3.11 A Word about Linear Equations (Review of Algebra)
Now that we understand how to price, or value, stocks and bonds, we need to turn to the source of the discount rate for stocks. (The following model may also be used for bonds, however, we have and will continue to depend in general instead on the market rate, the YTM, for the bond discount rate.) You will recall that the discount rate in the DDM has, so far, been an exogenous variable, i.e., one that is “given” or otherwise imported into the model, or formula, from the “outside,” so to speak. We will now turn to the derivation of the DDM’s discount rate, the Capital Asset Pricing Model.
In order to understand the Capital Asset Pricing Model, or “CAP-M,” which provides the discount rate used in the DDM, we must first recall how to utilize basic linear equations.
The standard formulation of a linear equation is:
y = a + bx
Where the terms are as follows:
| Dependent Variable | y |
| Independent Variable | x |
| Vertical Intercept | a |
| Slope | b |
| And where | b =Δy /Δx = “rise / run” |
We speak of “y” as being the dependent variable because, in a sense, its value depends on everything on the other side of the equal sign. The equation’s other independent variable, “x,” is exogenous.
You will note that the linear equation has a vertical intercept, “a,” i.e., the point at which the line intersects the vertical axis; the intercept may be less than or greater than zero. The intercept is a “constant” rather than a variable.
The slope, “b,” represents the steepness of the line. Think of slope in a physical sense, as in the slope of a hill. If the line is steep, it will take several steps up to move just a little distance forward horizontally; one shall need to exert more effort to climb a steep slope. If the slope is relatively flat, for every step one moves “up” along the vertical axis, one makes nearly equal progress forward. The slope of a line is a constant; the relative flatness of it is the same no matter where the slope is measured along its length. Equal effort is required in all locations in order to move forward. Slope is measured by the size of the increment of the movement along the vertical or “y” axis (“rise”) versus the equivalent movement along the “x” axis (“run”).
We may recognize and equation as linear when it has all the foregoing components. The equation will also be devoid of exponents, which would make the formula non-linear.