The deterministic model

The ultimate goal of investment is to make a profit, and the revenue from investing or loss depends on both the change in prices and the number of assets being held. Investors are usually interested in revenues that are highly relative to the size of the initial investments. Returns measure this mainly because returns are an asset. For example, a stock, a bond, or a portfolio of stocks and bonds are by definition expressed as changes, and price is expressed as a fraction of the initial price. Let's take a look at the gross returns example.

Gross returns

Let's assume that P_t is the investment amount at time _t. A simple gross return is expressed as follows:

Here, P_t+1 is the returned investment value and the return is R _t+1. For example, if P_t = 10 and P_t+1 = 10.6, then R_t+1 = 0.06 = 6%. Returns are scale-free, meaning that they do not depend on the units, but returns are dependent on the units of t (hour, day, and so on). In other words, if t is measured...