Exponential Growth
When we model the growth of a financial asset we are interested in its growth as a function of time. If we have a principal, 
at time zero, what will its value be at time, t; what is the function p(t)? It is often natural to start with exponential growth as this works with biologic systems--up to a point. The main idea behind exponential growth is that there is a growth rate which is proportional to the principal p at the time . The formula is then derived from a simple differential formula; r is a constant which represents the logarithmic rate of return.
Solving this by integrating, we get:
Taking exponents, where e is the base of the natural logarithm,
The graph below starts with an initial investment 
= 1000, at t = 0. You can move the slider to change the value of r, which is the logarithmic return to the base e. Other bases like 10 could be used, but today with all modern computers using natural logarithms, and with the mathematical relationship to the simple differential equation above it makes the most sense to use the base e. In the graphic, r, the log return, is shown to the right and the decimal return is calculated above. The conversion formula is:
Compound interest problems, where dr is the decimal return can be converted to this format as follows; the two formulas are equivalent.
Notice that r can also be negative, in which case the formula is no longer a growth formula, but a decay formula.
Take the opportunity with this very simple graphic to click on the icons with the + sign to see what other things you can do with this interactive module.
Below is the same formula plotted with p on a logarithmic scale now you have a linear problem where r is the slope of the plot; as you move the slider, all that changes is the slope of the line. You can see that you can calculate the logarithmic return from any two points in this equation by the following formula.
The above is pretty simple and a little boring, but let’s look at what happens if r is not constant but varies according to some statistical distribution with the rate of variation having some relationship with t. A simple model is to consider that r varies according to a normal distribution; this leads to a geometric Brownian motion, when we take the exponent of accumulated returns. r is the expected log return; n is the number of variations across the whole interval 20 t; since Brownian motion is scalable, the units of t could be what you decide, so the scale could be 20 years, 20 days, 20 minutes, etc. r scales in direct proportion to t and for short time scales should probably be zero. A bookmark has been added for this setting, if you click the + icon in the upper right corner of the graphic. σ, which is the standard deviation of the distribution of r, scales with 
. The thick line in the graphic is the expected curve of p, the other two continuous black lines show the curves for the expected value of r ± 2 σ. These latter curves need to be considered with the understanding that the outcome p at any given time is not normally distributed. The graphic is designed so that when you change the value of r the other graphics above will also change to have the same value of r so you can compare the results. As you change n, you will not change the expected outcomes as the scaling is adjusted by 
, what you change is the fineness of the variations, but you may see the scale on the y-axis change to accomodate a new range. If you select 100,000, the program will take a little while to evaluate because 100,000 line segments are being drawn on each of the 10 Brownian paths shown in the diagram.
The model of geometric Brownian motion serves as the basis for much of the mathematics still used in finance, including the Black-Scholes option model. The model makes a number of assumptions which need to be considered before you use it.
1. r is normally distributed with variance, 
2. Both r and σ, are stationary over the time interval.
3. The sequential random variables r, are independent of the preceding and subsequent random variables r.
4. n is presumed to scale proportionately to 
.
Unfortunately all of these assumptions are false for the distributions of returns seen in financial markets. Financial returns have distributions with significantly heavier tails than the normal distribution and heavy tailed return events cause large jumps in the price path. Financial returns are not stationary over time and the number of transactions per unit time varies dramatically over the course of a day. The scale factor, σ, in the model, varies significantly over long time frames, and its variation also shows strong serial dependence. The absolute value of r sequentially also shows strong serial dependence, indicating that the returns are not independent. Consider that the log returns derived from financial data are not the returns necessarily experienced by investors, rather they are generally calculated from sequential differences of log prices, sampled at fixed time intervals. Investors only actually experience a return when they close a transaction, i.e. selling a position after buying it or buying a position after short selling it. Thus the calculated returns shown in financial time series may not be driving investor behavior.
For further reading a very good discussion of Brownian motion and the mathematical basis for the Black-Scholes option formula can be found in Elementary Stochastic Calculus with Finance in View, by Thomas Mikosch.
Comments about this document or error reports would be greatly appreciated, please email mathestate@gmail.com. The document required only 22 lines of Mathematica code to create the graphics. More information about the computable document format can be found at http://www.wolfram.com/cdf/.
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© Copyright 2011 Robert H. Rimmer 23 July 2011
