Explain what the intercept, slope and coefficient of determination mean

CFA® Level I · Quantitative Methods · Applications of Simple Linear Regression in Finance · QM.8.b

The idea

The slope is the estimated change in the dependent variable for a one-unit change in the independent one, and it carries the units of both. The intercept is the fitted value when the independent variable is zero, which is often outside the range of the data and therefore not meaningful on its own. The coefficient of determination, R-squared, is the proportion of the dependent variable's variation explained by the model, running from zero to one. In a simple regression it is the square of the correlation between the two variables.

Formulas

Slope of a simple linear regression

slope = Cov ÷ variance of X

Cov: the covariance of X and Y
variance of X: the variance of the independent variable

The units are Y per unit of X, so a slope means nothing until both are named. The line always passes through the two means, which is what fixes the intercept once the slope is known.

Coefficient of determination

R² = explained variation ÷ total variation

explained variation: the part of the variation in Y the regression accounts for
total variation: the whole variation in Y around its mean

The share of the variation explained, between 0 and 1. In a simple regression it is the square of the correlation, and in no regression is it evidence that X causes Y.

A practice question

From the question bank, with the reason each wrong option is wrong.

What does the slope coefficient in a simple linear regression represent?

The expected change in the dependent variable per one-unit change in the independent variable — correct
The proportion of variation explainedWhy not: That is R-squared, which is unitless and bounded by zero and one. A slope is neither.
The value of the dependent variable when the independent variable is zeroWhy not: That is the intercept, which is where the line starts rather than how steeply it rises.

The slope is the rate of change the line describes. The last option is the intercept and the middle one is R-squared.

A harder one

A regression reports an R-squared of 0.36. What is the correlation between the two variables in magnitude?

0.13Why not: 0.36² — squared again when it needed to be square-rooted. R-squared is already the square of the correlation.
0.60 — correct
0.36Why not: The R-squared quoted back. In a simple regression the correlation is its square root, 0.60.

In a simple regression R-squared is the square of the correlation, so the correlation is √0.36 = 0.60. Its sign follows the slope.

The hardest tier

A regression of returns on a variable that cannot be zero in practice reports an intercept of −4%. How should this be read?

As evidence the model is misspecifiedWhy not: An intercept outside the data's range is expected, not diagnostic. It anchors the line and says nothing about whether the model is right.
As the expected return under normal conditionsWhy not: It is the fitted value when the independent variable is zero, which the question says never happens. Normal conditions sit somewhere in the middle of the data, not at that point.
As a fitted value at a point outside the data, with no direct interpretation — correct

The intercept anchors the line but describes a point the data never visit, so reading it as a forecast extrapolates beyond the evidence. It remains necessary for the fit without being meaningful on its own.

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