A linear relationship occurs when a change in one or more independent variables that have a power of one or zero affects a dependent variable. Linear relationships are represented on plots as straight lines. In statistics, linear regression is used to fit a linear equation through a set of data points which are linearly related. An example from financial theory is the security characteristic line, which describes the linear relationship between an asset’s and the market’s excess returns.
Linear relationships are typically described by linear equations written in the slope-intercept form y = mx + b. The independent variable x is plotted on the horizontal axis and the dependent variable y is plotted on the vertical axis. The constant m is the slope or steepness of the straight line. The constant b is called the y-intercept and is the value of y when the line crosses the vertical axis.
If a set of data points has a perfectly linear relationship, their plot will form a straight line. This rarely occurs with real world data, although a strong linear relationship may exist between two variables. Other times, the data is weakly linear, but a linear equation is still interesting since it is easy to work with and model. In both cases linear regression techniques, such as the least squares method, can be used to describe the relationship.
Studying the linear relationship between two variables can be useful when predicting future behaviors. For example, linear regression could be used on data concerning wage rates over the last ten years, considering wages as a function of time. Expected wage rates for a particular year can be calculated using the linear equation and this information may be used to budget for savings and retirement.
In the Capital Asset Pricing Model, the security characteristic line is derived by linear regression on a single asset’s historical data and describes the linear relationship between systematic and unsystematic risk. The independent variable is the market’s excess return, and the dependent variable is the asset’s excess return. The y-intercept called alpha measures an investment’s return given its risk. If alpha is positive the investment has overperformed, if negative it has underperformed, and if zero its returns are adequate given the investment’s risks.
The slope of the characteristic line is called beta and describes the asset’s sensitivity to changes in the market. A positive beta means that the asset’s price moves with the market. If beta is between zero and one, then the asset’s price will fluctuate as much as the market and can reduce a portfolio’s volatility. If the beta is greater than one, then the asset will outperform the market if the market increases, but will underperform the market if the market decreases, thus allowing for higher earnings or losses.
