed by Benoit Mandelbrot, uses his fractal theories to estimate volatility. This model is useful for estimating and displaying risk associated with a particular investment when at least four points are known. A significant drawback of Mandelbrot's model is its inability to predict future risk because it reflects only past performance of an investment. The second model uses phase curves to describe a recursive model which estimates future performance using parameter values derived from past performance.
Is it possible to roughly predict the volatility of the apparently random behavior of stock prices using models based on chaos theory?
This model will probably be written in Java. It will start out with a minimal interface for input of initial variables and output of results. We hope to combine the fractal and phase curve models to explain both the present and future risk of an investment. Though this model should be applicable to any investment, we plan to focus primarily on use for stock valuation. The user will input a series of dates and corresponding values of the investment and the program will output both a graphical and numerical estimation of risk. We hope to determine estimations of the Dow, S&P 500, and other indices to provide the user with a standard to which they can compare their results. To estimate the accuracy of our model, we hope to compare some results of trials for real investments with the actual risk of these investments. Then we will extend the model to allow for future predictions of risk for all investments.